1,840 research outputs found
Symmetry Detection of Rational Space Curves from their Curvature and Torsion
We present a novel, deterministic, and efficient method to detect whether a
given rational space curve is symmetric. By using well-known differential
invariants of space curves, namely the curvature and torsion, the method is
significantly faster, simpler, and more general than an earlier method
addressing a similar problem. To support this claim, we present an analysis of
the arithmetic complexity of the algorithm and timings from an implementation
in Sage.Comment: 25 page
Extended Gravity Cosmography
Cosmography can be considered as a sort of a model-independent approach to
tackle the dark energy/modified gravity problem. In this review, the success
and the shortcomings of the CDM model, based on General Relativity and
standard model of particles, are discussed in view of the most recent
observational constraints. The motivations for considering extensions and
modifications of General Relativity are taken into account, with particular
attention to and theories of gravity where dynamics is
represented by curvature or torsion field respectively. The features of
models are explored in metric and Palatini formalisms. We discuss the
connection between gravity and scalar-tensor theories highlighting the
role of conformal transformations in the Einstein and Jordan frames.
Cosmological dynamics of models is investigated through the
corresponding viability criteria. Afterwards, the equivalent formulation of
General Relativity (Teleparallel Equivalent General Relativity) in terms of
torsion and its extension to gravity is considered. Finally, the
cosmographic method is adopted to break the degeneracy among dark energy
models. A novel approach, built upon rational Pad\'e and Chebyshev polynomials,
is proposed to overcome limits of standard cosmography based on Taylor
expansion. The approach provides accurate model-independent approximations of
the Hubble flow. Numerical analyses, based on Monte Carlo Markov Chain
integration of cosmic data, are presented to bound coefficients of the
cosmographic series. These techniques are thus applied to reconstruct
and functions and to frame the late-time expansion history of the
universe with no \emph{a priori} assumptions on its equation of state. A
comparison between the CDM cosmological model with and
models is reported.Comment: 82 pages, 35 figures. Accepted for publication in IJMP
Involutions of polynomially parametrized surfaces
We provide an algorithm for detecting the involutions leaving a surface
defined by a polynomial parametrization invariant. As a consequence, the
symmetry axes, symmetry planes and symmetry center of the surface, if any, can
be determined directly from the parametrization, without computing or making
use of the implicit representation. The algorithm is based on the fact, proven
in the paper, that any involution of the surface comes from an involution of
the parameter space (the real plane, in our case); therefore, by determining
the latter, the former can be found. The algorithm has been implemented in the
computer algebra system Maple 17. Evidence of its efficiency for moderate
degrees, examples and a complexity analysis are also given
The unexpected resurgence of Weyl geometry in late 20-th century physics
Weyl's original scale geometry of 1918 ("purely infinitesimal geometry") was
withdrawn by its author from physical theorizing in the early 1920s. It had a
comeback in the last third of the 20th century in different contexts: scalar
tensor theories of gravity, foundations of gravity, foundations of quantum
mechanics, elementary particle physics, and cosmology. It seems that Weyl
geometry continues to offer an open research potential for the foundations of
physics even after the turn to the new millennium.Comment: Completely rewritten conference paper 'Beyond Einstein', Mainz Sep
2008. Preprint ELHC (Epistemology of the LHC) 2017-02, 92 pages, 1 figur
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