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This study presents the black hole accretion history of obscured active galactic nuclei (AGNs) identified from the JWST CEERS survey by Chien et al. (2024) using mid-infrared (MIR) SED fitting. We compute black hole accretion rates (BHARs) to estimate the black hole accretion density (BHAD), 
$\rho_{L_{\text{disk}}}$, across 
$0 \lt z \lt 4.25$. MIR luminosity functions (LFs) are also constructed for these sources, modeled with modified Schechter and double power law forms, and corresponding BHAD, 
$\rho_{\text{LF}}$, is derived by integrating the LFs and multiplying by the luminosity. Both 
$\rho_{\text{LF}}$ extend to luminosities as low as 
$10^7 \, {\rm L}_{\odot}$, two orders of magnitude fainter than pre-JWST studies. Our results show that BHAD peaks between redshifts 1 and 3, with the peak varying by method and model, 
$z \simeq$ 1 - 2 for 
$\rho_{L_{\text{disk}}}$ and the double power law, and 
$z \simeq$ 2 - 3 for the modified Schechter function. A scenario where AGN activity peaks before cosmic star formation would challenge existing black hole formation theories, but our present study, based on early JWST observations, provides an initial exploration of this possibility. At 
$z \sim 3$, 
$\rho_{\text{LF}}$ appears higher than X-ray estimates, suggesting that MIR observations are more effective in detecting obscured AGNs missed by X-ray observations. However, given the overlapping error bars, this difference remains within the uncertainties and requires confirmation with larger samples. These findings highlight the potential of JWST surveys to enhance the understanding of co-evolution between galaxies and AGNs.
Brown dwarfs are failed stars with very low mass (13–75 Jupiter mass) and an effective temperature lower than 2 500 K. Their mass range is between Jupiter and red dwarfs. Thus, they play a key role in understanding the gap in the mass function between stars and planets. However, due to their faint nature, previous searches are inevitably limited to the solar neighbourhood (20 pc). To improve our knowledge of the low mass part of the initial stellar mass function and the star formation history of the Milky Way, it is crucial to find more distant brown dwarfs. Using James Webb Space Telescope (JWST) COSMOS-Web data, this study seeks to enhance our comprehension of the physical characteristics of brown dwarfs situated at a distance of kpc scale. The exceptional sensitivity of the JWST enables the detection of brown dwarfs that are up to 100 times more distant than those discovered in the earlier all-sky infrared surveys. The large area coverage of the JWST COSMOS-Web survey allows us to find more distant brown dwarfs than earlier JWST studies with smaller area coverages. To capture prominent water absorption features around 2.7 
${\unicode{x03BC}}$m, we apply two colour criteria, 
$\text{F115W}-\text{F277W}+1\lt\text{F277W}-\text{F444W}$ and 
$\text{F277W}-\text{F444W}\gt\,0.9$. We then select point sources by CLASS_STAR, FLUX_RADIUS, and SPREAD_MODEL criteria. Faint sources are visually checked to exclude possibly extended sources. We conduct SED fitting and MCMC simulations to determine their physical properties and associated uncertainties. Our search reveals 25 T-dwarf candidates and 2 Y-dwarf candidates, more than any previous JWST brown dwarf searches. They are located from 0.3 to 4 kpc away from the Earth. The spatial number density of 900–1 050 K dwarf is 
$(2.0\pm0.9) \times10^{-6}\text{ pc}^{-3}$, 1 050–1 200 K dwarf is 
$(1.2\pm0.7) \times10^{-6}\text{ pc}^{-3}$, and 1 200–1 350 K dwarf is 
$(4.4\pm1.3) \times10^{-6}\text{ pc}^{-3}$. The cumulative number count of our brown dwarf candidates is consistent with the prediction from a standard double exponential model. Three of our brown dwarf candidates were detected by HST, with transverse velocities 
$12\pm5$, 
$12\pm4$, and 
$17\pm6$ km s
$^{-1}$. Along with earlier studies, the JWST has opened a new window of brown dwarf research in the Milky Way thick disk and halo.
We show that, for an arbitrary finite-dimensional quasi-reductive Lie superalgebra over 
${\mathbb {C}}$ with a triangular decomposition and a character 
$\zeta $ of the nilpotent radical, the associated Backelin functor 
$\Gamma _\zeta $ sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category 
${\mathcal {O}}$. In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor 
$\Gamma _\zeta $ and its target category, respectively, categorify a q-symmetrizing map and the corresponding q-symmetrized Fock space associated with a quasi-split quantum symmetric pair of type 
$AIII$.
