1,788 research outputs found
Symbol correspondences for spin systems
The present monograph explores the correspondence between quantum and
classical mechanics in the particular context of spin systems, that is,
SU(2)-symmetric mechanical systems. Here, a detailed presentation of quantum
spin-j systems, with emphasis on the SO(3)-invariant decomposition of their
operator algebras, is followed by an introduction to the Poisson algebra of the
classical spin system and a similarly detailed presentation of its
SO(3)-invariant decomposition. Subsequently, this monograph proceeds with a
detailed and systematic study of general quantum-classical symbol
correspondences for spin-j systems and their induced twisted products of
functions on the 2-sphere. This original systematic presentation culminates
with the study of twisted products in the asymptotic limit of high spin
numbers. In the context of spin systems, it shows how classical mechanics may
or may not emerge as an asymptotic limit of quantum mechanics.Comment: Research Monograph, 171 pages (book format, preliminary version
Even Dimensional Improper Affine Spheres
There are exactly two different types of bi-dimensional improper affine
spheres: the non-convex ones can be modeled by the center-chord transform of a
pair of planar curves while the convex ones can be modeled by a holomorphic
map. In this paper, we show that both constructions can be generalized to
arbitrary even dimensions: the former class corresponds to the center-chord
transform of a pair of Lagrangian submanifolds while the latter is related to
special K\"ahler manifolds. Furthermore, we show that the improper affine
spheres obtained in this way are solutions of certain exterior differential
systems. Finally, we also discuss the problem of realization of simple stable
Legendrian singularities as singularities of these improper affine spheres.Comment: 26 page
The Wigner caustic on shell and singularities of odd functions
We study the Wigner caustic on shell of a Lagrangian submanifold L of affine
symplectic space. We present the physical motivation for studying singularities
of the Wigner caustic on shell and present its mathematical definition in terms
of a generating family. Because such a generating family is an odd deformation
of an odd function, we study simple singularities in the category of odd
functions and their odd versal deformations, applying these results to classify
the singularities of the Wigner caustic on shell, interpreting these
singularities in terms of the local geometry of L.Comment: 24 page
A variational principle for actions on symmetric symplectic spaces
We present a definition of generating functions of canonical relations, which
are real functions on symmetric symplectic spaces, discussing some conditions
for the presence of caustics. We show how the actions compose by a neat
geometrical formula and are connected to the hamiltonians via a geometrically
simple variational principle which determines the classical trajectories,
discussing the temporal evolution of such ``extended hamiltonians'' in terms of
Hamilton-Jacobi-type equations. Simplest spaces are treated explicitly.Comment: 28 pages. Edited english translation of first author's PhD thesis
(2000
Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds
We study the global centre symmetry set (GCS) of a smooth closed submanifold M m ⊂ R n , n ≤ 2m. The GCS includes both the centre symmetry set defined by Janeczko , we present generating families for singularities of E λ (L) and prove that the caustic of any simple stable Lagrangian singularity in a 4m-dimensional Lagrangian fibre bundle is realizable as the germ of an affine equidistant of some L ⊂ R 2m . We characterize the criminant part of GCS(L) in terms of bitangent hyperplanes to L. Then, after presenting the appropriate equivalence relation to be used in this Lagrangian case, we classify the affine-Lagrangian stable singularities of GCS(L). In particular we show that, already for a smooth closed convex curve L ⊂ R 2 , many singularities of GCS(L) which are affine stable are not affineLagrangian stable
Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds
We define the Global Centre Symmetry set (GCS) of a smooth closed
m-dimensional submanifold M of R^n, , which is an affinely invariant
generalization of the centre of a k-sphere in R^{k+1}. The GCS includes both
the centre symmetry set defined by Janeczko and the Wigner caustic defined by
Berry. We develop a new method for studying generic singularities of the GCS
which is suited to the case when M is lagrangian in R^{2m} with canonical
symplectic form. The definition of the GCS, which slightly generalizes one by
Giblin and Zakalyukin, is based on the notion of affine equidistants, so, we
first study singularities of affine equidistants of Lagrangian submanifolds,
classifying all the stable ones. Then, we classify the affine-Lagrangian stable
singularities of the GCS of Lagrangian submanifolds and show that, already for
smooth closed convex curves in R^2, many singularities of the GCS which are
affine stable are not affine-Lagrangian stable.Comment: 26 pages, 2 figure
Ordenamento do Território e Planeamento Ambiental : investigação e prática
A secção Biologia é coordenada pelo Professor Universitário Armindo Rodrigues.O CIGPT fundado pelos Geógrafos Helena Calado e João Porteiro desenvolveu nas duas
últimas décadas um esforço considerável na investigação e prática do Ordenamento Territoria
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