12,089 research outputs found
Symplectic , singularities and Lagrangian tangency orders
We study the local symplectic algebra of curves. We use the method of
algebraic restrictions to classify symplectic singularities. We define
discrete symplectic invariants - the Lagrangian tangency orders. We use these
invariants to distinguish symplectic singularities of classical
singularities of planar curves, singularity and singularity. We
also give the geometric description of these symplectic singularities
Modified Eddington-inspired-Born-Infeld Gravity with a Trace Term
In this paper, a modified Eddington-inspired-Born-Infeld (EiBI) theory with a
pure trace term being added to the determinantal action is
analysed from a cosmological point of view. It corresponds to the most general
action constructed from a rank two tensor that contains up to first order terms
in curvature. This term can equally be seen as a conformal factor multiplying
the metric . This very interesting type of amendment has not been
considered within the Palatini formalism despite the large amount of works on
the Born-Infeld-inspired theory of gravity. This model can provide smooth
bouncing solutions which were not allowed in the EiBI model for the same EiBI
coupling. Most interestingly, for a radiation filled universe there are some
regions of the parameter space that can naturally lead to a de Sitter
inflationary stage without the need of any exotic matter field. Finally, in
this model we discover a new type of cosmic "quasi-sudden" singularity, where
the cosmic time derivative of the Hubble rate becomes very large but finite at
a finite cosmic time.Comment: 10 pages, 6 figures, RevTex4-1. References added and discussion
extended. Version accepted in EPJ
An embedding technique for the solution of reaction-fiffusion equations on algebraic surfaces with isolated singularities
In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry.\ud
We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction-diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities
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