50 research outputs found
Legendre Transform, Hessian Conjecture and Tree Formula
Let be a polynomial over (a field of characteristic 0) such that
the Hessian of is a nonzero constant. Let be the formal
Legendre Transform of . Then is well-defined as a formal power
series over . The Hessian Conjecture introduced here claims that
is actually a polynomial. This conjecture is shown to be true when K=\bb{R}
and the Hessian matrix of is either positive or negative definite
somewhere. It is also shown to be equivalent to the famous Jacobian Conjecture.
Finally, a tree formula for is derived; as a consequence, the tree
inversion formula of Gurja and Abyankar is obtained.Comment: 9 pages, references are update
Lorentz Group and Oriented MICZ-Kepler Orbits
The MICZ-Kepler orbits are the non-colliding orbits of the MICZ Kepler
problems (the magnetized versions of the Kepler problem). The oriented
MICZ-Kepler orbits can be parametrized by the canonical angular momentum
and the Lenz vector , with the parameter space
consisting of the pairs of 3D vectors with . The recent 4D perspective
of the Kepler problem yields a new parametrization, with the parameter space
consisting of the pairs of Minkowski vectors with ,
, .
This new parametrization of orbits implies that the MICZ-Kepler orbits of
different magnetic charges are related to each other by symmetries:
\emph{ acts transitively on both the
set of oriented elliptic MICZ-Kepler orbits and the set of oriented parabolic
MICZ-Kepler orbits}. This action extends to , the \emph{structure group} for the rank-two Euclidean Jordan
algebra whose underlying Lorentz space is the Minkowski space.Comment: 7 page
The Representation Aspect of the Generalized Hydrogen Atoms
Let be an integer. In the Enright-Howe-Wallach classification list
of the unitary highest weight modules of \widetilde{\mr{Spin}}(2, D+1), the
(nontrivial) Wallach representations in Case II, Case III, and the mirror of
Case III are special in the sense that they are precisely the ones that can be
realized by the Hilbert space of bound states for a generalized hydrogen atom
in dimension D. It has been shown recently that each of these special Wallach
representations can be realized as the space of L^2-sections of a canonical
hermitian bundle over the punctured {\bb R}^D. Here a simple algebraic
characterization of these special Wallach representations is found.Comment: 18 pages, simplified proo
The Poisson Realization of so(2, 2k+2) on Magnetic Leaves
Let ()
and : be the map sending to .
Denote by the pullback by of the canonical
principal -bundle . Let
be the associated co-adjoint bundle and
be the pullback bundle under projection
map . The canonical connection
on turns into a Poisson
manifold.
The main result here is that the real Lie algebra
can be realized as a Lie subalgebra of the Poisson algebra , where is a symplectic leave of
of special kind. Consequently, in view of the earlier result of the
author, an extension of the classical MICZ Kepler problems to dimension
is obtained. The hamiltonian, the angular momentum, the Lenz vector and the
equation of motion for this extension are all explicitly worked out.Comment: 14 page
The O(1)-Kepler Problems
Let be an integer. To each irreducible representation of
, an -Kepler problem in dimension is
constructed and analyzed. This system is super integrable and when it is
equivalent to a generalized MICZ-Kepler problem in dimension two. The dynamical
symmetry group of this system is
with the Hilbert space of bound states being the unitary
highest weight representation of
with highest weight
which occurs at the right-most nontrivial reduction point in the
Enright-Howe-Wallach classification diagram for the unitary highest weight
modules. (Here or 1 depending on whether is trivial or
not.) Furthermore, it is shown that the correspondence is the theta-correspondence for dual pair .Comment: Final published for