7,081 research outputs found

    Cosmological dynamics with non-minimally coupled scalar field and a constant potential function

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    Dynamical systems methods are used to investigate global behavior of the spatially flat Friedmann-Robertson-Walker cosmological model in gravitational theory with a non-minimally coupled scalar field and a constant potential function. We show that the system can be reduced to an autonomous three-dimensional dynamical system and additionally is equipped with an invariant manifold corresponding to an accelerated expansion of the universe. Using this invariant manifold we find an exact solution of the reduced dynamics. We investigate all solutions for all admissible initial conditions using theory of dynamical systems to obtain a classification of all evolutional paths. The right-hand sides of the dynamical system depend crucially on the value of the non-minimal coupling constant therefore we study bifurcation values of this parameter under which the structure of the phase space changes qualitatively. We found a special bifurcation value of the non-minimal coupling constant which is distinguished by dynamics of the model and may suggest some additional symmetry in matter sector of the theory.Comment: 39 pages, 8 multiple figs; v2. 41 pages, 8 multiple figs. published versio

    Algorithmic Properties of Sparse Digraphs

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    The notions of bounded expansion [Nesetril and Ossona de Mendez, 2008] and nowhere denseness [Nesetril and Ossona de Mendez, 2011], introduced by Nesetril and Ossona de Mendez as structural measures for undirected graphs, have been applied very successfully in algorithmic graph theory. We study the corresponding notions of directed bounded expansion and nowhere crownfulness on directed graphs, introduced by Kreutzer and Tazari [Kreutzer and Tazari, 2012]. The classes of directed graphs having those properties are very general classes of sparse directed graphs, as they include, on one hand, all classes of directed graphs whose underlying undirected class has bounded expansion, such as planar, bounded-genus, and H-minor-free graphs, and on the other hand, they also contain classes whose underlying undirected class is not even nowhere dense. We show that many of the algorithmic tools that were developed for undirected bounded expansion classes can, with some care, also be applied in their directed counterparts, and thereby we highlight a rich algorithmic structure theory of directed bounded expansion and nowhere crownful classes

    Consistent perturbations in an imperfect fluid

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    We present a new prescription for analysing cosmological perturbations in a more-general class of scalar-field dark-energy models where the energy-momentum tensor has an imperfect-fluid form. This class includes Brans-Dicke models, f(R) gravity, theories with kinetic gravity braiding and generalised galileons. We employ the intuitive language of fluids, allowing us to explicitly maintain a dependence on physical and potentially measurable properties. We demonstrate that hydrodynamics is not always a valid description for describing cosmological perturbations in general scalar-field theories and present a consistent alternative that nonetheless utilises the fluid language. We apply this approach explicitly to a worked example: k-essence non-minimally coupled to gravity. This is the simplest case which captures the essential new features of these imperfect-fluid models. We demonstrate the generic existence of a new scale separating regimes where the fluid is perfect and imperfect. We obtain the equations for the evolution of dark-energy density perturbations in both these regimes. The model also features two other known scales: the Compton scale related to the breaking of shift symmetry and the Jeans scale which we show is determined by the speed of propagation of small scalar-field perturbations, i.e. causality, as opposed to the frequently used definition of the ratio of the pressure and energy-density perturbations.Comment: 40 pages plus appendices. v2 reflects version accepted for publication in JCAP (new summary of notation, extra commentary on choice of gauge and frame, extra references to literature

    The Power of General Relativity

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    We study the cosmological and weak-field properties of theories of gravity derived by extending general relativity by means of a Lagrangian proportional to R1+δR^{1+\delta}. This scale-free extension reduces to general relativity when δ→0\delta \to 0. In order to constrain generalisations of general relativity of this power class we analyse the behaviour of the perfect-fluid Friedmann universes and isolate the physically relevant models of zero curvature. A stable matter-dominated period of evolution requires δ>0\delta >0 or δ<−1/4\delta <-1/4. The stable attractors of the evolution are found. By considering the synthesis of light elements (helium-4, deuterium and lithium-7) we obtain the bound −0.017<δ<0.0012.-0.017<\delta <0.0012. We evaluate the effect on the power spectrum of clustering via the shift in the epoch of matter-radiation equality. The horizon size at matter--radiation equality will be shifted by ∼1\sim 1% for a value of δ∼0.0005.\delta \sim 0.0005. We study the stable extensions of the Schwarzschild solution in these theories and calculate the timelike and null geodesics. No significant bounds arise from null geodesic effects but the perihelion precession observations lead to the strong bound δ=2.7±4.5×10−19\delta =2.7\pm 4.5\times 10^{-19} assuming that Mercury follows a timelike geodesic. The combination of these observational constraints leads to the overall bound 0≤δ<7.2×10−190\leq \delta <7.2\times 10^{-19} on theories of this type.Comment: 26 pages and 5 figures. Published versio

    Repetitive Reduction Patterns in Lambda Calculus with letrec (Work in Progress)

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    For the lambda-calculus with letrec we develop an optimisation, which is based on the contraction of a certain class of 'future' (also: virtual) redexes. In the implementation of functional programming languages it is common practice to perform beta-reductions at compile time whenever possible in order to produce code that requires fewer reductions at run time. This is, however, in principle limited to redexes and created redexes that are 'visible' (in the sense that they can be contracted without the need for unsharing), and cannot generally be extended to redexes that are concealed by sharing constructs such as letrec. In the case of recursion, concealed redexes become visible only after unwindings during evaluation, and then have to be contracted time and again. We observe that in some cases such redexes exhibit a certain form of repetitive behaviour at run time. We describe an analysis for identifying binders that give rise to such repetitive reduction patterns, and eliminate them by a sort of predictive contraction. Thereby these binders are lifted out of recursive positions or eliminated altogether, as a result alleviating the amount of beta-reductions required for each recursive iteration. Both our analysis and simplification are suitable to be integrated into existing compilers for functional programming languages as an additional optimisation phase. With this work we hope to contribute to increasing the efficiency of executing programs written in such languages.Comment: In Proceedings TERMGRAPH 2011, arXiv:1102.226

    Structurally Parameterized d-Scattered Set

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    In dd-Scattered Set we are given an (edge-weighted) graph and are asked to select at least kk vertices, so that the distance between any pair is at least dd, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following: - For any d≥2d\ge2, an O∗(dtw)O^*(d^{\textrm{tw}})-time algorithm, where tw\textrm{tw} is the treewidth of the input graph. - A tight SETH-based lower bound matching this algorithm's performance. These generalize known results for Independent Set. - dd-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if kk is an additional parameter. - A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness. - A 2O(td2)2^{O(\textrm{td}^2)}-time algorithm parameterized by tree-depth (td\textrm{td}), as well as a matching ETH-based lower bound, both for unweighted graphs. We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter ϵ>0\epsilon > 0, runs in time O∗((tw/ϵ)O(tw))O^*((\textrm{tw}/\epsilon)^{O(\textrm{tw})}) and returns a d/(1+ϵ)d/(1+\epsilon)-scattered set of size kk, if a dd-scattered set of the same size exists
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