arXiv subjects

A. Kumpera

Publications and source records attributed to A. Kumpera.

5 recordsLinked to original sources

arXiv2016

On the local equivalence of partial differential equations

In all the practical applications of partial differential equations, what is mostly needed and what is in fact hardest to obtains are the solutions of the system or, occasionally, some specific solutions. This work is based on a most enlightening Mémoire written by Élie Cartan in 1914 and that the majority ignores. We discuss a setting for the local equivalence problem and illustrate it by some examples. It should also be noted that any integration process or method is in fact a local equivalence problem involving a suitable model.

math.DG↗

arXiv2016

Group invariance of integrable Pfaffian systems

Let $\mathcal{S}$ be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra $g$ and consequently invariant under the local action of a Lie group $G$, we show that the vertical variational cohomology of $\mathcal{S}$ is equal to the Lie algebra cohomology of $g$ with values in the space of the horizontal cohomology in maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions to the existence of finite or infinitesimal actions leaving a given system invariant.

math.DG↗

arXiv2016

Non-Integrable Pfaffian Systems

We discuss a recurrent geometrical method, due to Élie Cartan and von Weber ([1],[11]) enabling us to determine, step by step, the maximal integral manifolds of a not necessarily integrable nor regular Pfaffian system. The dimensions of such integral manifolds can, of course, vary from point to point but more so can vary at a given point it depending upon the choice of their recurrent buildup. When the system is regular and integrable then, of course, we obtain the maximal integral leaves of the integral foliation. Attention is also given to those integrable systems that can be integrated by quadratures which was, in the 19th century, the dream of many. However, our main interest resides in enhancing the Jordan-Hölder integration procedure so as to construct the local maximal integral manifolds that find many applications.

math.DG↗

arXiv2015

An introduction to Lie groupoids

We discuss the basic properties of Lie groupoids, Lie algebroids and Lie pseudo-groups in view of applying these techniques to the analysis of Jordan-Hölder resolutions and, subsequently, to the integration of partial differential equations. The present introduction is an extension to Lie groupoids, as far as possible, of the so well known properties and techniques much useful in Lie groups theory. We mention as far as possible since, in the case of Lie groupoids, just the first Lie Theorem holds. As for the prolongation algorithm, it is extremely useful when dealing with groupoids whereas rather senseless in the case of Lie groups.

math.DG↗

arXiv2014

Automorphisms of Flag Systems

We first discuss the problems in the theory of ordinary differential equations that gave rise to the concept of a flag system and illustrate these with the Cartan criterion for Monge equations (1st order) as well as the Cartan statement concerning the local equivalence of Monge-Ampère type equations (2nd order). Next, we describe a prolongation functor operating on the infinitesimal symmetries (automorphisms) of the Darboux flag and extending these, isomorphically, to all the symmetries of any other flag. Hence, flag systems cannot be distinguished by their symmetry algebras and the local classification of these objects is approached by considering higher order isotropies of these algebras as well as the groupoids of $k-th$ order formal equivalences since the differential equations defining the latter provide precious information for the application of flag systems to differential equations (\textit{e.g.}, Cartan's criterion for non-linear Monge equations). In examining the behaviour of the isotropy algebras, that can either diminish or remain the same, when passing from a derived system $S_ν$ to the previous system $S_{ν-1}$, we obtain a full set of numerical invariants for the elementary flag systems that moreover specify the local models.

math.DG↗