98,707 research outputs found
The Continuous Spectrum in Discrete Series Branching Laws
If is a reductive Lie group of Harish-Chandra class, is a symmetric
subgroup, and is a discrete series representation of , the authors
give a condition on the pair which guarantees that the direct integral
decomposition of contains each irreducible representation of with
finite multiplicity. In addition, if is a reductive Lie group of
Harish-Chandra class, and is a closed, reductive subgroup of
Harish-Chandra class, the authors show that the multiplicity function in the
direct integral decomposition of is constant along `continuous
parameters'. In obtaining these results, the authors develop a new technique
for studying multiplicities in the restriction via convolution with
Harish-Chandra characters. This technique has the advantage of being useful for
studying the continuous spectrum as well as the discrete spectrum.Comment: International Journal of Mathematics, Volume 24, Number 7, 201
Direct and Inverse Variational Problems on Time Scales: A Survey
We deal with direct and inverse problems of the calculus of variations on
arbitrary time scales. Firstly, using the Euler-Lagrange equation and the
strengthened Legendre condition, we give a general form for a variational
functional to attain a local minimum at a given point of the vector space.
Furthermore, we provide a necessary condition for a dynamic
integro-differential equation to be an Euler-Lagrange equation (Helmholtz's
problem of the calculus of variations on time scales). New and interesting
results for the discrete and quantum settings are obtained as particular cases.
Finally, we consider very general problems of the calculus of variations given
by the composition of a certain scalar function with delta and nabla integrals
of a vector valued field.Comment: This is a preprint of a paper whose final and definite form will be
published in the Springer Volume 'Modeling, Dynamics, Optimization and
Bioeconomics II', Edited by A. A. Pinto and D. Zilberman (Eds.), Springer
Proceedings in Mathematics & Statistics. Submitted 03/Sept/2014; Accepted,
after a revision, 19/Jan/201
Discrete Lie Advection of Differential Forms
In this paper, we present a numerical technique for performing Lie advection
of arbitrary differential forms. Leveraging advances in high-resolution finite
volume methods for scalar hyperbolic conservation laws, we first discretize the
interior product (also called contraction) through integrals over Eulerian
approximations of extrusions. This, along with Cartan's homotopy formula and a
discrete exterior derivative, can then be used to derive a discrete Lie
derivative. The usefulness of this operator is demonstrated through the
numerical advection of scalar fields and 1-forms on regular grids.Comment: Accepted version; to be published in J. FoC
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