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Formal and finite order equivalences
We show that two families of germs of real-analytic subsets in are
formally equivalent if and only if they are equivalent of any finite order. We
further apply the same technique to obtain analogous statements for
equivalences of real-analytic self-maps and vector fields under conjugations.
On the other hand, we provide an example of two sets of germs of smooth curves
that are equivalent of any finite order but not formally equivalent
The Serpens filament: at the onset of slightly supercritical collapse
The Serpens filament, as one of the nearest infrared dark clouds, is regarded
as a pristine filament at a very early evolutionary stage of star formation. In
order to study its molecular content and dynamical state, we mapped this
filament in seven species. Among them, HCO, HNC, HCN, and CS show
self-absorption, while CO is most sensitive to the filamentary
structure. A kinematic analysis demonstrates that this filament forms a
velocity-coherent (trans-)sonic structure, a large part of which is one of the
most quiescent regions in the Serpens cloud. Widespread CO depletion is
found throughout the Serpens filament. Based on the Herschel dust-derived
H column density map, the line mass of the filament is
36--41~M~pc, and its full width at half maximum is
0.170.01~pc, while its length is ~1.6~pc. The inner radial column density
profile of this filament can be well fitted with a Plummer profile with an
exponent of 2.20.1, a scale radius of pc, and a central
density of ~cm. The Serpens filament appears
to be slightly supercritical. The widespread blue-skewed HNC and CS line
profiles and HCN hyperfine line anomalies across this filament indicate radial
infall in parts of the Serpens filament. CO velocity gradients also
indicate accretion flows along the filament. The velocity and density
structures suggest that such accretion flows are likely due to a longitudinal
collapse parallel to the filament's long axis. Both the radial infall rate and
the longitudinal accretion rate along the Serpens filament are lower than all
previously reported values in other filaments. This indicates that the Serpens
filament lies at an early evolutionary stage when collapse has just begun, or
that thermal and non-thermal support are effective in providing support against
gravity.Comment: 22 pages, 14 figures, 4 tables, accepted for publication in A&A; for
the draft showing figures with full resolution, see
http://gongyan2444.github.io/pdf/absfil.pd
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