4 research outputs found
Quantifying the homology of periodic cell complexes
A periodic cell complex, , has a finite representation as the quotient
space, , consisting of equivalence classes of cells identified under the
translation group acting on . We study how the Betti numbers and cycles of
are related to those of , first for the case that is a graph, and
then higher-dimensional cell complexes. When is a -periodic graph, it is
possible to define -weights on the edges of the quotient graph
and this information permits full recovery of homology generators for . 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
Let be a periodic cell complex endowed with a covering where
is a finite quotient space of equivalence classes under translations acting
on . We assume is embedded in a space whose homotopy type is a -torus
for some , which introduces "toroidal cycles" in which do not lift to
cycles in by . We study the behaviour of toroidal and non-toroidal
cycles for the case is 1-periodic, i.e. for some free
action of on . We show that toroidal cycles can be entirely
classified by endomorphisms on the homology of unit cells of , and moreover
that toroidal cycles have a sense of unimodality when studying the persistent
homology of .Comment: 1st revised version, only major change is in Section 3 to the theory
behind constructing the necessary endomorphism
Numerical Calibration of the HCNStar Formation Correlation
HCN(10) 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 ()
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 and that HCN luminosity is related to mass of dense gas of
with a conversion factor of . We also measure a new
empirical relationship between the SFR per global mean freefall time
() and the SFRHCN relationship, . The observed SFRHCN
correlation strongly constrains with a factor
of systematic uncertainty. The scatter in from
cloud to cloud within the Milky Way is a factor of a few. We conclude that
is an effective tracer of dense gas and that the IRHCN
correlation is a significant diagnostic of the microphysics of star formation
in dense gas