Vector Fields Of Type Research Articles

HAMILTONIAN FIELDS AND ENERGY IN CONTACT MANIFOLDS

To the contact distribution of a contact manifold we associate Hamiltonian type vector fields, called contact Hamiltonian fields. Their properties are investigated and the existence of such vector fields nowhere tangent to a given submanifold is proved. Time-depending contact Hamiltonian vector fields allow us to define the contact energy whose properties are studied. A class of submanifolds in relation to the study of contact Hamiltonian fields is also analyzed.

C k Solvability near the Characteristic set for a Class of Vector Fields of Infinite Type

C k Solvability near the Characteristic set for a Class of Vector Fields of Infinite Type

The center problem for a family of systems of differential equations having a nilpotent singular point

We study the analytic system of differential equations in the plane ( x ˙ , y ˙ ) t = ∑ i = 0 ∞ F q − p + 2 i s , where p , q ∈ N , p ⩽ q , s = ( n + 1 ) p − q > 0 , n ∈ N , and F i = ( P i , Q i ) t are quasi-homogeneous vector fields of type t = ( p , q ) and degree i, with F q − p = ( y , 0 ) t and Q q − p + 2 s ( 1 , 0 ) < 0 . The origin of this system is a nilpotent and monodromic isolated singular point. We prove for this system the existence of a Lyapunov function and we solve theoretically the center problem for such system. Finally, as an application of the theoretical procedure, we characterize the centers of several subfamilies.

Equivalence and semi-completude of foliations

Holomorphic vector fields of Siegel type with an isolated singularity at the origin are considered. It is proved that those vector fields, under suitable conditions always verified in dimension 3, admit a semi-complete representative. The method used gives new proof of the extension of the Theorem of Mattei–Moussu.

Solvability near the characteristic set for a class of planar vector fields of infinite type

On etudie la resolubilite des equations associees a un champ de vecteurs complexe L dans R 2 a coefficients de classe C ∞ ou C ω . On suppose que L est partout elliptique, sauf le long d'une courbe simple etfermee Σ. Sur Σ, on suppose que L est de type infini et que L A L s'annule a un ordre constant. Les equations considerees sont de la forme Lu = pu + f, ou f satisfait des conditions de compatibilite. On prouve, en particulier, que lorsque l'ordre d'annulation de L L est > 1, l'equation Lu = f est resoluble dans la categorie C ∞ mais pas dans la categorie C ω .

Normalizability, Synchronicity, and Relative Exactness for Vector Fields in C2

In this paper, we study the necessary and su.cient condition under which an orbitally normalizable vector field of saddle or saddle-node type in C2 is analytically conjugate to its formal normal form (i.e., normalizable) by a transformation fixing the leaves of the foliation locally. First, we express this condition in terms of the relative exactness of a certain 1-form derived from comparing the time-form of the vector field with the time-form of the normal form. Then we show that this condition is equivalent to a synchronicity condition: the vanishing of the integral of this 1-form along certain asymptotic cycles de.ned by the vector field. This can be seen as a generalization of the classical Poincare theorem saying that a center is isochronous (i.e., synchronous to the linear center) if and only if it is linearizable. The results, in fact, allow us in many cases to compare any two vector fields which differ by a multiplicative factor. In these cases we show that the two vector fields are analytically conjugate by a transformation fixing the leaves of the foliation locally if and only if their time-forms are synchronous.

On the product theory of singular integrals

We establish L^p -boundedness for a class of product singular integral operators on spaces \widetilde{M} = M_1 \times M_2\times \cdots \times M_n . Each factor space M_i is a smooth manifold on which the basic geometry is given by a control, or Carnot-Caratheodory, metric induced by a collection of vector fields of finite type. The standard singular integrals on M_i are non-isotropic smoothing operators of order zero. The boundedness of the product operators is then a consequence of a natural Littlewood-Paley theory on \widetilde M . This in turn is a consequence of a corresponding theory on each factor space. The square function for this theory is constructed from the heat kernel for the sub-Laplacian on each factor.

Differentiable conjugacy of the Poincaré type vector fields on $\mathbf \{R\}^3$

We prove that on R 3 , except for those germs of vector fields whose linear parts are conjugated to λx∂/∂x + λy∂/∂y + 2λz∂/∂z, any two Poincare type vector fields are at least C 3 conjugated to each other provided their linear approximations have the same eigenvalues and the nonlinear parts are generic.

A Crossed Module Representing the Godbillon–Vey Cocycle

We exhibit crossed modules of Lie algebras (i.e. certain four-term exact sequences) corresponding to the Godbillon–Vey generator for the Lie algebras of formal vector fields in one variable, of vector fields on the circle, of differentiable vector fields of holomorphic type and of holomorphic vector fields on a punctured Riemann surface.

Semiglobal solvability of a class of planar vector fields of infinite type

Semiglobal solvability of a class of planar vector fields of infinite type

The index at infinity for some vector fields with oscillating nonlinearities

This paper is devoted to the computation of the index at infinity for some asymptotically linear completely continuous vector fields $x-T(x)$, when the principal linear part $x-Ax$ is degenerate ($1$ is an eigenvalue of $A$), and the sublinear part is not asymptotically homogeneous (in particular do not satisfy Landesman-Lazer conditions). In this work we consider only the case of a one-dimensional degeneration of the linear part, i.e.s $1$ is a simple eigenvalue of $A$. For this case we formulate an abstract theorem and give some general examples for vector fields of Hammerstein type and for a two point boundary value problem.

Characterizing Yang–Mills Fields by Stochastic Parallel Transport

We give a new stochastic approach to Yang–Mills fields on vector bundles by studying the derivative of the stochastic parallel transport in the vector bundle under a variation of connection. We establish martingale criterions for variations transversal to the gauge orbit and for Yang–Mills fields using variations induced by the flow of vector fields of gradient type on the base manifold. In dimension four we link the local Yang–Mills action to the quadratic variation of the derivative process defined by scalings of the driving Brownian motion in a coordinate chart.

Global hypoellipticity for sums of squares of vector fields of infinite type.

Global hypoellipticity for sums of squares of vector fields of infinite type.

Global properties of a class of planar vector fields of infinite type

This paper studies some global and semi global properties of infinite type, planar, C-valued real analytic vector fields that are invariant under the rotation group. Results are proved on the integrability, kernel, range and classification of such operators.

GAUGE TRANSFORMATIONS OF TYPE (0,2) TENSOR FIELDS AND ASSOCIATED VARIATIONAL PRINCIPLES

ABSTRACT This article is concerned with variational problems whose field variables are functions on a product manifold M x G of two manifolds M and G. These field variables are denoted by ψhαj in local coordinates (h, j = 1,…,n = dim M, α = 1,…,r = dim G), and it is supposed that they behave as type (0,2) tensor fields under coordinate transformations on M. The dependance of ψhαj on the coordinates of G is specified by a generalized gauge transformation that depends on a map h: M → G. The requirement that the action integral be independent of the choice of this map imposes conditions on the matrices that define the gauge transformation in a manner that gives rise to a Lie algebra, which in turn imposes a Lie group structure on the manifold G. As in the case of standard gauge theories (whose gauge potentials consist of r type (0,1) vector fields on M) certain combinations of the first derivatives of the field variations ψhαj give rise to a set of r tensors on M, the so-called field strengths, that can be r...

Group of internal symmetry of the quantum theory of a vector field of general type

Group of internal symmetry of the quantum theory of a vector field of general type

Internal symmetries and the conservation laws for the classical theory of a vector field of general type

A study is made of a six-parameter internal-symmetry group of the classical theory of a vector field of general type (and its nonlinear generalizations). This group, which is not related to a change of the space-time coordinates, is interpreted physically.