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Stellar rotation plays a key role in the transfer of angular momentum, and a large sample of Be-type stars with reliable projected rotational velocities is crucial for understanding their formation and evolution. In this work, we derive the projected rotational velocities ($v$\,sin\,$i$) of 479 Be-type stars using the Fourier transform method, based on their LAMOST Medium-resolution Survey (MRS) spectra. Our results suggest that the Fourier transform method can provide reliable $v$\,sin\,$i$ values for Be-type stars by analyzing the \ion{He}{1}\,lines at 4922, 5015, 5047, and 6678 \,Åin their LAMOST MRS spectra. A K-S test indicates that Be-type stars with different H$α$ emission line morphologies exhibit different $v$\,sin\,$i$ distributions, and Be-type stars with double-peaked emission have a higher fraction of rapid rotators than those with single-peak emission. The $v$\,sin\,$i$ distributions of our Be-type stars in the field, OB associations, and clusters show no significant differences. The deconvolved $v$\,sin\,$i$ distribution of our entire Be-type star sample does not exhibit a bimodal distribution but rather a single peak at $v\approx260$\,km$\cdot$s$^{-1}$. Based on the analysis of 105 stars in our sample, we find that the mean equatorial rotational velocity is 0.74 times the critical velocity. Furthermore, we investigate the relationship between $v$\,sin\,$i$ and the H$α$ peak separation velocity for Be-type stars exhibiting double-peak H$α$ emission lines, using Pearson and Spearman rank correlation coefficients.

Black holes (BHs) can be studied via fundamentally different observational channels that probe complementary aspects of their physics. While BH imaging provides access to the quasistatic space-time geometry via the strong bending of light rays, gravitational wave (GW) observations probe the dynamical response of the space-time to time-dependent processes in the inspiral, merger and ringdown phases. Both messengers---electromagnetic imaging probes and ringdown GW spectroscopy---provide access to essentially the same region---the one between the BH event horizon and the photon region---but they do it via conceptually different methods, encoding different physical information. However, it has been shown in the literature that physical quantities supposedly exclusive of each such messenger are actually tightly related to each another via a correspondence that occurs in the eikonal limit (i.e. large values of the multipole number <inline-formula><mml:math><mml:mo>ℓ</mml:mo></mml:math></inline-formula>) of the geometric-optics approximation. In this paper we clarify the actual identification of observables within such a correspondence and test its accuracy for modified spherically symmetric BH geometries proposed in the literature. We find that even for low values of <inline-formula><mml:math><mml:mo>ℓ</mml:mo></mml:math></inline-formula> the correspondence is surprisingly accurate in relating the real and imaginary parts of quasinormal modes in the GW ringdown phase with the critical impact parameter and Lyapunov exponent of nearly bound light trajectories for every such model analyzed. We discuss the applicability of such a result both for each messenger individually, and also for foreseeable tests of BHs combining both messengers.

Beam mismatch in polarization-sensitive bolometer pairs produces temperature-to-polarization (T → P) leakage that contaminates cosmic microwave background (CMB) polarization maps and biases constraints on the tensor-to-scalar ratio. Conventional deprojection methods rely on the specific Gaussian beam model and filter the time-ordered data, introducing some extra E → B mixing that must be corrected. In this work, we demonstrate that a deep learning approach can efficiently remove these systematics directly at the map level. Using an Ali CMB Polarization Telescope (AliCPT)-like telescope configuration and differential beam parameters drawn from BICEP/Keck measurements, we generate 4000 full-sky mock Q and U maps containing both the true primordial CMB signal and the T → P leakage induced by beam mismatch. We divide each full-sky map into 48 square patches of size 512 × 512 and consider 12 of them that enclose the observation field of AliCPT. For each of these square regions, a multi-patch hierarchical convolutional neural network based on the U-Net architecture is trained on 3600 randomly selected sky patches extracted from the contaminated maps; the corresponding uncontaminated patches serve as ground truth. On an independent test set, the trained network recovers the true polarization maps with a mean absolute deviation of 0.024 μK in Q and 0.023 μK in U. The reconstructed BB and EE angular power spectra match the input cosmology to within residuals of order 10<SUP>−4</SUP> μK<SUP>2</SUP> and 10<SUP>−3</SUP> μK<SUP>2</SUP>, respectively. Operating directly on maps, our neural network model avoids the filtering-induced E → B mixing and achieves millisecond-level inference time per sample, making it a promising alternative to traditional techniques under the simulation assumptions considered here and potentially applicable to next-generation CMB experiments such as AliCPT and LiteBIRD.