In this work, we are interested in building the fully discrete scheme for stochastic fractional diffusion equation driven by fractional Brownian sheet which is temporally and spatially fractional with Hurst parameters belonging to (0, 1/2]. We first provide the regularity of the solution. Then we employ the Wong-Zakai approximation to regularize the rough noise and discuss the convergence of the approximation. Next, the finite element and backward Euler convolution quadrature methods are used to discretize spatial and temporal operators for the obtained regularized equation, and the detailed error analyses are developed. Finally, some numerical examples are presented to confirm the theory.

The existence of intermediate-mass black holes (IMBHs) still poses challenges to theoretical and observational astronomers. Several candidates have been proposed, including the one in the IRS13 cluster in the Galactic centre, where the evidence is based on the velocity dispersion of its members; however, none have been confirmed to date. We aim to gain insights into the presence of an IMBH in the Galactic centre through a numerical study of the dynamical interplay between an IMBH and star clusters (SCs) in the vicinity of a supermassive black hole (SMBH). We used high-precision $N$-body models of IRS13-like SCs in the Galactic centre, and of more massive SCs that fall into the centre of the Galaxy from larger distances. We find that at IRS13's physical distance of $0.4\ an IRS13-sized SC cannot remain gravitationally bound even if it contains an IMBH of thousands of Msun . Thus, IRS13 appears to be an incidental present-day clumping of stars. Furthermore, we show that the velocity dispersion of tidally disrupted SCs (the likely origin of IRS13) can be fully accounted for by the tidal forces of the central SMBH; the IMBH's influence is not essential.

Wolf-Rayet stars (WRs) represent one of the final evolutionary stages of massive stars and are thought to be the immediate progenitors of stellar-mass black holes. Their multiplicity characteristics form an important anchor point in single and binary population models for predicting gravitational-wave progenitors. Recent spectroscopic campaigns have suggested incompatible multiplicity fractions and period distributions for N- and C-rich Galactic WRs (WNs and WCs) at both short and long orbital periods, in contradiction with evolutionary model predictions. In this work, we employed long-baseline infrared interferometry to investigate the multiplicity of WRs at long periods and explored the nature of their companions. We present a magnitude-limited ($K<9$; $V<14$) survey of 39 Galactic WRs, including 11 WN, 15 WC, and 13 H-rich WN (WNh) stars. We used the $K$-band instrument GRAVITY at the Very Large Telescope Interferometer (VLTI) in Chile. The sensitivity of GRAVITY at spatial scales of sim 1--200 milliarcseconds and flux contrast of $1<!PCT!>$ allowed an exploration of periods in the range $10^ $ d and companions down to sim 5$\,M_ We carried out a companion search for all our targets, with the aim of either finding wide companions or calculating detection limits. We also explored the rich GRAVITY dataset beyond a multiplicity search to look for other interesting properties of the WR sample. We detected wide companions with VLTI/GRAVITY for only four stars in our sample: WR 48, WR 89, WR 93, and WR 115. Combining our results with spectroscopic studies, we arrived at observed multiplicity fractions of $f^ WN obs WC obs and WNh obs The multiplicity fractions and period distributions of WNs and WCs are consistent in our sample. For single WRs, we placed upper limits on the mass of potential companions down to sim 5$\,M_ for WNs and WCs, and sim 7$\,M_ for WNh stars. In addition, we also found other features in the GRAVITY dataset, such as (i) a diffuse extended component contributing significantly to the $K$-band flux in over half the WR sample; (ii) five known spectroscopic binaries resolved in differential phase data, which constitutes an alternative detection method for close binaries; and (iii) spatially resolved winds in four stars: WR 16, WR 31a, WR 78, and WR 110. Our survey reveals a lack of intermediate- (a few hundred days) and long- (a few years to decades) period WR systems. The 200d peak in the period distributions of WR+OB and BH+OB binaries predicted by Case B mass-transfer binary evolution models is not seen in our data. The rich companionship of their O-type progenitors in this separation range suggests that the WR progenitor stars expand and interact with their companions, most likely through unstable mass transfer, resulting in either a short-period system or a merger.

X-ray surveys provide the most efficient means for the detection of active galactic nuclei (AGNs). However, they do face difficulties in detecting the most heavily obscured Compton-thick AGNs. The BAT detector on board the mission, operating in the very hard 14-195 keV band, has provided the largest samples of Compton-thick AGN in the Local Universe. However, even these flux-limited samples may miss the most obscured sources among the Compton-thick AGN population. A robust way to find these local sources is to systematically study volume-limited AGN samples detected in the IR or the optical part of the spectrum. Here, we utilise a local sample (<100 Mpc) of mid-IR-selected AGNs, unbiased against obscuration, to determine the fraction of Compton-thick sources in the Local Universe. When available, we acquired X-ray spectral information for the sources in our sample from previously published studies. In addition, to maximise the X-ray spectral information for the sources in our sample, we analysed eleven unexplored and observations, for the first time. In this way, we identified four new Compton-thick sources. Our results reveal an increased fraction of Compton-thick AGNs among the sources that have not been detected by BAT of 44 <!PCT!>. Overall, we have estimated a 25-30<!PCT!> share of Compton-thick sources in the Local Universe among mid-IR-selected AGNs. We find no evidence for any evolution of the AGN Compton-thick fraction with luminosity.