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To reveal the origin of the mini-starbursts in the Milky Way, we carried out large-scale CO observations toward the RCW 106 giant molecular cloud (GMC) complex using the NANTEN2 4-m radio telescope operated by Nagoya University. We also analyzed the Mopra Southern Galactic plane CO survey and Herschel infrared continuum archival data. The RCW 106 GMC complex contains the radial velocity components of $-68$ km s$^{-1}$ and $-50$ km s$^{-1}$ reported by Nguyen et al. (2015). Focusing on the RCW 106 East and West region with the massive star formation having the bright infrared dust emission, we found that these regions have three different velocity components with $\sim 10$ km s$^{-1}$ differences. The two out of three velocity components show morphological correspondence with the infrared cold dust emission and connect with the bridge feature on a position-velocity diagram. Therefore, two molecular clouds (MCs) with $\sim 10$ km s$^{-1}$ differences are likely to be physically associated with massive star-forming regions in the GMC complex. Based on these observational results, we argue that mini-starbursts and massive star/cluster formation in the RCW 106 GMC complex are induced by supersonic cloud-cloud collisions in an agglomerate of molecular gas on the Scutum-Centurus arm., Comment: 25 pages, 13 figures, 3 tables, accepted for publication in The Astronomical Journal (AJ)

In this article, we show that each two metric fibrations with a common base and a common fiber have isomorphic magnitude homology, and even more, the same magnitude homotopy type. That can be considered as a generalization of a fact proved by T. Leinster that the magnitude of a metric fibration with finitely many points is a product of those of the base and the fiber. We also show that the definition of the magnitude homotopy type due to the second and the third authors is equivalent to the geometric realization of Hepworth and Willerton's pointed simplicial set., Comment: 12 pages

Asao-Ivanov showed that magnitude homology is a Tor functor, hence we can compute it by giving a projective resolution of a certain module. In this article, we compute magnitude homology by constructing a minimal projective resolution. As a consequence, we determine magnitude homology of geodetic metric spaces. We show that it is a free $\mathbb Z$-module, and give a recursive algorithm for constructing all cycles. As a corollary, we show that a finite geodetic metric space is diagonal if and only if it contains no 4-cuts. Moreover, we give explicit computations for cycle graphs, Petersen graph, Hoffman-Singleton graph, and a missing Moore graph. It includes another approach to the computation for cycle graphs, which has been studied by Hepworth--Willerton and Gu., Comment: 29 pages

We prove that the magnitude (co)homology of an enriched category can, under some technical assumptions, be described in terms of derived functors between certain abelian categories. We show how this statement is specified for the cases of quasimetric spaces, finite quasimetric spaces, and finite digraphs. For quasimetric spaces, we define the notion of a distance module over a quasimetric space, define the functor of (co)invariants of a distance module and show that the magnitude (co)homology can be presented via its derived functors. As a corollary we obtain that the magnitude cohomology of a quasimetric space can be presented in terms of Ext functors in the category of distance modules. For finite quasimetric spaces, we show that magnitude (co)homology can be presented in terms of Tor and Ext functors over a certain graded algebra, called the distance algebra of the quasimetric space. For finite digraphs, the distance algebra is a bound quiver algebra. In addition, we show that the magnitude cohomology algebra of a finite quasimetric space can be described as a Yoneda algebra.

We have studied the evolution of the central hundred pc region of barred galaxies by performing numerical simulations realizing multi-phase nature of gas. Our simulations have shown that a stellar bar produces an oval gas ring namely the $x$-2 ring within $1~{\rm kpc}$ as the bar grows. The ring is self-gravitationally unstable enough to trigger formations of gas clouds. Although the gas clouds initially rotate in the $x$-2 ring, cloud-cloud collisions and/or energy injections into the gas ring by Type II supernovae deviate some of the clouds from the ring orbit. After the deviation, the deviated clouds repeat collisions with the other clouds, which rotate in the $x$-2 ring, during several rotations. These processes effectively reduce the angular momentum of the deviated gas cloud. As a result, the gas cloud finally falls into the galactic center, and episodic gas supply to the galactic center takes place., Comment: 10 pages, 9 figures, accepted for publication in PASJ

Cloud-cloud collisions (CCCs) are expected to compress gas and trigger star formation. However, it is not well understood how the collisions and the induced star formation affect galactic-scale properties. By developing an on-the-fly algorithm to identify CCCs at each timestep in a galaxy simulation and a model that relates CCC-triggered star formation to collision speeds, we perform simulations of isolated galaxies to study the evolution of galaxies and giant molecular clouds (GMCs) with prescriptions of self-consistent CCC-driven star formation and stellar feedback. We find that the simulation with the CCC-triggered star formation produces slightly higher star formation rates and a steeper Kennicutt-Schmidt relation than that with a more standard star formation recipe, although collision speeds and frequencies are insensitive to the star formation models. In the simulation with the CCC model, about 70 per cent of the stars are born via CCCs, and colliding GMCs with masses of $\approx 10^{5.5}\,M_{\odot}$ are the main drivers of CCC-driven star formation. In the simulation with the standard star formation recipe, about 50 per cent of stars are born in colliding GMCs even without the CCC-triggered star formation model. These results suggest that CCCs may be one of the most important star formation processes in galaxy evolution. Furthermore, we find that a post-processing analysis of CCCs, as used in previous studies in galaxy simulations, may lead to slightly greater collision speeds and significantly lower collision frequencies than the on-the-fly analysis., Comment: 18 pages, 16 figures, published in MNRAS

In this paper, we study `a fibration of metric spaces' that was originally introduced by Leinster in the study of the magnitude and called metric fibrations. He showed that the magnitude of a metric fibration splits into the product of those of the fiber and the base, which is analogous to the Euler characteristic and topological fiber bundles. His idea and our approach is based on Lawvere's suggestion of viewing a metric space as an enriched category. Actually, the metric fibration turns out to be the restriction of the enriched Grothendieck fibrations to metric spaces. We give a complete classification of metric fibrations by several means, which is parallel to that of topological fiber bundles. That is, the classification of metric fibrations is reduced to that of `principal fibrations', which is done by the `1-Cech cohomology' in an appropriate sense. Here we introduce the notion of torsors in the category of metric spaces, and the discussions are analogous to the sheaf theory. Further, we can define the `fundamental group $\pi^m_1(X)$' of a metric space $X$, which is a group object in metric spaces, such that the conjugation classes of homomorphisms $Hom(\pi^m_1(X), G)$ corresponds to the isomorphism classes of `principal $G$-fibrations' over $X$. Namely, it is classified like topological covering spaces., Comment: 26 pages, 1 figure

In this article, we give a framework for studying the Euler characteristic and its categorification of objects across several areas of geometry, topology and combinatorics. That is, the magnitude theory of filtered sets enriched categories. It is a unification of the Euler characteristic of finite categories and it the magnitude of metric spaces, both of which are introduced by Leinster. Our definitions cover a class of metric spaces which is broader than the original ones, so that magnitude (co)weighting of infinite metric spaces can be considered. We give examples of the magnitude from various research areas containing the Poincar\'{e} polynomial of ranked posets and the growth function of finitely generated groups. In particular, the magnitude homology gives categorifications of them. We also discuss homotopy invariance of the magnitude homology and its variants. Such a homotopy includes digraph homotopy and r-closeness of Lipschitz maps. As a benefit of our categorical view point, we generalize the notion of Grothendieck fibrations of small categories to our enriched categories, whose restriction to metric spaces is a notion called metric fibration that is initially introduced by Leinster. It is remarkable that the magnitude of such a fibration is a product of those of the fiber and the base. We especially study fibrations of graphs, and give examples of graphs with the same magnitude but are not isomorphic., Comment: 35 pages, 1 figure, Fixed English for the second version

The dependence of star formation efficiency (SFE) on galactic structures, especially whether the SFE in the bar region is lower than those in the other regions, has recently been debated. We report the SFEs of 18 nearby gas-rich massive star-forming barred galaxies with a large apparent bar major axis ($\geqq 75^{\prime\prime}$). We statistically measure the SFE by distinguishing the center, bar-end, and bar regions for the first time. The molecular gas surface density is derived from archival CO(1-0) and/or CO(2-1) data by assuming a constant CO-to-H$_2$ conversion factor ($\alpha_{\rm CO}$), and the star formation rate surface density is derived from a linear combination of far-ultraviolet and mid-infrared intensities. The angular resolution is $15^{\prime\prime}$, which corresponds to $0.3 - 1.8~\rm kpc$. We find that the ratio of the SFE in the bar to that in the disk was systematically lower than unity (typically $0.6-0.8$), which means that the star formation in the bar is systematically suppressed. Our results are inconsistent with similar recent statistical studies that reported that SFE tends to be independent of galactic structures. This inconsistency can be attributed to the differences in the definition of the bar region, spatial resolution, $\alpha_{\rm CO}$, and sample galaxies. Furthermore, we find a negative correlation between SFE and velocity width of the CO spectrum, which is consistent with the idea that the large dynamical effects, such as strong shocks, large shear, and fast cloud-cloud collisions caused by the noncircular motion of the bar, result in a low SFE., Comment: 27 pages, 11 figures, accepted for publication in ApJ

We study the effects of the magnetic fields on the formation of massive, self-gravitationally bound cores (MBCs) in high-speed cloud-cloud collisions (CCCs). Extending our previous work (Sakre et al. 2021), we perform magnetohydrodynamic simulations following the high-speed (20 - 40 km s$^{-1}$) collisions between two magnetized (4 $\mu$G initially), turbulent clouds of different sizes in the range of 7 - 20 pc. We show that a magnetic field effect hinders the core growth, particularly after a short-duration collision during which cores cannot get highly bound. In such a case, a shocked region created by the collision rapidly expands to the ambient medium owing to the enhanced magnetic pressure, resulting in the destruction of the highly unbound cores and suppression of gas accretion to massive cores. This negative effect on the MBC formation is a phenomenon not seen in the past hydrodynamic simulations of similar CCC models. Together with our previous work, we conclude that the magnetic fields provide the two competing effects on the MBC formation in CCC; while they promote the mass accumulation into cores during a collision, they operate to destroy cores or hinder the core growth after the collision. The duration of collision determines which effect prevails, providing the maximum collision speed for the MBC formation with given colliding clouds. Our results agree with the observed trend among CCC samples in the corresponding column density range; clouds with higher relative velocity require higher column density for the formation of massive stars (Enokiya et al. 2021)., Comment: Submitted to MNRAS journal

We give a proof that, under relatively mild conditions, fully-connected feed-forward deep random neural networks converge to a Gaussian mixture distribution as only the width of the last hidden layer goes to infinity. We conducted experiments for a simple model which supports our result. Moreover, it gives a detailed description of the convergence, namely, the growth of the last hidden layer gets the distribution closer to the Gaussian mixture, and the other layer successively get the Gaussian mixture closer to the normal distribution., Comment: 14 pages + supplemental materials

In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials ${\mathrm MH}^{\ell}_k(G) \longrightarrow {\mathrm MH}^{\ell-1}_{k-1}(G)$ between magnitude homologies of a digraph $G$, which make them chain complexes. Then we show that its homology ${\mathcal MH}^{\ell}_k(G)$ is non-trivial and homotopy invariant in the context of `homotopy theory of digraphs' developed by Grigor'yan--Muranov--S.-T. Yau et al (G-M-Ys in the following). It is remarkable that the diagonal part of our homology ${\mathcal MH}^{k}_k(G)$ is isomorphic to the reduced path homology $\tilde{H}_k(G)$ also introduced by G-M-Ys. Further, we construct a spectral sequence whose first page is isomorphic to magnitude homology ${\mathrm MH}^{\ell}_k(G)$, and the second page is isomorphic to our homology ${\mathcal MH}^{\ell}_k(G)$. As an application, we show that the diagonality of magnitude homology implies triviality of reduced path homology. We also show that $\tilde{H}_k(g) = 0$ for $k \geq 2$ and $\tilde{H}_1(g) \neq 0$ if any edges of an undirected graph $g$ is contained in a cycle of length $\geq 5$., Comment: version 3 (refined some proofs, added some figures, and fixed some typos. To appear in Bulletin of the London Mathematical Society)

CO(2-1) emission is often used as a tracer of the giant molecular clouds (GMCs) as an alternative to CO(1-0) emission in recent years. Therefore, understanding the environmental dependence of the line ratio of CO(2-1)/CO(1-0), $R_{21}$, on GMC scale is important to accurately estimate the mass of the GMCs. We thus measured the $R_{21}$ in the strongly barred galaxy NGC1300, where star formation activity strongly depends on galactic structure, on $\sim 100$ pc scale. CO images were obtained from ALMA and Nobeyama 45-m telescope. The resultant typical $R_{21}$ in NGC1300 is $0.57 \pm 0.06$. We find environmental variations in $R_{21}$; it is the highest in the bar-end region ($0.72 \pm 0.08$), followed by arm ($0.60 \pm 0.07$) and bar regions ($0.50 \pm 0.06$). GMCs with H$\alpha$ emission show a systematically higher ratio ($0.67 \pm 0.07$) than those without H$\alpha$ ($0.47 \pm 0.05$). In the bar region, where massive star formation is suppressed, H$\alpha$ emission is not associated with most GMCs, resulting in the lowest $R_{21}$. These results raise a possibility that properties of GMCs derived from CO(2-1) observations with the assumption of a constant $R_{21}$ are different from those derived from CO(1-0) observations. Furthermore, we find the $R_{21}$ measured on kpc scale tends to be lower than that of the GMCs probably due to the presence of an extended diffuse molecular gas in NGC1300., Comment: 18 pages, 14 figures, accepted for publication in ApJ

Extracting informative features from images has been of capital importance in computer vision. In this paper, we propose a way to extract such features from images by a method based on algebraic topology. To that end, we construct a weighted graph from an image, which extracts local information of an image. By considering this weighted graph as a pseudo-metric space, we construct a Vietoris-Rips complex with a parameter $\varepsilon$ by a well-known process of algebraic topology. We can extract information of complexity of the image and can detect a sub-image with a relatively high concentration of information from this Vietoris-Rips complex. The parameter $\varepsilon$ of the Vietoris-Rips complex produces robustness to noise. We empirically show that the extracted feature captures well images' characteristics., Comment: 10 pages