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Papers by Sebastian Walcher

Research paper thumbnail of Wechselwirkung von Populationen in einem begrenzten Lebensraum : Modellierung, Simulation und mathematische Analyse im Unterricht

Research paper thumbnail of Mathematics in school and readiness for university

Research paper thumbnail of Orbital symmetries of first order ODEs

Research paper thumbnail of On Sums of Vector Fields

Results in Mathematics, 1997

ABSTRACT

Research paper thumbnail of On monocomposition algebras

Proceedings of the American Mathematical Society, 1995

Research paper thumbnail of On algebras of rank three

Http Dx Doi Org 10 1080 00927879908826635, Jun 27, 2007

Research paper thumbnail of Error Estimatesfor Linear Compartmental Systems

SIAM Journal on Matrix Analysis and Applications, 2002

ABSTRACT

Research paper thumbnail of Motion in a Symmetric Potential on the Hyperbolic Plane

Canadian Journal of Mathematics, 2013

Research paper thumbnail of Dynamical systems and sigma-symmetries

A deformation of the standard prolongation operation, defined on sets of vector fields in involut... more A deformation of the standard prolongation operation, defined on sets of vector fields in involution rather than on single ones, was recently introduced and christened "\sigma-prolongation"; correspondingly one has "\sigma-symmetries" of differential equations. These can be used to reduce the equations under study, but the general reduction procedure under \sigma-symmetries fails for equations of order one. In this note we discuss how \sigma-symmetries can be used to reduce dynamical systems, i.e. sets of first order ODEs in the form dx^a/dt = f^a (x).

Research paper thumbnail of A generalization of lambda-symmetry reduction for systems of ODEs: sigma-symmetries

We consider a deformation of the prolongation operation, defined on sets of vector fields and inv... more We consider a deformation of the prolongation operation, defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones. This maintains the "invariants by differentiation" property, and can hence be used to reduce ODEs satisfying suitable invariance conditions in a fully algorithmic way, similarly to what happens for standard prolongations and symmetries.

Research paper thumbnail of Orbital reducibility and a generalization of lambda symmetries

Journal of Lie Theory, Oct 11, 2012

We review the notion of reducibility and we introduce and discuss the notion of orbital reducibil... more We review the notion of reducibility and we introduce and discuss the notion of orbital reducibility for autonomous ordinary differential equations of first order. The relation between (orbital) reducibility and (orbital) symmetry is investigated and employed to construct (orbitally) reducible systems. By standard identifications, the notions extend to non-autonomous ODEs of first and higher order. Moreover we thus obtain a generalization of the lambda symmetries of Muriel and Romero. Several examples are given.

Research paper thumbnail of On differential equations in normal form

Research paper thumbnail of A generalization of ?-symmetry reduction for systems of ODEs: s-symmetries

J Phys a Math Theor, 2012

Research paper thumbnail of A radical for arbitrary algebras

Commun Algebra, 1995

ABSTRACT

Research paper thumbnail of A characterization of regular jordan pairs and its application to riccati differential equations

Research paper thumbnail of On the Poincare Problem

J Differential Equations, 2000

Research paper thumbnail of On a class of inversions

Http Dx Doi Org 10 1080 00927879208824469, Jun 27, 2007

Research paper thumbnail of �ber polynomiale, insbesondere Riccatische, Differentialgleichungen mit Fundamentall�sungen

Research paper thumbnail of Local Darboux first integrals of analytic differential systems

Bulletin Des Sciences Mathematiques, 2014

ABSTRACT

Research paper thumbnail of MultiParameter Symmetries of First Order Ordinary Dieren tial Equations

This is a collection of results on the use of innitesimal orbital symmetries of rst-order ordinar... more This is a collection of results on the use of innitesimal orbital symmetries of rst-order ordinary dieren tial equations. Some of these results are classical, dating back to Lie and Bianchi, and some new results are added.

Research paper thumbnail of Wechselwirkung von Populationen in einem begrenzten Lebensraum : Modellierung, Simulation und mathematische Analyse im Unterricht

Research paper thumbnail of Mathematics in school and readiness for university

Research paper thumbnail of Orbital symmetries of first order ODEs

Research paper thumbnail of On Sums of Vector Fields

Results in Mathematics, 1997

ABSTRACT

Research paper thumbnail of On monocomposition algebras

Proceedings of the American Mathematical Society, 1995

Research paper thumbnail of On algebras of rank three

Http Dx Doi Org 10 1080 00927879908826635, Jun 27, 2007

Research paper thumbnail of Error Estimatesfor Linear Compartmental Systems

SIAM Journal on Matrix Analysis and Applications, 2002

ABSTRACT

Research paper thumbnail of Motion in a Symmetric Potential on the Hyperbolic Plane

Canadian Journal of Mathematics, 2013

Research paper thumbnail of Dynamical systems and sigma-symmetries

A deformation of the standard prolongation operation, defined on sets of vector fields in involut... more A deformation of the standard prolongation operation, defined on sets of vector fields in involution rather than on single ones, was recently introduced and christened "\sigma-prolongation"; correspondingly one has "\sigma-symmetries" of differential equations. These can be used to reduce the equations under study, but the general reduction procedure under \sigma-symmetries fails for equations of order one. In this note we discuss how \sigma-symmetries can be used to reduce dynamical systems, i.e. sets of first order ODEs in the form dx^a/dt = f^a (x).

Research paper thumbnail of A generalization of lambda-symmetry reduction for systems of ODEs: sigma-symmetries

We consider a deformation of the prolongation operation, defined on sets of vector fields and inv... more We consider a deformation of the prolongation operation, defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones. This maintains the "invariants by differentiation" property, and can hence be used to reduce ODEs satisfying suitable invariance conditions in a fully algorithmic way, similarly to what happens for standard prolongations and symmetries.

Research paper thumbnail of Orbital reducibility and a generalization of lambda symmetries

Journal of Lie Theory, Oct 11, 2012

We review the notion of reducibility and we introduce and discuss the notion of orbital reducibil... more We review the notion of reducibility and we introduce and discuss the notion of orbital reducibility for autonomous ordinary differential equations of first order. The relation between (orbital) reducibility and (orbital) symmetry is investigated and employed to construct (orbitally) reducible systems. By standard identifications, the notions extend to non-autonomous ODEs of first and higher order. Moreover we thus obtain a generalization of the lambda symmetries of Muriel and Romero. Several examples are given.

Research paper thumbnail of On differential equations in normal form

Research paper thumbnail of A generalization of ?-symmetry reduction for systems of ODEs: s-symmetries

J Phys a Math Theor, 2012

Research paper thumbnail of A radical for arbitrary algebras

Commun Algebra, 1995

ABSTRACT

Research paper thumbnail of A characterization of regular jordan pairs and its application to riccati differential equations

Research paper thumbnail of On the Poincare Problem

J Differential Equations, 2000

Research paper thumbnail of On a class of inversions

Http Dx Doi Org 10 1080 00927879208824469, Jun 27, 2007

Research paper thumbnail of �ber polynomiale, insbesondere Riccatische, Differentialgleichungen mit Fundamentall�sungen

Research paper thumbnail of Local Darboux first integrals of analytic differential systems

Bulletin Des Sciences Mathematiques, 2014

ABSTRACT

Research paper thumbnail of MultiParameter Symmetries of First Order Ordinary Dieren tial Equations

This is a collection of results on the use of innitesimal orbital symmetries of rst-order ordinar... more This is a collection of results on the use of innitesimal orbital symmetries of rst-order ordinary dieren tial equations. Some of these results are classical, dating back to Lie and Bianchi, and some new results are added.

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