Strong deflection gravitational lensing by a modified Hayward black hole

A modified Hayward black hole is a nonsingular black hole. It is proposed to form when the pressure generated by quantum gravity can stop matter's collapse as the matter reaches Planck density. Strong deflection gravitational lensing happening nearby its event horizon might provide some clues of these quantum effects in its central core. We investigate observables of the strong deflection lensing, including angular separations, brightness differences and time delays between its relativistic images, and estimate their values for the supermassive black hole in the Galactic center. We find that it is possible to distinguish the modified Hayward black hole from a Schwarzschild one, but it demands very high resolution beyond current stage.Comment: 10 pages, 1 figur

Strong field gravitational lensing by a charged Galileon black hole

Strong field gravitational lensings are dramatically disparate from those in the weak field by representing relativistic images due to light winds one to infinity loops around a lens before escaping. We study such a lensing caused by a charged Galileon black hole, which is expected to have possibility to evade no-hair theorem. We calculate the angular separations and time delays between different relativistic images of the charged Galileon black hole. All these observables can potentially be used to discriminate a charged Galileon black hole from others. We estimate the magnitudes of these observables for the closest supermassive black hole Sgr A*. The strong field lensing observables of the charged Galileon black hole can be close to those of a tidal Reissner-Nordstr\"{o}m black hole or those of a Reissner-Nordstr\"{o}m black hole. It will be helpful to distinguish these black holes if we can separate the outermost relativistic images and determine their angular separation, brightness difference and time delay, although it requires techniques beyond the current limit

3-Factor-criticality in double domination edge critical graphs

A vertex subset $S$ of a graph $G$ is a double dominating set of $G$ if $|N[v]\cap S|\geq 2$ for each vertex $v$ of $G$, where $N[v]$ is the set of the vertex $v$ and vertices adjacent to $v$. The double domination number of $G$, denoted by $\gamma_{\times 2}(G)$, is the cardinality of a smallest double dominating set of $G$. A graph $G$ is said to be double domination edge critical if $\gamma_{\times 2}(G+e)<\gamma_{\times 2}(G)$ for any edge $e \notin E$. A double domination edge critical graph $G$ with $\gamma_{\times 2}(G)=k$ is called $k$-$\gamma_{\times 2}(G)$-critical. A graph $G$ is $r$-factor-critical if $G-S$ has a perfect matching for each set $S$ of $r$ vertices in $G$. In this paper we show that $G$ is 3-factor-critical if $G$ is a 3-connected claw-free $4$-$\gamma_{\times 2}(G)$-critical graph of odd order with minimum degree at least 4 except a family of graphs.Comment: 14 page

Nonaxial-octupole Y_{32} correlations in N = 150 isotones from multidimensional constrained covariant density functional theories

The non-axial reflection-asymmetric $\beta_{32}$ shape in some transfermium nuclei with N=150, namely $^{246}$Cm, $^{248}$Cf, $^{250}$Fm, and $^{252}$No are investigated with multidimensional constrained covariant density functional theories. By using the density-dependent point coupling covariant density functional theory with the parameter set DD-PC1 in the particle-hole channel, it is found that, for the ground states of $^{248}$Cf and $^{250}$Fm, the non-axial octupole deformation parameter $\beta_{32} > 0.03$ and the energy gain due to the $\beta_{32}$ distortion is larger than 300 keV. In $^{246}$Cm and $^{252}$No, shallow $\beta_{32}$ minima are found. The occurrence of the non-axial octupole $\beta_{32}$ correlations is mainly from a pair of neutron orbitals $[734]9/2$ ($\nu j_{15/2}$) and $[622]5/2$ ($\nu g_{9/2}$) which are close to the neutron Fermi surface and a pair of proton orbitals $[521]3/2$ ($\pi f_{7/2}$) and $[633]7/2$ ($\pi i_{13/2}$) which are close to the proton Fermi surface. The dependence of the non-axial octupole effects on the form of energy density functional and on the parameter set is also studied.Comment: 5 pages, 2 figure