267 research outputs found
Large scale detection of half-flats in CAT(0)-spaces
For a k-flat F inside a locally compact CAT(0)-space X, we identify various
conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our
conditions are formulated in terms of the ultralimit of X. As applications, we
obtain (1) constraints on the behavior of quasi-isometries between tocally
compact CAT(0)-spaces, (2) constraints on the possible non-positively curved
Riemannian metrics supported by certain manifolds, and (3) a correspondence
between metric splittings of a complete, simply connected, non-positively
curved Riemannian manifold and the metric splittings of its asymptotic cones.
Furthermore, combining our results with the Ballmann, Burns-Spatzier rigidity
theorem and the classical Mostow rigidity theorem, we also obtain (4) a new
proof of Gromov's rigidity theorem for higher rank locally symmetric spaces.Comment: 21 pages. This article is a substantially improved version of our
earlier preprint arXiv:0801.3636. It features more general results, with
shorter, cleaner proofs. Applications remain the sam
Matter Lagrangians Coupled with Connections
We shall here consider extended theories of gravitation in the metric-affine
formalism with matter coupled directly to the connection. A sufficiently
general procedure will be exhibited to solve the resulting field equation
associated to the connection. As special cases one has the no-coupling case
(which is standard in f(R) literature) as well as the cases already analyzed in
ref.[1].Comment: Refs adde
First Order Extended Gravity and the Dark Side of the Universe: the General Theory
General Relativity is not the definitive theory of Gravitation due to several
shortcomings which are coming out both from theoretical and experimental
viewpoints. At large scales (astrophysical and cosmological scales) the
attempts to match it with the today observational data lead to invoke Dark
Energy and Dark Matter as the bulk components of the cosmic fluid. Since no
final evidence, at fundamental level, exists for such ingredients, it is clear
that General Relativity presents shortcomings at infrared scales. On the other
hand, the attempts to formulate theories more general than the Einstein one
give rise to mathematical difficulties that need workarounds which, in turn,
generate problems from the interpretative viewpoint. We present here a
completely new approach to the mathematical objects in terms of which a theory
of Gravitation may be written in a first-order `a la Palatini formalism, and
introduce the concept of Dark Metric which could completely bypass the
introduction of disturbing concepts as Dark Energy and Dark Matter.Comment: Proceedings of the Conference "The Invisible Universe" Paris, June
29-July 3, 2009 10 page
The physical foundations for the geometric structure of relativistic theories of gravitation. From General Relativity to Extended Theories of Gravity through Ehlers-Pirani-Schild approach
We discuss in a critical way the physical foundations of geometric structure
of relativistic theories of gravity by the so-called Ehlers-Pirani-Schild
formalism. This approach provides a natural interpretation of the observables
showing how relate them to General Relativity and to a large class of Extended
Theories of Gravity. In particular we show that, in such a formalism, geodesic
and causal structures of space-time can be safely disentangled allowing a
correct analysis in view of observations and experiment. As specific case, we
take into account the case of f(R) gravity.Comment: 11 pages, 2 figure
DISPLACEMENTS OF AUTOMORPHISMS OF FREE GROUPS II: CONNECTIVITY OF LEVEL SETS AND DECISION PROBLEMS
This is the second of two papers in which we investigate the properties of
displacement functions of automorphisms of free groups (more generally, free products)
on the Culler-Vogtmann Outer space CVn and its simplicial bordification. We develop a
theory for both reducible and irreducible autormorphisms. As we reach the bordification
of CVn we have to deal with general deformation spaces, for this reason we developed
the theory in such generality. In our previous first paper we studied general properties of the
displacement functions, such as well-orderability of the spectrum and the topological
characterization of min-points via partial train tracks (possibly at infinity).
This paper is devoted to proving that for any automorphism (reducible or not) any
level set of the displacement function is connected. Here, by the “level set” we intend to
indicate the set of points displaced by at most some amount, rather than exactly some
amount; this is sometimes called a “sub-level set”.
As an application, this result provides a stopping procedure for brute force search
algorithms in CVn. We use this to reprove two known algorithmic results: the conjugacy
problem for irreducible automorphisms and detecting irreducibility of automorphisms
Covariant Lagrangian Formulation of Chern-Simons and BF Theories
We investigate the covariant formulation of Chern-Simons theories in a
general odd dimension which can be obtained by introducing a vacuum connection
field as a reference. Field equations, Noether currents and superpotentials are
computed so that results are easily compared with the well-known results in
dimension 3. Finally we use this covariant formulation of Chern-Simons theories
to investigate their relation with topological BF theories.Comment: 23 pages, refs. adde
Thermal behavior of Quantum Cellular Automaton wires
We investigate the effect of a finite temperature on the behavior of logic
circuits based on the principle of Quantum Cellular Automata (QCA) and of
ground state computation. In particular, we focus on the error probability for
a wire of QCA cells that propagates a logic state. A numerical model and an
analytical, more approximate, model are presented for the evaluation of the
partition function of such a system and, consequently, of the desired
probabilities. We compare the results of the two models, assessing the limits
of validity of the analytical approach, and provide estimates for the maximum
operating temperature.Comment: 15 pages, 7 figures, uses revte
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