316,011 research outputs found
Realizing modules over the homology of a DGA
Let A be a DGA over a field and X a module over H_*(A). Fix an
-structure on H_*(A) making it quasi-isomorphic to A. We construct an
equivalence of categories between A_{n+1}-module structures on X and length n
Postnikov systems in the derived category of A-modules based on the bar
resolution of X. This implies that quasi-isomorphism classes of A_n-structures
on X are in bijective correspondence with weak equivalence classes of
rigidifications of the first n terms of the bar resolution of X to a complex of
A-modules. The above equivalences of categories are compatible for different
values of n. This implies that two obstruction theories for realizing X as the
homology of an A-module coincide.Comment: 24 page
Extending the halo mass resolution of -body simulations
We present a scheme to extend the halo mass resolution of N-body simulations
of the hierarchical clustering of dark matter. The method uses the density
field of the simulation to predict the number of sub-resolution dark matter
haloes expected in different regions. The technique requires as input the
abundance of haloes of a given mass and their average clustering, as expressed
through the linear and higher order bias factors. These quantities can be
computed analytically or, more accurately, derived from a higher resolution
simulation as done here. Our method can recover the abundance and clustering in
real- and redshift-space of haloes with mass below at to better than 10%. We demonstrate the
technique by applying it to an ensemble of 50 low resolution, large-volume
-body simulations to compute the correlation function and covariance matrix
of luminous red galaxies (LRGs). The limited resolution of the original
simulations results in them resolving just two thirds of the LRG population. We
extend the resolution of the simulations by a factor of 30 in halo mass in
order to recover all LRGs. With existing simulations it is possible to generate
a halo catalogue equivalent to that which would be obtained from a -body
simulation using more than 20 trillion particles; a direct simulation of this
size is likely to remain unachievable for many years. Using our method it is
now feasible to build the large numbers of high-resolution large volume mock
galaxy catalogues required to compute the covariance matrices necessary to
analyse upcoming galaxy surveys designed to probe dark energy.Comment: 11 pages, 7 Figure
Definable transformation to normal crossings over Henselian fields with separated analytic structure
We are concerned with rigid analytic geometry in the general setting of
Henselian fields with separated analytic structure, whose theory was
developed by Cluckers--Lipshitz--Robinson. It unifies earlier work and
approaches of numerous mathematicians. Separated analytic structures admit
reasonable relative quantifier elimination in a suitable analytic language.
However, the rings of global analytic functions with two kinds of variables
seem not to have good algebraic properties such as Noetherianity or excellence.
Therefore the usual global resolution of singularities from rigid analytic
geometry is no longer at our disposal. Our main purpose is to give a definable
version of the canonical desingularization algorithm (the hypersurface case)
due to Bierstone--Milman so that both these powerful tools are available in the
realm of non-Archimedean analytic geometry at the same time. It will be carried
out within a category of definable, strong analytic manifolds and maps, which
is more flexible than that of affinoid varieties and maps. Strong analytic
objects are those definable ones that remain analytic over all fields
elementarily equivalent to . This condition may be regarded as a kind of
symmetry imposed on ordinary analytic objects. The strong analytic category
makes it possible to apply a model-theoretic compactness argument in the
absence of the ordinary topological compactness. On the other hand, our
closedness theorem enables application of resolution of singularities to
topological problems involving the topology induced by valuation. Eventually,
these three results will be applied to such issues as the existence of
definable retractions or extending continuous definable functions.Comment: This is the final version published in the journal Symmetry-Basel,
2019, 11, 93
The Boardman-Vogt resolution of operads in monoidal model categories
We extend the W-construction of Boardman and Vogt to operads of an arbitrary
monoidal model category with suitable interval, and show that it provides a
cofibrant resolution for well-pointed sigma-cofibrant operads. The standard
simplicial resolution of Godement as well as the cobar-bar chain resolution are
shown to be particular instances of this generalised W-construction
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