15,465 research outputs found
Contact Lie algebras of vector fields on the plane
The paper is devoted to the complete classification of all real Lie algebras
of contact vector fields on the first jet space of one-dimensional submanifolds
in the plane. This completes Sophus Lie's classification of all possible Lie
algebras of contact symmetries for ordinary differential equations. As a main
tool we use the abstract theory of filtered and graded Lie algebras. We also
describe all differential and integral invariants of new Lie algebras found in
the paper and discuss the infinite-dimensional case.Comment: 20 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol3/paper1.abs.htm
Tangential Extremal Principles for Finite and Infinite Systems of Sets, I: Basic Theory
In this paper we develop new extremal principles in variational analysis that
deal with finite and infinite systems of convex and nonconvex sets. The results
obtained, unified under the name of tangential extremal principles, combine
primal and dual approaches to the study of variational systems being in fact
first extremal principles applied to infinite systems of sets. The first part
of the paper concerns the basic theory of tangential extremal principles while
the second part presents applications to problems of semi-infinite programming
and multiobjective optimization
Integrability via geometry: dispersionless differential equations in three and four dimensions
We prove that the existence of a dispersionless Lax pair with spectral
parameter for a nondegenerate hyperbolic second order partial differential
equation (PDE) is equivalent to the canonical conformal structure defined by
the symbol being Einstein-Weyl on any solution in 3D, and self-dual on any
solution in 4D. The first main ingredient in the proof is a characteristic
property for dispersionless Lax pairs. The second is the projective behaviour
of the Lax pair with respect to the spectral parameter. Both are established
for nondegenerate determined systems of PDEs of any order. Thus our main result
applies more generally to any such PDE system whose characteristic variety is a
quadric hypersurface.Comment: 26 pages, v2 substantially expanded and restructured, with new
theorem on projective property of Lax pair in 3
- …