2,683 research outputs found
On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem
We prove that the number of limit cycles generated by a small
non-conservative perturbation of a Hamiltonian polynomial vector field on the
plane, is bounded by a double exponential of the degree of the fields. This
solves the long-standing tangential Hilbert 16th problem. The proof uses only
the fact that Abelian integrals of a given degree are horizontal sections of a
regular flat meromorphic connection (Gauss-Manin connection) with a
quasiunipotent monodromy group.Comment: Final revisio
Non-integrability of density perturbations in the FRW universe
We investigate the evolution equation of linear density perturbations in the
Friedmann-Robertson-Walker universe with matter, radiation and the cosmological
constant. The concept of solvability by quadratures is defined and used to
prove that there are no "closed form" solutions except for the known Chernin,
Heath, Meszaros and simple degenerate ones. The analysis is performed applying
Kovacic's algorithm. The possibility of the existence of other, more general
solutions involving special functions is also investigated.Comment: 13 pages. The latest version with added references, and a relevant
new paragraph in section I
Time and dark matter from the conformal symmetries of Euclidean space
The quotient of the conformal group of Euclidean 4-space by its Weyl subgroup
results in a geometry possessing many of the properties of relativistic phase
space, including both a natural symplectic form and non-degenerate Killing
metric. We show that the general solution posesses orthogonal Lagrangian
submanifolds, with the induced metric and the spin connection on the
submanifolds necessarily Lorentzian, despite the Euclidean starting pont. By
examining the structure equations of the biconformal space in an orthonormal
frame adapted to its phase space properties, we also find that two new tensor
fields exist in this geometry, not present in Riemannian geometry. The first is
a combination of the Weyl vector with the scale factor on the metric, and
determines the timelike directions on the submanifolds. The second comes from
the components of the spin connection, symmetric with respect to the new
metric. Though this field comes from the spin connection it transforms
homogeneously. Finally, we show that in the absence of conformal curvature or
sources, the configuration space has geometric terms equivalent to a perfect
fluid and a cosmological constant.Comment: 26 pages, no figures. Appreciable introductory material added.
Results substantially strengthened and explained. New results concerning dark
matter and dark energy candidates added to this versio
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