Quotients with respect to strongly LL-subgyrogroups

A topological gyrogroup is a gyrogroup endowed with a compatible topology such that the multiplication is jointly continuous and the inverse is continuous. In this paper, we study the quotient gyrogroups in topological gyrogroups with respect to strongly $L$-subgyrogroups, and prove that let $(G, \tau,\oplus)$ be a topological gyrogroup and $H$ a closed strongly $L$-subgyrogroup of $G$, then the natural homomorphism $\pi$ from a topological gyrogroup $G$ to its quotient topology on $G/H$ is an open and continuous mapping, and $G/H$ is a homogeneous $T_1$-space. We also establish that for a locally compact strongly $L$-subgyrogroup $H$ of a topological gyrogroup $G$, the natural quotient mapping $\pi$ of $G$ onto the quotient space $G/H$ is a locally perfect mapping. This leads us to some interesting results on how properties of $G$ depend on the properties of $G/H$. Some classical results in topological groups are generalized.Comment: 10. arXiv admin note: substantial text overlap with arXiv:2003.08843 by other authors; text overlap with arXiv:2204.02079 by other author

Semileptonic decays of BcB_c meson to S-wave charmonium states in the perturbative QCD approach

Inspired by the recent measurement of the ratio of $B_c$ branching fractions to $J/\psi \pi^+$ and $J/\psi \mu^+\nu_{\mu}$ final states at the LHCb detector, we study the semileptonic decays of $B_c$ meson to the S-wave ground and radially excited 2S and 3S charmonium states with the perturbative QCD approach. After evaluating the form factors for the transitions $B_c\rightarrow P,V$, where $P$ and $V$ denote pseudoscalar and vector S-wave charmonia, respectively, we calculate the branching ratios for all these semileptonic decays. The theoretical uncertainty of hadronic input parameters are reduced by utilizing the light-cone wave function for $B_c$ meson. It is found that the predicted branching ratios range from $10^{-6}$ up to $10^{-2}$ and could be measured by the future LHCb experiment. Our prediction for the ratio of branching fractions $\frac{\mathcal {BR}(B_c^+\rightarrow J/\Psi \pi^+)}{\mathcal {BR}(B_c^+\rightarrow J/\Psi \mu^+\nu_{\mu})}$ is in good agreement with the data. For $B_c\rightarrow V l \nu_l$ decays, the relative contributions of the longitudinal and transverse polarization are discussed in different momentum transfer squared regions. These predictions will be tested on the ongoing and forthcoming experiments.Comment: 12 pages, 3 figures, 5 table

On the continuity of the inverse in (strongly) paratopological gyrogroups

In this paper, we consider the continuity of the inverse in (strongly) paratopological gyrogroups. The conclusions are established as follows: (1)\, Every pseudocompact Tychonoff strongly paratopological gyrogroup is a topological gyrogroup; (2) Suppose that $(G, \tau,\oplus)$ is a locally compact strongly paratopological gyrogroup with a symmetric neighborhood base $\mathscr{B}$ at 0. Then $G$ is a strongly topological gyrogroup

Non-LTE Calculations of the Fe I 6173 {\AA} Line in a Flaring Atmosphere

The Fe I 6173 {\AA} line is widely used in the measurements of vector magnetic fields by instruments including the Helioseismic and Magnetic Imager (HMI). We perform non-local thermodynamic equilibrium calculations of this line based on radiative hydrodynamic simulations in a flaring atmosphere. We employ both a quiet-Sun atmosphere and a penumbral atmosphere as the initial one in our simulations. We find that, in the quiet-Sun atmosphere, the line center is obviously enhanced during an intermediate flare. The enhanced emission is contributed from both radiative backwarming in the photosphere and particle beam heating in the lower chromosphere. A blue asymmetry of the line profile also appears due to an upward mass motion in the lower chromosphere. If we take a penumbral atmosphere as the initial atmosphere, the line has a more significant response to the flare heating, showing a central emission and an obvious asymmetry. The low spectral resolution of HMI would indicate some loss of information but the enhancement and line asymmetry are still kept. By calculating polarized line profiles, we find that the Stokes I and V profiles can be altered as a result of flare heating. Thus the distortion of this line has a crucial influence on the magnetic field measured from this line, and one should be cautious in interpreting the magnetic transients observed frequently in solar flares.Comment: 12 pages, 5 figures, accepted by ApJ