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Let $\tau$ be an uncountable cardinal. The notion of a \emph{$\tau$-fine} topological group was introduced in 2021. More recently, H. Zhang et al. generalized this concept by defining pseudo-$\tau$-fine topological groups to study certain factorization properties of continuous functions on topological groups. It is known that $\tau$-fineness cannot coexist with precompactness in topological groups with uncountable character. In this paper, we investigate this problem further. We prove that, in topological groups with uncountable pseudocharacter, precompactness can coexist with pseudo-$\tau$-fineness for some bounded $\tau$ but pseudocompactness can never.

In recent years, the issue of radon emanation and diffusion has become a critical concern for rare decay experiments, such as JUNO and PandaX-4T. This paper introduces a detector design featuring a symmetric radon detector cavity for the quantitative assessment of membrane materials' radon blocking capabilities. The performance of this design is evaluated through the application of Fick's Law and the diffusion equation considering material solubility. Our detector has completed measurements of radon diffusion coefficients for four types of membrane materials currently used in experiments, which also confirms the rationality of this detector design. The findings are instrumental in guiding the selection and evaluation of optimal materials for radon shielding to reduce radon background, contributing to boost sensitivities of rare event research., Comment: 7 pages, 10 figures and 2 tables

Resonant scattering (RS) of X-ray line emission in supernova remnants (SNRs) may modify the observed line profiles and fluxes and has potential impact on estimating the physical properties of the hot gas and hence on understanding the SNR physics, but has not been theoretically modeled ever. Here we present our Monte-Carlo simulation of RS effect on X-ray resonant-line emission, typified by O VII He$\alpha$ r line, from SNRs. We employ the physical conditions characterized by the Sedov-Taylor solution and some basic parameters similar to those in Cygnus Loop. We show that the impact of RS effect is most significant near the edge of the remnant. The line profiles are predicted to be asymmetric because of different temperatures and photon production efficiencies of the expanding gas at different radii. We also predict the surface brightness of the line emission would decrease in the outer projected region but is slightly enhanced in the inner. The G-ratio of the OVII He$\alpha$ triplet can be effectively elevated by RS in the outer region. We show that RS effect of the O VII He$\alpha$ r line in the southwestern boundary region of Cygnus Loop is non-negligible. The observed OVII G-ratio $\sim$1.8 of the region could be achieved with RS taken into account for properly elevated O abundance from the previous estimates. Additional simulation performed for the SNRs in ejecta-dominated phase like Cas A shows that RS in the shocked ejecta may have some apparently effects on the observational properties of oxygen resonant lines., Comment: 19 pages, 17 figures, accepted by ApJ

Recent years have witnessed a hot wave of deep neural networks in various domains; however, it is not yet well understood theoretically. A theoretical characterization of deep neural networks should point out their approximation ability and complexity, i.e., showing which architecture and size are sufficient to handle the concerned tasks. This work takes one step on this direction by theoretically studying the approximation and complexity of deep neural networks to invariant functions. We first prove that the invariant functions can be universally approximated by deep neural networks. Then we show that a broad range of invariant functions can be asymptotically approximated by various types of neural network models that includes the complex-valued neural networks, convolutional neural networks, and Bayesian neural networks using a polynomial number of parameters or optimization iterations. We also provide a feasible application that connects the parameter estimation and forecasting of high-resolution signals with our theoretical conclusions. The empirical results obtained on simulation experiments demonstrate the effectiveness of our method.

Recent years have emerged a surge of interest in SNNs owing to their remarkable potential to handle time-dependent and event-driven data. The performance of SNNs hinges not only on selecting an apposite architecture and fine-tuning connection weights, similar to conventional ANNs, but also on the meticulous configuration of intrinsic structures within spiking computations. However, there has been a dearth of comprehensive studies examining the impact of intrinsic structures. Consequently, developers often find it challenging to apply a standardized configuration of SNNs across diverse datasets or tasks. This work delves deep into the intrinsic structures of SNNs. Initially, we unveil two pivotal components of intrinsic structures: the integration operation and firing-reset mechanism, by elucidating their influence on the expressivity of SNNs. Furthermore, we draw two key conclusions: the membrane time hyper-parameter is intimately linked to the eigenvalues of the integration operation, dictating the functional topology of spiking dynamics, and various hyper-parameters of the firing-reset mechanism govern the overall firing capacity of an SNN, mitigating the injection ratio or sampling density of input data. These findings elucidate why the efficacy of SNNs hinges heavily on the configuration of intrinsic structures and lead to a recommendation that enhancing the adaptability of these structures contributes to improving the overall performance and applicability of SNNs. Inspired by this recognition, we propose two feasible approaches to enhance SNN learning. These involve leveraging self-connection architectures and employing stochastic spiking neurons to augment the adaptability of the integration operation and firing-reset mechanism, respectively. We verify the effectiveness of the proposed methods from perspectives of theory and practice.

Galactic supernova remnants (SNRs) and their environments provide the nearest laboratories to study SN feedback. We performed molecular observations toward SNR W49B, the most luminous Galactic SNR in the X-ray band, aiming to explore signs of multiple feedback channels of SNRs on nearby molecular clouds (MCs). We found very broad HCO+ lines with widths of dv = 48--75 km/s in the SNR southwest, providing strong evidence that W49B is perturbing MCs at a systemic velocity of $V_{LSR}=61$--65 km/s, and placing W49B at a distance of $7.9\pm 0.6$ kpc. We observed unusually high-intensity ratios of HCO+ J=1-0/CO J=1-0 not only at shocked regions ($1.1\pm 0.4$ and $0.70\pm 0.16$), but also in quiescent clouds over 1 pc away from the SNR's eastern boundary (> 0.2). By comparing with the magnetohydrodynamics shock models, we interpret that the high ratio in the broad-line regions can result from a cosmic-ray (CR) induced chemistry in shocked MCs, where the CR ionization rate is enhanced to around 10--100 times of the Galactic level. The high HCO+/CO ratio outside the SNR is probably caused by the radiation precursor, while the luminous X-ray emission of W49B can explain a few properties in this region. The above results provide observational evidence that SNRs can strongly influence the molecular chemistry in and outside the shock boundary via their shocks, CRs, and radiation. We propose that the HCO+/CO ratio is a potentially useful tool to probe an SNR's multichannel influence on MCs., Comment: 19 pages, 11 figures, ApJ in press

The notion of sobriety is extended to the realm of topological spaces valued in a commutative and unital quantale, via an adjunction between a category of quantale modules and the category of quantale-valued topological spaces. Relations between such sober spaces and quantale-valued domains based on flat ideals are investigated., Comment: 22 pages. arXiv admin note: text overlap with arXiv:2102.02939

This paper considers some extensions of the notion of filter to the quantale-valued context, including saturated prefilter, $\top$-filter and bounded saturated prefilter. The question is whether these constructions give rise to monads on the category of sets. It is shown that the answer depends on the structure of the quantale. Specifically, if the quantale is the unit interval equipped with a continuous t-norm, then these constructions give rise to monads if and only if the implication operator corresponding to that t-norm is continuous at each point off the diagonal., Comment: 30 pages