744 research outputs found
Geometric and Extensor Algebras and the Differential Geometry of Arbitrary Manifolds
We give in this paper which is the third in a series of four a theory of
covariant derivatives of representatives of multivector and extensor fields on
an arbitrary open set U of M, based on the geometric and extensor calculus on
an arbitrary smooth manifold M. This is done by introducing the notion of a
connection extensor field gamma defining a parallelism structure on U, which
represents in a well defined way the action on U of the restriction there of
some given connection del defined on M. Also we give a novel and intrinsic
presentation (i.e., one that does not depend on a chosen orthonormal moving
frame) of the torsion and curvature fields of Cartan's theory. Two kinds of
Cartan's connection operator fields are identified, and both appear in the
intrinsic Cartan's structure equations satisfied by the Cartan's torsion and
curvature extensor fields. We introduce moreover a metrical extensor g in U
corresponding to the restriction there of given metric tensor \slg defined on M
and also introduce the concept a geometric structure (U,gamma,g) for U and
study metric compatibility of covariant derivatives induced by the connection
extensor gamma. This permits the presentation of the concept of gauge
(deformed) derivatives which satisfy noticeable properties useful in
differential geometry and geometrical theories of the gravitational field.
Several derivatives operators in metric and geometrical structures, like
ordinary and covariant Hodge coderivatives and some duality identities are
exhibit.Comment: This paper is an improved version of material contained in
math.DG/0501560, math.DG/0501561, math.DG/050200
Geometric Algebras and Extensors
This is the first paper in a series (of four) designed to show how to use
geometric algebras of multivectors and extensors to a novel presentation of
some topics of differential geometry which are important for a deeper
understanding of geometrical theories of the gravitational field. In this first
paper we introduce the key algebraic tools for the development of our program,
namely the euclidean geometrical algebra of multivectors Cl(V,G_{E}) and the
theory of its deformations leading to metric geometric algebras Cl(V,G) and
some special types of extensors. Those tools permit obtaining, the remarkable
golden formula relating calculations in Cl(V,G) with easier ones in Cl(V,G_{E})
(e.g., a noticeable relation between the Hodge star operators associated to G
and G_{E}). Several useful examples are worked in details fo the purpose of
transmitting the "tricks of the trade".Comment: This paper (to appear in Int. J. Geom. Meth. Mod. Phys. 4 (6) 2007)
is an improved version of material appearing in math.DG/0501556,
math.DG/0501557, math.DG/050155
Recruitment of Staufen2 Enhances Dendritic Localization of an Intron-Containing CaMKIIα mRNA
Regulation of mRNA localization is a conserved cellular process observed in many types of cells and organisms. Asymmetrical mRNA distribution plays a particularly important role in the nervous system, where local translation of localized mRNA represents a key mechanism in synaptic plasticity. CaMKIIα is a very abundant mRNA detected in neurites, consistent with its crucial role at glutamatergic synapses. Here, we report the presence of CaMKIIα mRNA isoforms that contain intron i16 in dendrites, RNA granules, and synaptoneurosomes from primary neurons and brain. This subpopulation of unspliced mRNA preferentially localizes to distal dendrites in a synaptic-activity-dependent manner. Staufen2, a well-established marker of RNA transport in dendrites, interacts with intron i16 sequences and enhances its distal dendritic localization, pointing to the existence of intron-mediated mechanisms in the molecular pathways that modulate dendritic transport and localization of synaptic mRNAs.This work was funded by grants from the Ministry of Economy and Competitiveness of Spain (BFU2014-52591-R CG) and the European Union (FEDER) to C.G
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