4 research outputs found

    Quantifying the homology of periodic cell complexes

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    A periodic cell complex, KK, has a finite representation as the quotient space, q(K)q(K), consisting of equivalence classes of cells identified under the translation group acting on KK. We study how the Betti numbers and cycles of KK are related to those of q(K)q(K), first for the case that KK is a graph, and then higher-dimensional cell complexes. When KK is a dd-periodic graph, it is possible to define Zd\mathbb{Z}^d-weights on the edges of the quotient graph and this information permits full recovery of homology generators for KK. The situation for higher-dimensional cell complexes is more subtle and studied in detail using the Mayer-Vietoris spectral sequence.Comment: 1st revised version, only major change to the content of the original version is the addition of the new "Theorem 3" and "Corollary 2

    Computing 1-Periodic Persistent Homology with Finite Windows

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    Let KK be a periodic cell complex endowed with a covering q:K→Gq:K\to G where GG is a finite quotient space of equivalence classes under translations acting on KK. We assume GG is embedded in a space whose homotopy type is a dd-torus for some dd, which introduces "toroidal cycles" in GG which do not lift to cycles in KK by qq . We study the behaviour of toroidal and non-toroidal cycles for the case KK is 1-periodic, i.e. G=K/ZG=K/\mathbb{Z} for some free action of Z\mathbb{Z} on KK. We show that toroidal cycles can be entirely classified by endomorphisms on the homology of unit cells of KK, and moreover that toroidal cycles have a sense of unimodality when studying the persistent homology of GG.Comment: 1st revised version, only major change is in Section 3 to the theory behind constructing the necessary endomorphism

    Numerical Calibration of the HCN−-Star Formation Correlation

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    HCN(1−-0) emission traces dense gas and correlates very strongly with star formation rates (SFRs) on scales from small Milky Way clouds to whole galaxies. The observed correlation offers strong constraints on the efficiency of star formation in dense gas, but quantitative interpretation of this constraint requires a mapping from HCN emission to gas mass and density. In this paper we provide the required calibration by postprocessing high-resolution simulations of dense, star-forming clouds to calculate their HCN emission (LHCNL_{\rm HCN}) and to determine how that emission is related to the underlying gas density distribution and star formation efficiency. We find that HCN emission traces gas with a luminosity-weighted mean number density of 0.8−1.7×104 cm−30.8-1.7 \times 10^4\,{\rm cm}^{-3} and that HCN luminosity is related to mass of dense gas of ≳104 cm−3\gtrsim 10^4\,{\rm cm}^{-3} with a conversion factor of αHCN≈14 M⊙/(K km s−1 pc2){\alpha}_{\rm HCN} \approx 14\,\rm M_{\odot}/(K\,km\,s^{-1}\,{pc}^2). We also measure a new empirical relationship between the SFR per global mean freefall time (ϵff{\epsilon}_{\rm ff}) and the SFR−-HCN relationship, SFR/LHCN=2.0×10−7 (ϵff/0.01)1.1 M⊙ yr−1/(K km s−1 pc2){\rm SFR}/L_{\rm HCN} = 2.0 \times 10^{-7}\,({\epsilon}_{\rm ff}/0.01)^{1.1}\,\rm M_{\odot}\,{yr}^{-1}/(K\,km\,s^{-1}\,{pc}^2). The observed SFR−-HCN correlation strongly constrains ϵff≈1%{\epsilon}_{\rm ff} \approx 1\% with a factor of ∼3\sim 3 systematic uncertainty. The scatter in ϵff{\epsilon}_{\rm ff} from cloud to cloud within the Milky Way is a factor of a few. We conclude that LHCNL_{\rm HCN} is an effective tracer of dense gas and that the IR−-HCN correlation is a significant diagnostic of the microphysics of star formation in dense gas
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