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

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

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

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