22,118 research outputs found
Contact Geometry of Hyperbolic Equations of Generic Type
We study the contact geometry of scalar second order hyperbolic equations in
the plane of generic type. Following a derivation of parametrized
contact-invariants to distinguish Monge-Ampere (class 6-6), Goursat (class 6-7)
and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence
method to study the generic case. An intriguing feature of this class of
equations is that every generic hyperbolic equation admits at most a
nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp:
normal forms for the contact-equivalence classes of these maximally symmetric
generic hyperbolic equations are derived and explicit symmetry algebras are
presented. Moreover, these maximally symmetric equations are Darboux
integrable. An enumeration of several submaximally symmetric (eight and
seven-dimensional) generic hyperbolic structures is also given.Comment: This is a contribution to the Special Issue "Elie Cartan and
Differential Geometry", published in SIGMA (Symmetry, Integrability and
Geometry: Methods and Applications) at http://www.emis.de/journals/SIGM
Geometrodynamics in a spherically symmetric, static crossflow of null dust
The spherically symmetric, static spacetime generated by a crossflow of
non-interacting radiation streams, treated in the geometrical optics limit
(null dust) is equivalent to an anisotropic fluid forming a radiation
atmosphere of a star. This reference fluid provides a preferred / internal
time, which is employed as a canonical coordinate. Among the advantages we
encounter a new Hamiltonian constraint, which becomes linear in the momentum
conjugate to the internal time (therefore yielding a functional Schr\"{o}dinger
equation after quantization), and a strongly commuting algebra of the new
constraints.Comment: Section on boundary behavior and fall-off conditions of canonical
variables added. New references, 1 new figure, 12 pages. Version accepted in
Phys.Rev.
Crystal Graphs and -Analogues of Weight Multiplicities for the Root System
We give an expression of the -analogues of the multiplicities of weights
in irreducible \sl_{n+1}-modules in terms of the geometry of the crystal
graph attached to the corresponding U_q(\sl_{n+1})-modules. As an
application, we describe multivariate polynomial analogues of the
multiplicities of the zero weight, refining Kostant's generalized exponents.Comment: LaTeX file with epic, eepic pictures, 17 pages, November 1994, to
appear in Lett. Math. Phy
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