Spherically symmetric equilibrium configurations of perfect fluid obeying a polytropic equation of state are studied in spacetimes with a repulsive cosmological constant. The configurations are specified in terms of three parameters-the polytropic index n, the ratio of central pressure and central energy density of matter σ, and the ratio of energy density of vacuum and central density of matter λ. The static equilibrium configurations are determined by two coupled first-order nonlinear differential equations that are solved by numerical methods with the exception of polytropes with n = 0 corresponding to the configurations with uniform distribution of energy density, when the solution is given in terms of elementary functions. The geometry of the polytropes is conveniently represented by embedding diagrams of both the ordinary space geometry and the optical reference geometry reflecting some dynamical properties of the geodesic motion. The polytropes are represented by radial profiles of energy density, pressure, mass, and metric coefficients. For all tested values of n > 0, the static equilibrium configurations with fixed parameters n, σ, are allowed only up to a critical value of the cosmological parameter λc = λc(n, σ). In the case of n > 3, the critical value λc tends to zero for special values of σ. The gravitational potential energy and the binding energy of the polytropes are determined and studied by numerical methods. We discuss in detail the polytropes with extension comparable to those of the dark matter halos related to galaxies, i.e., with extension ℓ > 100 kpc and mass M > 10 12 M⊙. For such largely extended polytropes the cosmological parameter relating the vacuum energy to the central density has to be larger than λ = ρvac/ρc ∼ 10 −9 . We demonstrate that extension of the static general relativistic polytropic configurations cannot exceed the so called static radius related to their external spacetime, supporting the idea that the static radius represents a natural limit on extension of gravitationally bound configurations in an expanding universe dominated by the vacuum energy.PACS numbers: 98.80. Es,

We demonstrate that in the framework of standard general relativity polytropic spheres with properly fixed polytropic index n and relativistic parameter σ, giving ratio of the central pressure pc to the central energy density ρc, can contain region of trapped null geodesics. Such trapping polytropes can exist for n > 2.138 and they are generally much more extended and massive than the observed neutron stars. We show that in the n-σ parameter space the region of allowed trapping increase with polytropic index for interval of physical interest 2.138 < n < 4. Space extension of the region of trapped null geodesics increases with both increasing n and σ > 0.677 from the allowed region. In order to relate the trapping phenomenon to astrophysically relevant situations, we restrict validity of the polytropic configurations to their extension rextr corresponding to the gravitational mass M ∼ 2M of the most massive observed neutron stars. Then for the central density ρc ∼ 10 15 g cm −3 the trapped regions are outside rextr for all values of 2.138 < n < 4, for the central density ρc ∼ 5 × 10 15 g cm −3 the whole trapped regions are located inside of rextr for 2.138 < n < 3.1, while for ρc ∼ 10 16 g cm −3 the whole trapped regions are inside of rextr for all values of 2.138 < n < 4, guaranteeing astrophysically plausible trapping for all considered polytropes. The region of trapped null geodesics is located closely to the polytrope centre and could have relevant influence on cooling of such polytropes or for binding of gravitational waves in their interior.

Abstract. We study behaviour of gravitational waves in the recently introduced general relativistic polytropic spheres containing a region of trapped null geodesics extended around radius of the stable null circular geodesic that can exist for the polytropic index N > 2.138 and the relativistic parameter, giving ratio of the central pressure p c to the central energy density ρ c , higher than σ = 0.677. In the trapping zones of such polytropes, the effective potential of the axial gravitational wave perturbations resembles those related to the ultracompact uniform density objects, giving thus similar long-lived axial gravitational modes. These longlived linear perturbations are related to the stable circular null geodesic and due to additional non-linear phenomena could lead to conversion of the trapping zone to a black hole. We give in the eikonal limit examples of the long-lived gravitational modes, their oscillatory frequencies and slow damping rates, for the trapping zones of the polytropes with N ∈ (2.138, 4). However, in the trapping polytropes the long-lived damped modes exist only for very large values of the multipole number ℓ > 50, while for smaller values of ℓ the numerical calculations indicate existence of fast growing unstable axial gravitational modes. We demonstrate that for polytropes with N ≥ 3.78, the trapping region is by many orders smaller than extension of the polytrope, and the mass contained in the trapping zone is about 10 −3 of the total mass of the polytrope. Therefore, the gravitational instability of such trapping zones could serve as a model explaining creation of central supermassive black holes in galactic halos or galaxy clusters.

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We performed an extensive simulation study to compare the relative performance of many price-jump indicators with respect to false positive and false negative probabilities. We simulated twenty different time series specifications with different intraday noise volatility patterns and price-jump specifications. The double McNemar (1947) non-parametric test has been applied on constructed artificial time series to compare fourteen different price-jump indicators that are widely used in the literature. The results suggest large differences in terms of performance among the indicators, but we were able to identify the best-performing indicators. In the case of false positive probability, the best-performing price-jump indicator is based on thresholding with respect to centiles. In the case of false negative probability, the best indicator is based on bipower variation. Abstrakt Provedli jsme extenzivní simulační studii a srovnali výkonnost mnoha různých indikátorů cenových skoků na základě kritéria rozlišné pravděpodobnosti falešné pozitivní a falešné negativní identifikace. Simulovali jsme dvacet různých typů časových řad s různě specifikovaným chováním vnitrodenní volatility tvořené bílým šumem a ne-normálními cenovými skoky. Na simulovaných časových řadách jsme aplikovali dvojitý McNemarův (1947) neparametrický test a porovnali14 různých indikátorů cenových skoků, které se nejvíce používají ve finanční ekonometrii. Výsledky ukazují velmi odlišné vlastnosti indikátorů, přičemž jsme byli schopni identifikovat ty nejpřesnější z nich. V případě kritéria porovnávajícího pravděpodobnosti falešné pozitivní indikace cenových skoků se jako nejlepší nástroj ukázal indikátor využívající filtrování pomocí centilů. V případě pravděpodobností falešné negativní indikace nejlépe fungoval indikátor založený na dvojčlenném rozptylu (bipower variance).

The method based on the Horský-Mitskievitch conjecture is applied to the Levi-Civita vacuum metric. It is shown, that every Killing vector is connected with a particular class of Einstein-Maxwell fields and each of those classes is found explicitly. Some of obtained classes are quite new. Radial geodesic motion in constructed space-times is discussed and graphically illustrated in the Appendix.