Apparent horizons in D-dimensional Robinson-Trautman spacetime

We derive the higher dimensional generalization of Penrose-Tod equation describing apparent horizons in Robinson-Trautman spacetimes. New results concerning the existence and uniqueness of its solutions in four dimensions are proven. Namely, previous results of Tod are generalized to nonvanishing cosmological constant.Comment: 4 pages, 1 figure, to appear in ERE 2008 conference proceedings, to be published by AI

The wave equation on axisymmetric stationary black hole backgrounds

Understanding the behaviour of linear waves on black hole backgrounds is a central problem in general relativity, intimately connected with the nonlinear stability of the black hole spacetimes themselves as solutions to the Einstein equations--a major open question in the subject. Nonetheless, it is only very recently that even the most basic boundedness and quantitative decay properties of linear waves have been proven in a suitably general class of black hole exterior spacetimes. This talk will review our current mathematical understanding of waves on black hole backgrounds, beginning with the classical boundedness theorem of Kay and Wald on exactly Schwarzschild exteriors and ending with very recent boundedness and decay theorems (proven in collaboration with Igor Rodnianski) on a wider class of spacetimes. This class of spacetimes includes in particular slowly rotating Kerr spacetimes, but in the case of the boundedness theorem is in fact much larger, encompassing general axisymmetric stationary spacetimes whose geometry is sufficiently close to Schwarzschild and whose Killing fields span the null generator of the horizon.Comment: 20 pages, 6 pages, to appear in the proceedings of the Spanish Relativity Meeting, Salamanca 200

Accelerated expansion through interaction

Interactions between dark matter and dark energy with a given equation of state are known to modify the cosmic dynamics. On the other hand, the strength of these interactions is subject to strong observational constraints. Here we discuss a model in which the transition from decelerated to accelerated expansion of the Universe arises as a pure interaction phenomenon. Various cosmological scenarios that describe a present stage of accelerated expansion, like the LCDM model or a (generalized) Chaplygin gas, follow as special cases for different interaction rates. This unifying view on the homogeneous and isotropic background level is accompanied by a non-adiabatic perturbation dynamics which can be seen as a consequence of a fluctuating interaction rate.Comment: 4 pages, to appear in the Proceedings of the Spanish Relativity Meeting ERE2008 in Salamanca, September 200

The weak call-by-value λ-calculus is reasonable for both time and space

We study the weak call-by-value -calculus as a model for computational complexity theory and establish the natural measures for time and space Ð the number of beta-reduction steps and the size of the largest term in a computation Ð as reasonable measures with respect to the invariance thesis of Slot and van Emde Boas from 1984. More precisely, we show that, using those measures, Turing machines and the weak call-by-value -calculus can simulate each other within a polynomial overhead in time and a constant factor overhead in space for all computations terminating in (encodings of) łtruež or łfalsež. The simulation yields that standard complexity classes like , NP, PSPACE, or EXP can be defined solely in terms of the -calculus, but does not cover sublinear time or space. Note that our measures still have the well-known size explosion property, where the space measure of a computation can be exponentially bigger than its time measure. However, our result implies that this exponential gap disappears once complexity classes are considered instead of concrete computations. We consider this result a first step towards a solution for the long-standing open problem of whether the natural measures for time and space of the -calculus are reasonable. Our proof for the weak call-by-value -calculus is the first proof of reasonability (including both time and space) for a functional language based on natural measures and enables the formal verification of complexity-theoretic proofs concerning complexity classes, both on paper and in proof assistants. The proof idea relies on a hybrid of two simulation strategies of reductions in the weak call-by-value -calculus by Turing machines, both of which are insufficient if taken alone. The first strategy is the most naive one in the sense that a reduction sequence is simulated precisely as given by the reduction rules; in particular, all substitutions are executed immediately. This simulation runs within a constant overhead in space, but the overhead in time might be exponential. The second strategy is heap-based and relies on structure sharing, similar to existing compilers of eager functional languages. This strategy only has a polynomial overhead in time, but the space consumption might require an additional factor of log, which is essentially due to the size of the pointers required for this strategy. Our main contribution is the construction and verification of a space-aware interleaving of the two strategies, which is shown to yield both a constant overhead in space and a polynomial overhead in time

The Weak Call-By-Value {\lambda}-Calculus is Reasonable for Both Time and Space

We study the weak call-by-value $\lambda$-calculus as a model for computational complexity theory and establish the natural measures for time and space -- the number of beta-reductions and the size of the largest term in a computation -- as reasonable measures with respect to the invariance thesis of Slot and van Emde Boas [STOC~84]. More precisely, we show that, using those measures, Turing machines and the weak call-by-value $\lambda$-calculus can simulate each other within a polynomial overhead in time and a constant factor overhead in space for all computations that terminate in (encodings) of 'true' or 'false'. We consider this result as a solution to the long-standing open problem, explicitly posed by Accattoli [ENTCS~18], of whether the natural measures for time and space of the $\lambda$-calculus are reasonable, at least in case of weak call-by-value evaluation. Our proof relies on a hybrid of two simulation strategies of reductions in the weak call-by-value $\lambda$-calculus by Turing machines, both of which are insufficient if taken alone. The first strategy is the most naive one in the sense that a reduction sequence is simulated precisely as given by the reduction rules; in particular, all substitutions are executed immediately. This simulation runs within a constant overhead in space, but the overhead in time might be exponential. The second strategy is heap-based and relies on structure sharing, similar to existing compilers of eager functional languages. This strategy only has a polynomial overhead in time, but the space consumption might require an additional factor of $\log n$, which is essentially due to the size of the pointers required for this strategy. Our main contribution is the construction and verification of a space-aware interleaving of the two strategies, which is shown to yield both a constant overhead in space and a polynomial overhead in time

Frequency-domain controller design by linear programming

In this thesis, a new framework to design controllers in the frequency domain is proposed. The method is based on the shaping of the open-loop transfer function in the Nyquist diagram. A line representing a lower approximation for the crossover frequency and a line representing a new linear robustness margin guaranteeing lower bounds for the classical robustness margins are defined and used as constraints. A linear programming approach is proposed to tune fixed-order linearly parameterized controllers for stable single-input single-output linear time-invariant plants. Two optimization problems are proposed and solved by linear programming. In the first one, the new robustness margin is maximized given a lower approximation of the crossover frequency, whereas in the second one, the closed-loop performance in terms of load disturbance rejection, output disturbance rejection and tracking is maximized subject to constraints on the new robustness margin. The method can directly consider multi-model systems. Moreover, this new framework can be used directly with frequency-domain data. Thus, it can also consider systems with frequency-domain uncertainties. Using the same framework, an extension of the method is proposed to tune fixed-order linearly parameterized gain-scheduled controllers for stable single-input single-output linear parameter varying plants. This method directly computes a linear parameter varying controller from a linear parameter varying model or from a set of frequency-domain data in different operating points and no interpolation is needed. In terms of closed-loop performance, this approach leads to extremely good results. However, the global stability cannot be guaranteed for fast parameter variations and should be analyzed a posteriori. Nevertheless, for certain classes of switched systems and linear parameter varying systems, it is also possible to guarantee the stability within the design framework. This can be accomplished by adding constraints based on the phase difference of the characteristic polynomials of the closed-loop systems. This frequency-domain methodology has been tested on numerous simulation examples and implemented experimentally on a high-precision double-axis positioning system. The results show the effectiveness and simplicity of the proposed methodologies

The Interaction Rate in Holographic Models of Dark Energy

Observational data from supernovae type Ia, baryon acoustic oscillations, gas mass fraction in galaxy clusters, and the growth factor are used to reconstruct the the interaction rate of the holographic dark energy model recently proposed by Zimdahl and Pav\'{o}n [1] in the redshift interval $0 < z < 1.8$. It shows a reasonable behavior as it increases with expansion from a small or vanishing value in the far past and begins decreasing at recent times. This suggests that the equation of state parameter of dark energy does not cross the phantom divide line.Comment: 8 pages, 2 figures. Key words: cosmology, holography, late accelerated expansion, dark energy. To appear in the Proceedings of the Spanish Relativity Meeting held in Salamanca (Spain) in September 2008. Uses AIP styl

On the dark energy rest frame and the CMB

Dark energy is usually parametrized as a perfect fluid with negative pressure and a certain equation of state. Besides, it is supposed to interact very weakly with the rest of the components of the universe and, as a consequence, there is no reason to expect it to have the same large-scale rest frame as matter and radiation. Thus, apart from its equation of state $w$ and its energy density $\Omega_{DE}$ one should also consider its velocity as a free parameter to be determined by observations. This velocity defines a cosmological preferred frame, so the universe becomes anisotropic and, therefore, the CMB temperature fluctuations will be affected, modifying mainly the dipole and the quadrupole.Comment: 4 pages. Contribution to the proceedings of Spanish Relativity Meeting 2008, Salamanca, Spain, 15-19 September 200

Exact f(R)f(R)-cosmological model coming from the request of the existence of a Noether symmetry

We present an $f(R)$-cosmological model with an exact analytic solution, coming from the request of the existence of a Noether symmetry, which is able to describe a dust-dominated decelerated phase before the current accelerated phase of the universe.Comment: 4 pages, 2 figures, Contribution to the proceedings of Spanish Relativity Meeting 2008, Salamanca, Sapin, 15-19 September 200

Application of initial data sequences to the study of Black Hole dynamical trapping horizons

Non-continuous "jumps" of Apparent Horizons occur generically in 3+1 (binary) black hole evolutions. The dynamical trapping horizon framework suggests a spacetime picture in which these "Apparent Horizon jumps" are understood as spatial cuts of a single spacetime hypersurface foliated by (compact) marginally outer trapped surfaces. We present here some work in progress which makes use of uni-parametric sequences of (axisymmetric) binary black hole initial data for exploring the plausibility of this spacetime picture. The modelling of Einstein evolutions by sequences of initial data has proved to be a successful methodological tool in other settings for the understanding of certain qualitative features of evolutions in restricted physical regimes.Comment: Contribution to the proceedings volume of the Spanish Relativity Meeting 2008: Physics and Mathematics of Gravitation, Salamanca, Spain, 15-19 Sep 200