18,303 research outputs found
An area formula in metric spaces
We present an area formula for continuous mappings between metric spaces,
under minimal regularity assumptions. In particular, we do not require any
notion of differentiability. This is a consequence of a measure theoretic
notion of Jacobian, defined as the density of a suitable "pull-back measure"
Towards a theory of area in homogeneous groups
A general approach to compute the spherical measure of submanifolds in
homogeneous groups is provided. We focus our attention on the homogeneous
tangent space, that is a suitable weighted algebraic expansion of the
submanifold. This space plays a central role for the existence of blow-ups.
Main applications are area-type formulae for new classes of smooth
submanifolds. We also study various classes of distances, showing how their
symmetries lead to simpler area and coarea formulas. Finally, we establish the
equality between spherical measure and Hausdorff measure on all horizontal
submanifolds.Comment: 60 page
Study of Crystal-field Effects in Rare-earth (RE) - Transition-metal Intermetallic Compounds and in RE-based Laser Crystals
Rare-earth (RE) based compounds and alloys are of great interest both for
their fundamental physical properties and for applications. In order to tailor
the required compounds for a specific task, one must be able to predict the
energy level structure and transition intensities for any magnetic ion in any
crystalline environment. The crystal-field (CF) analysis is one of the most
powerful theoretical methods to deal with the physics of magnetic ions. In the
present work, this technique is used to analyze peculiar physical properties of
some materials employed in the production of new-generation solid-state laser
and high-performance permanent magnets.Comment: 6 pages, 2 figures; extended abstract of PhD thesis (final version
with updated references
Contact equations, Lipschitz extensions and isoperimetric inequalities
We characterize locally Lipschitz mappings and existence of Lipschitz
extensions through a first order nonlinear system of PDEs. We extend this study
to graded group-valued Lipschitz mappings defined on compact Riemannian
manifolds. Through a simple application, we emphasize the connection between
these PDEs and the Rumin complex. We introduce a class of 2-step groups,
satisfying some abstract geometric conditions and we show that Lipschitz
mappings taking values in these groups and defined on subsets of the plane
admit Lipschitz extensions. We present several examples of these groups, called
Allcock groups, observing that their horizontal distribution may have any
codimesion. Finally, we show how these Lipschitz extensions theorems lead us to
quadratic isoperimetric inequalities in all Allcock groups.Comment: This version has additional references and a revisited introductio
Blow-up of regular submanifolds in Heisenberg groups and applications
We obtain a blow-up theorem for regular submanifolds in the Heisenberg group,
where intrinsic dilations are used. Main consequence of this result is an
explicit formula for the density of (p+1)-dimensional spherical Hausdorff
measure restricted to a p-dimensional submanifold with respect to the
Riemannian surface measure. We explicitly compute this formula in some simple
examples and we present a lower semicontinuity result for the spherical
Hausdorff measure with respect to the weak convergence of currents. Another
application is the proof of an intrinsic coarea formula for vector-valued
mappings on the Heisenberg group
A new differentiation, shape of the unit ball and perimeter measure
We present a new blow-up method that allows for establishing the first
general formula to compute the perimeter measure with respect to the spherical
Hausdorff measure in noncommutative nilpotent groups. This result leads us to
an unexpected relationship between the area formula with respect to a distance
and the profile of its corresponding unit ball.Comment: 17 page
Nonexistence of horizontal Sobolev surfaces in the Heisenberg group
Involutivity is a well known necessary condition for integrability of smooth
tangent distributions. We show that this condition is still necessary for
integrability with Sobolev surfaces. We specialize our study to the left
invariant horizontal distribution of the first Heisenberg group \H^1. Here we
answer a question raised in a paper by Z.M.Balogh, R.Hoefer-Isenegger,
J.T.Tyson
Friezes of type D
In this article, we establish a link between the values of a frieze of type D
and some values of a particular frieze of type A. This link allows us to
compute, independently of each other, all the cluster variables in the cluster
algebra associated with a quiver Q of type D
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