Covariant holography of a tachyonic accelerating universe

We apply the holographic principle to a flat dark energy dominated Friedmann-Robertson-Walker spacetime filled with a tachyon scalar field with constant equation of state $w=p/\rho$, both for $w>-1$ and $w<-1$. By using a geometrical covariant procedure, which allows the construction of holographic hypersurfaces, we have obtained for each case the position of the preferred screen and have then compared these with those obtained by using the holographic dark energy model with the future event horizon as the infrared cutoff. In the phantom scenario, one of the two obtained holographic screens is placed on the big rip hypersurface, both for the covariant holographic formalism and the holographic phantom model. It is also analyzed whether the existence of these preferred screens allows a mathematically consistent formulation of fundamental theories based on the existence of a S matrix at infinite distances.Comment: 7 pages, 4 figures, one reference added, matches published versio

On distinguished orbits of reductive representations

Let $G$ be a real reductive Lie group and ${\tau}:G \longrightarrow GL(V)$ be a real reductive representation of $G$ with (restricted) moment map m_{\ggo}: V-{0} \longrightarrow \ggo. In this work, we introduce the notion of "nice space" of a real reductive representation to study the problem of how to determine if a $G$-orbit is "distinguished" (i.e. it contains a critical point of the norm squared of m_{\ggo}). We give an elementary proof of the well-known convexity theorem of Atiyah-Guillemin-Sternberg in our particular case and we use it to give an easy-to-check sufficient condition for a $G$-orbit of a element in a nice space to be distinguished. In the case where $G$ is algebraic and $\tau$ is a rational representation, the above condition is also necessary (making heavy use of recent results of M. Jablonski), obtaining a generalization of Nikolayevsky's nice basis criterium. We also provide useful characterizations of nice spaces in terms of the weights of $\tau$. Finally, some applications to ternary forms and minimal metrics on nilmanifolds are presented.Comment: 27 pages (with an appendix), 2 figures, 5 tables. This is a preliminary version; comments, criticisms and suggestions are welcom

Classification of Nilsoliton metrics in dimension seven

The aim of this paper is to classify Ricci soliton metrics on $7$-dimensional nilpotent Lie groups. It can be considered as a continuation of our paper in [Transformation Groups, Volume 17, Number 3 (2012), 639--656]. To this end, we use the classification of $7$-dimensional real nilpotent Lie algebras given by Ming-Peng Gong in his Ph.D thesis and some techniques from the results of Michael Jablonski in [Rocky Mtn. Journal of Math. 42 (2012), 1521--1549] and [M\"{u}nster J. Math. 3 (2010), 67--88], and of Yuri Nikolayevsky in [Trans. Amer. Math. Soc. 363 (2011), 3935--3958]. Of the $9$ one-parameter families and $140$ isolated $7$-dimensional indecomposable real nilpotent Lie algebras, we have $99$ nilsoliton metrics given in an explicit form and $7$ one-parameter families admitting nilsoliton metrics. Our classification is the result of a case-by-case analysis, so many illustrative examples are carefully developed to explain how to obtain the main result.Comment: 21 pages. Some maple files with information on this paper are available in my homepage (sites.google.com/site/efernandezculma); anyone can use them (by crediting the author appropriately

A microscopy technique based on bio-impedance sensors

It is proposed a microscopy for cell culture applications based on impedance sensors. The imagined signals are measured with the Electrical Cell-Substrate Spectroscopy (ECIS) technique, by identifying the cell area. The proposed microscopy allows real-time monitoring inside the incubator, reducing the contamination risk by human manipulation. It requires specific circuits for impedance measurements, a two-dimensional sensor array (pixels), and employing electrical models to decode efficiently the measured signals. Analogue Hardware Description Language (AHDL) circuits for cell-microelectrode enables the use of geometrical and technological data into the system design flow. A study case with 8x8 sensor array is reported, illustrating the evolution and power of the proposed image acquisition.Junta de Andalucía P0-TIC-538

Exact G2G_2-structures on unimodular Lie algebras

We consider seven-dimensional unimodular Lie algebras $\mathfrak{g}$ admitting exact $G_2$-structures, focusing our attention on those with vanishing third Betti number $b_3(\mathfrak{g})$. We discuss some examples, both in the case when $b_2(\mathfrak{g})\neq0$, and in the case when the Lie algebra $\mathfrak{g}$ is (2,3)-trivial, i.e., when both $b_2(\mathfrak{g})$ and $b_3(\mathfrak{g})$ vanish. These examples are solvable, as $b_3(\mathfrak{g})=0$, but they are not strongly unimodular, a necessary condition for the existence of lattices on the simply connected Lie group corresponding to $\mathfrak{g}$. More generally, we prove that any seven-dimensional (2,3)-trivial strongly unimodular Lie algebra does not admit any exact $G_2$-structure. From this, it follows that there are no compact examples of the form $(\Gamma\backslash G,\varphi)$, where $G$ is a seven-dimensional simply connected Lie group with (2,3)-trivial Lie algebra, $\Gamma\subset G$ is a co-compact discrete subgroup, and $\varphi$ is an exact $G_2$-structure on $\Gamma\backslash G$ induced by a left-invariant one on $G$.Comment: Final version; to appear in Monatshefte f\"ur Mathemati