scholarly journals Projective Metrizability and Formal Integrability

2011 ◽  
Author(s):  
Ioan Bucataru
Keyword(s):  
Formal Integrability

scholarly journals On formal integrability of evolution equations and local geometry of surfaces

2001 ◽  
Vol 15 (2) ◽  
pp. 183-199 ◽  
Author(s):  
Mikhail V. Foursov ◽  
Peter J. Olver ◽  
Enrique G. Reyes
Keyword(s):  
Evolution Equations ◽  
Local Geometry ◽  
Formal Integrability

scholarly journals Formal Inverse Integrating Factor and the Nilpotent Center Problem

2016 ◽  
Vol 26 (01) ◽  
pp. 1650015 ◽  
Author(s):  
Isaac A. García
Keyword(s):  
Vector Fields ◽  
Center Problem ◽  
Formal Integrability ◽  
Number Formula ◽  
Formal Inverse ◽  
General Object ◽  
Formal Power ◽  
Nilpotent Center ◽  
The Relationship ◽  
Integrating Factors

We are interested in deepening the knowledge of methods based on formal power series applied to the nilpotent center problem of planar local analytic monodromic vector fields [Formula: see text]. As formal integrability is not enough to characterize such a center we use a more general object, namely, formal inverse integrating factors [Formula: see text] of [Formula: see text]. Although by the existence of [Formula: see text] it is not possible to describe all nilpotent centers strata, we simplify, improve and also extend previous results on the relationship between these concepts. We use in the performed analysis the so-called Andreev number [Formula: see text] with [Formula: see text] associated to [Formula: see text] which is invariant under orbital equivalency of [Formula: see text]. Besides the leading terms in the [Formula: see text]-quasihomogeneous expansions that [Formula: see text] can have, we also prove the following: (i) If [Formula: see text] is even and there exists [Formula: see text] then [Formula: see text] has a center; (ii) if [Formula: see text], the existence of [Formula: see text] characterizes all the centers; (iii) if there is a [Formula: see text] with minimum “vanishing multiplicity” at the singularity then, generically, [Formula: see text] has a center.

scholarly journals On the Formal Integrability Problem for Planar Differential Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Antonio Algaba ◽  
Cristóbal García ◽  
Jaume Giné
Keyword(s):  
Algebraic Curves ◽  
Necessary Conditions ◽  
Differential Systems ◽  
Analytic Integrability ◽  
Integrability Problem ◽  
Formal Integrability

We study the analytic integrability problem through the formal integrability problem and we show its connection, in some cases, with the existence of invariant analytic (sometimes algebraic) curves. From the results obtained, we consider some families of analytic differential systems inℂ2, and imposing the formal integrability we find resonant centers obviating the computation of some necessary conditions.

scholarly journals Nilpotent centres via inverse integrating factors

2016 ◽  
Vol 27 (5) ◽  
pp. 781-795 ◽  
Author(s):  
ANTONIO ALGABA ◽  
CRISTÓBAL GARCÍA ◽  
JAUME GINÉ
Keyword(s):  
Vector Fields ◽  
Integrating Factor ◽  
The Family ◽  
Formal Integrability ◽  
Formal Inverse ◽  
Inverse Integrating Factor ◽  
Integrating Factors

In this paper, we are interested in the nilpotent centre problem of planar analytic monodromic vector fields. It is known that the formal integrability is not enough to characterize such centres. More general objects are considered as the formal inverse integrating factors. However, the existence of a formal inverse integrating factor is not sufficient to describe all the nilpotent centres. For the family studied in this paper, it is enough.

Restriction and dimensional reduction of differential operators

2019 ◽  
Vol 16 (06) ◽  
pp. 1950094
Author(s):  
Petko A. Nikolov ◽  
Nikola P. Petrov
Keyword(s):  
Vector Bundle ◽  
Dimensional Reduction ◽  
Differential Operators ◽  
Important Application ◽  
Lie Derivative ◽  
Jet Bundles ◽  
Natural Embedding ◽  
Formal Integrability ◽  
Self Consistency ◽  
Connected Lie Group

We consider the restriction of a differential operator (DO) [Formula: see text] acting on the sections [Formula: see text] of a vector bundle [Formula: see text] with base [Formula: see text], in the language of jet bundles. When the base of [Formula: see text] is restricted to a submanifold [Formula: see text], all information about derivatives in directions that are not tangent to [Formula: see text] is lost. To restrict [Formula: see text] to a DO [Formula: see text] acting on sections [Formula: see text] of the restricted bundle [Formula: see text] (with [Formula: see text] the natural embedding), one must choose an auxiliary DO [Formula: see text] and express the derivatives non-tangent to [Formula: see text] from the kernel of [Formula: see text]. This is equivalent to choosing a splitting of certain short exact sequence of jet bundles. A property of [Formula: see text] called formal integrability is crucial for restriction’s self-consistency. We give an explicit example illustrating what can go wrong if [Formula: see text] is not formally integrable. As an important application of this methodology, we consider the dimensional reduction of DOs invariant with respect to the action of a connected Lie group [Formula: see text]. The splitting relation comes from the Lie derivative of the action, which is formally integrable. The reduction of the action of another group is also considered.

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