Related papers: Normalization of rationally integrable systems
We study in this paper logarithmic derivatives associated to derivations on graded complete Lie algebra, as well as the existence of inverses. These logarithmic derivatives, when invertible, generalize the exp-log correspondence between a…
Quantum fields are generally taken to be operator-valued distributions, linear functionals of test functions into an algebra of operators; here the effective dynamics of an interacting quantum field is taken to be nonlinearly modified by…
For an $n$--dimensional local analytic differential system $\dot x=Ax+f(x)$ with $f(x)=O(|x|^2)$, the Poincar\'e nonintegrability theorem states that if the eigenvalues of $A$ are not resonant, the system does not have an analytic or a…
As shown in a previous paper, whenever a rational vector field on $\mathbb C^n$, $n>2$, is Liouvillian integrable, then it admits a first integral obtained by two successive integrations from a one-form with coefficients in a finite…
Recent years have witnessed the introduction and development of extremely fast rational function algorithms. Many ideas in this realm arose from polynomial-based linear-algebraic algorithms. However, polynomial approximation is occasionally…
This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases…
Poincar\'e proved nonexistence of formal first integrals near a nonresonant singularity of analytic autonomous differential systems. In the resonant case with one zero eigenvalue and others nonresonant, there remains an open problem on…
In this paper we deal with analytic nonautonomous vector fields with a periodic time-dependancy, that we study near an equilibrium point. In a first part, we assume that the linearized system is split in two invariant subspaces E0 and E1.…
In this note, we give a new simple system of global parameters on the moduli space of rational functions, and clarify the relation to the parameters indicating location of fixed points and the indices at them. As a byproduct, we solve a…
We present an approach for variational regularization of inverse and imaging problems for recovering functions with values in a set of vectors. We introduce regularization functionals, which are derivative-free double integrals of such…
In this paper, we introduce an alternative method for applying averaging theory of orders $1$ and $2$ in the plane. This is done by combining Taylor expansions of the displacement map with the integral form of the…
Toric ideals to hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say m, into account, we introduce and study invariant filtrations and their equivariant Hilbert series. We…
The study of the behavior of solutions of ODEs often benefits from deciding on a convenient choice of coordinates. This choice of coordinates may be used to "simplify" the functional expressions that appear in the vector field in order that…
We consider a family of vector fields and we assume a horizontal regularity on their derivatives. We discuss the notion of commutator showing that different definitions agree. We apply our results to the proof of a ball-box theorem and…
In this paper we prove the existence of a simultaneous local normalization for couples $(X,\mathcal{G})$, where $X$ is a vector field which vanishes at a point and $\mathcal{G}$ is a singular underlying geometric structure which is…
Singular complex analytic vector fields on the Riemann surfaces enjoy several geometric properties (singular means that poles and essential singularities are admissible). We describe relations between singular complex analytic vector fields…
This paper offers a new perspective to ease the challenge of domain generalization, which involves maintaining robust results even in unseen environments. Our design focuses on the decision-making process in the final classifier layer.…
The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn…
Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an…
Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue…