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Search Results: 1 - 10 of 537 matches for " Galliano Valent "
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Bianchi type II,III and V diagonal Einstein metrics re-visited
Galliano Valent
Physics , 2010, DOI: 10.1007/s10714-009-0774-1
Abstract: We present, for both minkowskian and euclidean signatures, short derivations of the diagonal Einstein metrics for Bianchi type II, III and V. For the first two cases we show the integrability of the geodesic flow while for the third case a somewhat unusual bifurcation phenomenon takes place: for minkowskian signature elliptic functions are essential in the metric while for euclidean signature only elementary functions appear.
Explicit integrable systems on two dimensional manifolds with a cubic first integral
Galliano Valent
Physics , 2010,
Abstract: A few years ago Selivanova gave an existence proof for some integrable models, in fact geodesic flows on two dimensional manifolds, with a cubic first integral. However the explicit form of these models hinged on the solution of a nonlinear third order ordinary differential equation which could not be obtained. We show that an appropriate choice of coordinates allows for integration and gives the explicit local form for the full family of integrable systems. The relevant metrics are described by a finite number of parameters and lead to a large class of models on the manifolds ${\mb S}^2, {\mb H}^2$ and $P^2({\mb R})$ containing as special cases examples due to Goryachev, Chaplygin, Dullin, Matveev and Tsiganov.
Bianchi III and V Einstein metrics
Galliano Valent
Physics , 2008,
Abstract: We present diagonal Einstein metrics for Bianchi III and V, both for minkowskian and euclidean signatures and we show that the Einstein Bianchi III metrics have an integrable geodesic flow.
Further results on non-diagonal Bianchi type III vacuum metrics
Galliano Valent
Physics , 2011, DOI: 10.1007/s10714-012-1340-9
Abstract: We present the derivation, for these vacuum metrics, of the Painlev\'e VI equation first obtained by Christodoulakis and Terzis, from the field equations for both minkowskian and euclidean signatures. This allows a complete discussion and the precise connection with some old results due to Kinnersley. The hyperk\"ahler metrics are shown to belong to the Multi-Centre class and for the cases exhibiting an integrable geodesic flow the relevant Killing tensors are given. We conclude by the proof that for the Bianchi B family, excluding type III, there are no hyperk\"ahler metrics.
Reply to Professor Yehia comments arXiv:1305.0026
Galliano Valent
Physics , 2013,
Abstract: This is a reply to Professor Yehia Comments in arXiv:1305.0026
Global structure and geodesics for Koenigs superintegrable systems
Galliano Valent
Physics , 2015,
Abstract: Starting from the framework defined by Matveev and Shevchishin we derive the local and the global structure for the four types of super-integrable Koenigs metrics. These dynamical systems are always defined on non-compact manifolds, namely $\,{\mb R}^2\,$ and $\,{\mb H}^2$. The study of their geodesic flows is made easier using their linear and quadratic integrals. Using Carter (or minimal) quantization we show that the formal superintegrability is preserved at the quantum level and in two cases, for which all of the geodesics are closed, it is even possible to compute the discrete spectrum of the quantum hamiltonian.
On a class of integrable systems with a quartic first integral
Galliano Valent
Physics , 2013, DOI: 10.1134/S1560354713040060
Abstract: We generalize, to some extent, the results on integrable geodesic flows on two dimensional manifolds with a quartic first integral in the framework laid down by Selivanova and Hadeler. The local structure is first determined by a direct integration of the differential system which expresses the conservation of the quartic observable and is seen to involve a finite number of parameters. The global structure is studied in some details and leads to a class of models living on the manifolds S^2, H^2 or R^2. As special cases we recover Kovalevskaya's integrable system and a generalization of it due to Goryachev.
Zoll and Tannery metrics from a superintegrable geodesic flow
Galliano Valent
Physics , 2014, DOI: 10.1007/s11005-014-0712-3
Abstract: We prove that for Matveev and Shevchishin superintegrable system, with a linear and a cubic integral, the metrics defined on S^2 and on Tannery's orbifold T^2 are either Zoll or Tannery metrics.
Integrability versus separability for the multi-centre metrics
Galliano Valent
Mathematics , 2003, DOI: 10.1007/s00220-003-1002-6
Abstract: The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra conserved quantity quadratic in the momenta, induced by a Killing-St\" ackel tensor. Our systematic approach brings to light a subclass of metrics which correspond to new classically integrable dynamical systems. Within this subclass we analyze on the one hand the separation of coordinates in the Hamilton-Jacobi equation and on the other hand the construction of some new Killing-Yano tensors.
From asymptotics to spectral measures: determinate versus indeterminate moment problems
Galliano Valent
Mathematics , 2005, DOI: 10.1007/s00009-006-0068-8
Abstract: In the field of orthogonal polynomials theory, the classical Markov theorem shows that for determinate moment problems the spectral measure is under control of the polynomials asymptotics. The situation is completely different for indeterminate moment problems, in which case the interesting spectral measures are to be constructed using Nevanlinna theory. Nevertheless it is interesting to observe that some spectral measures can still be obtained from weaker forms of Markov theorem. The exposition will be illustrated by orthogonal polynomials related to elliptic functions: in the determinate case by examples due to Stieltjes and some of their generalizations and in the indeterminate case by more recent examples.
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