On the Kernel of a Polynomial of Scalar Derivations

Savin Treanţă

doi:10.3390/math8040515

Abstract

In this paper, by using a vector variable, the procedure of characteristic systems allows us to describe the kernel of a polynomial of scalar derivations by solving Cauchy Problems for the corresponding system of ODEs. Moreover, a gradient representation for the associated Cauchy Problem solution is derived.

Highlights

Introduction and Problem FormulationThe gradient-type representations for some solutions, Lie algebras, gradient systems in a Lie algebra, algebraic representation of gradient systems and their integral manifolds, have been studied for a long time, with remarkable results, by Vârsan [1] and Barbu et al [2]
For other different but related viewpoints on this subject, the reader is directed to Friedman [4], Sussmann [5], Crandall and Souganidis [6], Sontag [7], Bressan and Shen [8], Evans [9], Brezis [10], Parveen and Akram [11], Treanţă and Vârsan [12], Treanţă [13]
By using a vector variable, the procedure of characteristic systems allows us to describe the kernel of a polynomial of scalar derivations by solving Cauchy Problems for the corresponding system of ODEs

Summary

Introduction

Introduction and Problem FormulationThe gradient-type representations for some solutions, Lie algebras, gradient systems in a Lie algebra, algebraic representation of gradient systems and their integral manifolds, have been studied for a long time, with remarkable results, by Vârsan [1] and Barbu et al [2]. By using a vector variable, the procedure of characteristic systems allows us to describe the kernel of a polynomial of scalar derivations by solving Cauchy Problems for the corresponding system of ODEs. a gradient representation for the associated Cauchy As the main motivation of this study, the mathematical framework developed in this work can be extended for the study of some higher-order hyperbolic, parabolic or Hamilton–Jacobi equations involving a finite set of derivations.

Results

Conclusion