We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer ...G-functions of
|
x
|
2
, or generalized hypergeometric functions of
-
|
x
|
2
, multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator
(
1
-
|
x
|
2
)
+
α
/
2
(
-
Δ
)
α
/
2
with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper (Dyda et al., Eigenvalues of the fractional Laplace operator in the unit ball,
2015
,
arXiv:1509.08533
).
The main purpose of this article is to obtain (weighted) fractional Hardy inequalities with a remainder and fractional Hardy–Sobolev–Maz'ya inequalities valid for 1<p<2.
Abstract
We develop a general method for the self-consistent calculation of the hydrodynamics of an astrophysical object irradiated by a radiation field with an arbitrary strength and spectral energy ...distribution (SED). Using the xstar photoionization code, we calculate heating and cooling rates as a function of gas photoionization parameter and temperature for several examples of SEDs: bremsstrahlung, blackbody, hard and soft state XRBs, Type 1 and Type 2 AGN. As an application of our method, we study the hydrodynamics of 1D spherical winds heated by a uniform radiation field using the code athena++. We find that in all cases explored a wind settles into a transonic, steady state. The wind evolves along the radiative heating equilibrium curve until adiabatic cooling effects become important and the flow departs from radiative equilibrium. If the flow is heated very rapidly, for example as in a thermally unstable regime, the corresponding column density of gas is low. Perhaps one of the most intriguing results of our work is the two-stage acceleration of the wind that happens when there are two thermally unstable regions and the flux is relatively high. The efficiency with which the radiation field transfers energy to the wind is dependent on the SED of the external source, particularly the relative flux of soft X-rays. These results suggest that detailed photoionization calculations are essential not only to predict spectra but also to properly capture the flow dynamics.
When regulating ritual slaughter, we need to take into consideration issues related both to the protection of freedom of conscience and religion, the freedom of business activity, and the humane ...treatment of animals. The economic arguments have an influence on the way ritual slaughter is regulated and generally lead to departing from the original religious purpose of such legal regulations, in favour of protecting economic interests.
Abstract
Observations of ionized outflows from active galactic nuclei (AGNs) provide evidence of energy and momentum transfer from the AGN radiation to the plasma. The AGN radiation is very ...energetic. Therefore, at distances of parsec scale, where gravity is relatively weak, energy transfer alone can lead to outflow. Much closer to the black hole, gravity dominates thermal energy and the gas is in the so-called “cold” regime. Only magnetic or radiation forces can lead to outflow. However, it is unclear when the radiation force is efficient in overcoming gravity because of its dependence on the spectral energy distribution (SED) of the radiation and opacity. In this work, we survey the parameter space of radiation forces due to spectral lines resulting from blackbody SEDs with temperatures ranging from ∼10
4
to 10
6
K. The objective was to identify the radiation temperature above which line driving becomes inefficient. We find that the temperature ≲4 × 10
5
K marks such a transition. We also self-consistently calculate heating and cooling balance to estimate gas temperature and identify the transition where thermal driving becomes comparable to line driving. We summarize hydrodynamical simulations of radial outflows to illustrate how wind properties change during the transition from line to thermal driving and their dependence on outflow parameters and SED.
As our main result we prove a variant of the fractional Hardy-Sobolev-Maz'ya inequality for half spaces. This result contains a complete answer to a recent open question by Musina and Nazarov. In the ...proof we apply a new version of the fractional Hardy-Sobolev inequality that we establish also for more general unbounded John domains than half spaces.
We prove that smooth C∞ functions are dense in weighted fractional Sobolev spaces on an arbitrary open set, under some mild conditions on the weight. We also obtain a similar result in non-weighted ...spaces defined by some kernel similar to x↦|x|−d−sp. One may consider the results to be a version of the Meyers–Serrin theorem.
The diversity of styles of the architectural environment at the outset of the 20 th century gave a unique possibility to express the national identity of a building in its forms and details. The ...Ukrainian identity was not only present in the buildings of the Lviv but also equally competed with other styles, thanks to the creative methods of interpretation of compositional solutions that were traditional for the Ukrainian architecture and design, implemented by the company of Ivan Levynskyi. Buildings designed and constructed for the Ukrainian community contributed greatly to the preservation and purposeful formation of the Ukrainian architectural image of Lviv. They present a valuable foundation for the creative rethinking and use in conditions of the current architectural process in Ukraine.
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