scholarly journals Principal Frequency of $p$-Sub-Laplacians for General Vector Fields

2021 ◽  
Vol 40 (1) ◽  
pp. 97-109
Author(s):  
Michael Ruzhansky ◽  
Bolys Sabitbek ◽  
Durvudkhan Suragan
Keyword(s):  
Vector Fields ◽  
Principal Frequency ◽  
General Vector

Renormalization of vector fields and Diophantine invariant tori

2008 ◽  
Vol 28 (5) ◽  
pp. 1559-1585 ◽  
Author(s):  
HANS KOCH ◽  
SAŠA KOCIĆ
Keyword(s):  
Renormalization Group ◽  
Vector Field ◽  
Vector Fields ◽  
Invariant Tori ◽  
Analytic Manifold ◽  
Hamiltonian Flows ◽  
Divergence Free ◽  
General Vector

AbstractWe extend the renormalization group techniques that were developed originally for Hamiltonian flows to more general vector fields on 𝕋d×ℝℓ. Each Diophantine vector ω∈ℝd determines an analytic manifold 𝒲 of infinitely renormalizable vector fields, and each vector field on 𝒲 is shown to have an elliptic invariant d-torus with frequencies ω1,ω2,…,ωd. Analogous manifolds for particular classes of vector fields (Hamiltonian, divergence-free, symmetric, reversible) are obtained simply by restricting 𝒲 to the corresponding subspace. We also discuss non-degeneracy conditions, and the resulting reduction in the number of parameters needed in parametrized families to guarantee the existence of invariant tori.

scholarly journals On the generalization of the Darboux theorem

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Kaveh Eftekharinasab
Keyword(s):  
Symplectic Form ◽  
Vector Fields ◽  
Projective Limit ◽  
Smoothness Condition ◽  
General Context ◽  
Classifying Space ◽  
Banach Manifolds ◽  
Dependent Vector ◽  
Darboux Theorem ◽  
General Vector

Darboux theorem to more general context of Frechet manifolds we face an obstacle:  in general vector fields do not have local flows. Recently, Fr\'{e}chet geometry has been developed in terms of projective limit of Banach manifolds. In this framework under an appropriate Lipchitz condition The Darboux theorem asserts that a symplectic  manifold $(M^{2n},\omega)$ is locally symplectomorphic to $(R^{2n}, \omega_0)$, where $\omega_0$  is the standard symplectic form on  $R^{2n}$. This theorem was proved by Moser in 1965, the idea of proof, known as the Moser’s trick, works in many situations. The Moser tricks is to construct an appropriate isotopy $ \ff_t $  generated by a time-dependent vector field $ X_t  $ on $M$ such that $ \ff_1^{*} \omega = \omega_0$. Nevertheless,  it was showed by Marsden that Darboux theorem is not valid for weak symplectic Banach manifolds. However, in 1999 Bambusi showed that if we  associate to each point of a Banach manifold a suitable Banach space (classifying space) via a given symplectic form then the Moser trick can be applied to obtain the theorem if the  classifying space does not depend on the point of the manifold and a suitable smoothness condition holds.  If we want to try to generalize the local flows exist and with some restrictive conditions the Darboux theorem was proved by Kumar.  In this paper we consider the category of so-called bounded Fr\'{e}chet manifolds and prove that in this category vector fields have local flows and following the idea of Bambusi we associate to each point of a manifold a Fr\'{e}chet space independent of the choice of the point and with the assumption of bounded smoothness on vector fields  we prove the Darboux theorem.

Parallel Computation of Complex Antennas around the Coated Object Using Iterative Vector Fields Technique

2014 ◽  
Vol E97.C (7) ◽  
pp. 661-669
Author(s):  
Ying YAN ◽  
Xunwang ZHAO ◽  
Yu ZHANG ◽  
Changhong LIANG ◽  
Zhewang MA
Keyword(s):  
Parallel Computation ◽  
Vector Fields
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