758 research outputs found
Projective dynamics and classical gravitation
Given a real vector space V of finite dimension, together with a particular
homogeneous field of bivectors that we call a "field of projective forces", we
define a law of dynamics such that the position of the particle is a "ray" i.e.
a half-line drawn from the origin of V. The impulsion is a bivector whose
support is a 2-plane containing the ray. Throwing the particle with a given
initial impulsion defines a projective trajectory. It is a curve in the space
of rays S(V), together with an impulsion attached to each ray. In the simplest
example where the force is identically zero, the curve is a straight line and
the impulsion a constant bivector. A striking feature of projective dynamics
appears: the trajectories are not parameterized.
Among the projective force fields corresponding to a central force, the one
defining the Kepler problem is simpler than those corresponding to other
homogeneities. Here the thrown ray describes a quadratic cone whose section by
a hyperplane corresponds to a Keplerian conic. An original point of view on the
hidden symmetries of the Kepler problem emerges, and clarifies some remarks due
to Halphen and Appell. We also get the unexpected conclusion that there exists
a notion of divergence-free field of projective forces if and only if dim V=4.
No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure
The Lie-Poisson structure of the reduced n-body problem
The classical n-body problem in d-dimensional space is invariant under the
Galilean symmetry group. We reduce by this symmetry group using the method of
polynomial invariants. As a result we obtain a reduced system with a
Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The
reduction preserves the natural form of the Hamiltonian as a sum of kinetic
energy that depends on velocities only and a potential that depends on
positions only. Hence we proceed to construct a Poisson integrator for the
reduced n-body problem using a splitting method.Comment: 26 pages, 2 figure
Mise en correspondance semi-dense de points de vue éloignés et non calibrés
Cet article propose une méthode générale de mise en correspondance semi-dense d'une paire d'images couleur prises de points de vue éloignés et non calibrés. AprÚs une initialisation de la géométrie épipolaire et des appariements par le descripteur d'images SIFT, la contrainte épipolaire est récursivement resserrée en vue d'affiner la mise en correspondance. En fin du processus itératif, une étape de densification par corrélation affine permet d'obtenir entre 1733 et 10717 appariements entre deux images de résolution comprise entre 800x600 et 1024x768. Le processus complet s'exécute en environ 2 minutes sur un ordinateur de type Pentium IV à 3GHz, sans optimisation particuliÚre
Projective dynamics and first integrals
We present the theory of tensors with Young tableau symmetry as an efficient
computational tool in dealing with the polynomial first integrals of a natural
system in classical mechanics. We relate a special kind of such first
integrals, already studied by Lundmark, to Beltrami's theorem about
projectively flat Riemannian manifolds. We set the ground for a new and simple
theory of the integrable systems having only quadratic first integrals. This
theory begins with two centered quadrics related by central projection, each
quadric being a model of a space of constant curvature. Finally, we present an
extension of these models to the case of degenerate quadratic forms.Comment: 39 pages, 2 figure
Euler configurations and quasi-polynomial systems
In the Newtonian 3-body problem, for any choice of the three masses, there
are exactly three Euler configurations (also known as the three Euler points).
In Helmholtz' problem of 3 point vortices in the plane, there are at most three
collinear relative equilibria. The "at most three" part is common to both
statements, but the respective arguments for it are usually so different that
one could think of a casual coincidence. By proving a statement on a
quasi-polynomial system, we show that the "at most three" holds in a general
context which includes both cases. We indicate some hard conjectures about the
configurations of relative equilibrium and suggest they could be attacked
within the quasi-polynomial framework.Comment: 21 pages, 6 figure
Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
We study the commutation relations within the Pauli groups built on all
decompositions of a given Hilbert space dimension , containing a square,
into its factors. Illustrative low dimensional examples are the quartit ()
and two-qubit () systems, the octit (), qubit/quartit () and three-qubit () systems, and so on. In the single qudit case,
e.g. , one defines a bijection between the maximal
commuting sets [with the sum of divisors of ] of Pauli
observables and the maximal submodules of the modular ring ,
that arrange into the projective line and a independent set
of size [with the Dedekind psi function]. In the
multiple qudit case, e.g. , the Pauli graphs rely on
symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if
) and GQ(3,3) (if ). More precisely, in dimension ( a
prime) of the Hilbert space, the observables of the Pauli group (modulo the
center) are seen as the elements of the -dimensional vector space over the
field . In this space, one makes use of the commutator to define
a symplectic polar space of cardinality , that
encodes the maximal commuting sets of the Pauli group by its totally isotropic
subspaces. Building blocks of are punctured polar spaces (i.e. a
observable and all maximum cliques passing to it are removed) of size given by
the Dedekind psi function . For multiple qudit mixtures (e.g.
qubit/quartit, qubit/octit and so on), one finds multiple copies of polar
spaces, ponctured polar spaces, hypercube geometries and other intricate
structures. Such structures play a role in the science of quantum information.Comment: 18 pages, version submiited to J. Phys. A: Math. Theo
Functional diversity of sharks and rays is highly vulnerable and supported by unique species and locations worldwide
Elasmobranchs (sharks, rays and skates) are among the most threatened marine vertebrates, yet their global functional diversity remains largely unknown. Here, we use a trait dataset of >1000 species to assess elasmobranch functional diversity and compare it against other previously studied biodiversity facets (taxonomic and phylogenetic), to identify species- and spatial- conservation priorities. We show that threatened species encompass the full extent of functional space and disproportionately include functionally distinct species. Applying the conservation metric FUSE (Functionally Unique, Specialised, and Endangered) reveals that most top-ranking species differ from the top Evolutionarily Distinct and Globally Endangered (EDGE) list. Spatial analyses further show that elasmobranch functional richness is concentrated along continental shelves and around oceanic islands, with 18 distinguishable hotspots. These hotspots only marginally overlap with those of other biodiversity facets, reflecting a distinct spatial fingerprint of functional diversity. Elasmobranch biodiversity facets converge with fishing pressure along the coast of China, which emerges as a critical frontier in conservation. Meanwhile, several components of elasmobranch functional diversity fall in high seas and/or outside the global network of marine protected areas. Overall, our results highlight acute vulnerability of the worldâs elasmobranchsâ functional diversity and reveal global priorities for elasmobranch functional biodiversity previously overlooked
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