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Vector theories in cosmology
This article provides a general study of the Hamiltonian stability and the
hyperbolicity of vector field models involving both a general function of the
Faraday tensor and its dual, , as well as a Proca potential
for the vector field, . In particular it is demonstrated that theories
involving only do not satisfy the hyperbolicity conditions. It is then
shown that in this class of models, the cosmological dynamics always dilutes
the vector field. In the case of a nonminimal coupling to gravity, it is
established that theories involving or are generically
pathologic. To finish, we exhibit a model where the vector field is not diluted
during the cosmological evolution, because of a nonminimal vector
field-curvature coupling which maintains second-order field equations. The
relevance of such models for cosmology is discussed.Comment: 17 pages, no figur
Generalization of a criterion for semistable vector bundles
It is known that a vector bundle E on a smooth projective curve Y defined
over an algebraically closed field is semistable if and only if there is a
vector bundle F on Y such that the cohomologies of E\otimes F vanish. We extend
this criterion for semistability to vector bundles on curves defined over
perfect fields. Let X be a geometrically irreducible smooth projective curve
defined over a perfect field k, and let E be a vector bundle on X. We prove
that E is semistable if and only if there is a vector bundle F on such that
the cohomologies of E\otimes F vanish. We also give an explicit bound for the
rank of
The model theory of Commutative Near Vector Spaces
In this paper we study near vector spaces over a commutative from a model
theoretic point of view. In this context we show regular near vector spaces are
in fact vector spaces. We find that near vector spaces are not first order
axiomatisable, but that finite block near vector spaces are. In the latter case
we establish quantifier elimination, and that the theory is controlled by which
elements of the pointwise additive closure of are automorphisms of the near
vector space
-jets of difeomorphisms preserving orbits of vector fields
Let be a smooth vector field defined in a neighborhood of the origin in
, , and let be its local flow. Denote by the
set of germs of diffeomorphisms preserving
orbits of and let be the identity component of with
respect to -topology. Then every contains a subset
consisting of mappings of the form , where is a smooth function. It was proved earlier by the author that
if is a linear vector field, then . In this paper we
present a class of vector fields for which and
coincide on the level of -jets. We also establish a parameter rigidity
of linear vector fields and "reduced" Hamiltonian vector fields of real
homogeneous polynomials in two variables.Comment: 34 pages. version 5. Many misprints are corrected and some minor
changes are mad
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