Spectral Stability of the Neumann Laplacian

We prove the equivalence of Hardy- and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform H\"older category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount. AMS subject classifications: 35P15, 35J25, 47A75, 47B25, 26D10, 46E35. Keywords: Neumann Laplacian, Sobolev inequalities, Hardy inequalities, spectral stability, H\"older continuity.Comment: 23 page

Spectral monitoring of AGNs: Preliminary results for Ark 564 and Arp 102B

We present preliminary results of the long term spectral monitoring of two active galactic nuclei with different broad line shapes: Ark 564 and Arp 102B. Ark 564 is a bright nearby narrow line Syfert 1 (NLS1) galaxy with relatively narrow permitted optical emission lines and a high FeII/H${\beta}$ ratio, while Arp 102B is a nearby broad-line radio galaxy with broad double-peaked Balmer emission lines. The spectra of Ark 564 were observed during 11-year period (1999-2009) and the spectra of Arp 102B in the 12-year period (1998-2009), with SAO 6-m and 1-m telescopes (Russia) and the GHAO 2.1-m telescope (Cananea, Mexico).Comment: Presented on "8th Serbian Conference on Spectral Line Shapes in Astrophysics". In revised version minor changes in the tex

On boundedness of the generalized Riesz potential in local Morrey-type spaces

For all admissible values of the numerical parameters sharp sufficient conditions on the functional parameters are obtained ensuring the boundedness of the generalized Riesz potential from one general local Morrey-type space to another one, which, for a certain range of the numerical parameters, coincide with the necessary ones

Optical study of the polar BM CrB in low accretion state

This paper presents a spectral and photometric study of the poorly studied polar BM CrB. Three states of the polar brightness and signs of transition from one-pole to two-pole accretion mode were found by an analysis of ZTF data. It is shown that the transition from the low state to the high state changes the longitude of the main accretion spot (by $\approx 17^{\circ}$) and increases its elongation (by $\approx 10^{\circ}$). The spectra contain Zeeman absorptions of the H$\alpha$ line which are formed at a magnetic field strength of $15.5\pm1$ MG. These absorptions are likely produced by a cold halo extending from the accretion spot at $\approx {^1/_4}$ of the white dwarf radius. Modeling of the behavior of the H$\alpha$ emission line shows that the main source of emission is the part of the accretion stream near the Lagrangian point L$_1$, which is periodically eclipsed by the donor star. The spectra exhibit a cyclotron component formed in the accretion spot. Its modeling by a simple accretion spot model gives constraints on the magnetic field strength $B=15-40$ MG and the temperature $T_e\gtrsim15$ keV

Long-term variability of the optical spectra of NGC 4151: II. Evolution of the broad Ha and Hb emission-line profiles

Results of the long-term (11 years, from 1996 to 2006) H$\alpha$ and H$\beta$ line variations of the active galactic nucleus of NGC 4151 are presented. High quality spectra (S/N>50 and R~8A) of H$\alpha$ and H$\beta$ were investigated. We analyzed line profile variations during monitoring period. Comparing the line profiles of H$\alpha$ and H$\beta$, we studied different details (bumps, absorption features) in the line profiles. The variations of the different H$\alpha$ and H$\beta$ line profile segments have been investigated. Also, we analyzed the Balmer decrement for whole line and for line segments. We found that the line profiles were strongly changing during the monitoring period, showing blue and red asymmetries. This indicates a complex BLR geometry of NGC 4151 with, at least, three kinematically distinct regions: one that contributes to the blue line wing, one to the line core and one to the red line wing. Such variation can be caused by an accelerating outflow starting very close to the black hole, where the red part may come from the region {closer to the black hole than the blue part, which is coming} from the region having the highest outflow velocities. Taking into account the fact that the BLR of NGC 4151 has a complex geometry (probably affected by an outflow) and that a portion of the broad line emission seems to have not a pure photoionization origin, one can ask the question whether the study of the BLR by reverberation mapping may be valid in the case of this galaxy.Comment: 24 pages, 18 figures, accepted for publications in A&