5,788 research outputs found
The Theory of the Interleaving Distance on Multidimensional Persistence Modules
In 2009, Chazal et al. introduced -interleavings of persistence
modules. -interleavings induce a pseudometric on (isomorphism
classes of) persistence modules, the interleaving distance. The definitions of
-interleavings and generalize readily to multidimensional
persistence modules. In this paper, we develop the theory of multidimensional
interleavings, with a view towards applications to topological data analysis.
We present four main results. First, we show that on 1-D persistence modules,
is equal to the bottleneck distance . This result, which first
appeared in an earlier preprint of this paper, has since appeared in several
other places, and is now known as the isometry theorem. Second, we present a
characterization of the -interleaving relation on multidimensional
persistence modules. This expresses transparently the sense in which two
-interleaved modules are algebraically similar. Third, using this
characterization, we show that when we define our persistence modules over a
prime field, satisfies a universality property. This universality result
is the central result of the paper. It says that satisfies a stability
property generalizing one which is known to satisfy, and that in
addition, if is any other pseudometric on multidimensional persistence
modules satisfying the same stability property, then . We also show
that a variant of this universality result holds for , over arbitrary
fields. Finally, we show that restricts to a metric on isomorphism
classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in
Foundations of Computational Mathematics. 36 page
Large-scale exact diagonalizations reveal low-momentum scales of nuclei
Ab initio methods aim to solve the nuclear many-body problem with controlled
approximations. Virtually exact numerical solutions for realistic interactions
can only be obtained for certain special cases such as few-nucleon systems.
Here we extend the reach of exact diagonalization methods to handle model
spaces with dimension exceeding on a single compute node. This allows
us to perform no-core shell model (NCSM) calculations for 6Li in model spaces
up to and to reveal the 4He+d halo structure of this
nucleus. Still, the use of a finite harmonic-oscillator basis implies
truncations in both infrared (IR) and ultraviolet (UV) length scales. These
truncations impose finite-size corrections on observables computed in this
basis. We perform IR extrapolations of energies and radii computed in the NCSM
and with the coupled-cluster method at several fixed UV cutoffs. It is shown
that this strategy enables information gain also from data that is not fully UV
converged. IR extrapolations improve the accuracy of relevant bound-state
observables for a range of UV cutoffs, thus making them profitable tools. We
relate the momentum scale that governs the exponential IR convergence to the
threshold energy for the first open decay channel. Using large-scale NCSM
calculations we numerically verify this small-momentum scale of finite nuclei.Comment: Minor revisions.Accepted for publication in Physical Review
Modeling of Covalent Bonding in Solids by Inversion of Cohesive Energy Curves
We provide a systematic test of empirical theories of covalent bonding in
solids using an exact procedure to invert ab initio cohesive energy curves. By
considering multiple structures of the same material, it is possible for the
first time to test competing angular functions, expose inconsistencies in the
basic assumption of a cluster expansion, and extract general features of
covalent bonding. We test our methods on silicon, and provide the direct
evidence that the Tersoff-type bond order formalism correctly describes
coordination dependence. For bond-bending forces, we obtain skewed angular
functions that favor small angles, unlike existing models. As a
proof-of-principle demonstration, we derive a Si interatomic potential which
exhibits comparable accuracy to existing models.Comment: 4 pages revtex (twocolumn, psfig), 3 figures. Title and some wording
(but no content) changed since original submission on 24 April 199
Class and rank of differential modules
A differential module is a module equipped with a square-zero endomorphism.
This structure underpins complexes of modules over rings, as well as
differential graded modules over graded rings. We establish lower bounds on the
class--a substitute for the length of a free complex--and on the rank of a
differential module in terms of invariants of its homology. These results
specialize to basic theorems in commutative algebra and algebraic topology. One
instance is a common generalization of the equicharacteristic case of the New
Intersection Theorem of Hochster, Peskine, P. Roberts, and Szpiro, concerning
complexes over noetherian commutative rings, and of a theorem of G. Carlsson on
differential graded modules over graded polynomial rings.Comment: 27 pages. Minor changes; mainly stylistic. To appear in Inventiones
Mathematica
Categorification of persistent homology
We redevelop persistent homology (topological persistence) from a categorical
point of view. The main objects of study are diagrams, indexed by the poset of
real numbers, in some target category. The set of such diagrams has an
interleaving distance, which we show generalizes the previously-studied
bottleneck distance. To illustrate the utility of this approach, we greatly
generalize previous stability results for persistence, extended persistence,
and kernel, image and cokernel persistence. We give a natural construction of a
category of interleavings of these diagrams, and show that if the target
category is abelian, so is this category of interleavings.Comment: 27 pages, v3: minor changes, to appear in Discrete & Computational
Geometr
Persistent Homology Over Directed Acyclic Graphs
We define persistent homology groups over any set of spaces which have
inclusions defined so that the corresponding directed graph between the spaces
is acyclic, as well as along any subgraph of this directed graph. This method
simultaneously generalizes standard persistent homology, zigzag persistence and
multidimensional persistence to arbitrary directed acyclic graphs, and it also
allows the study of more general families of topological spaces or point-cloud
data. We give an algorithm to compute the persistent homology groups
simultaneously for all subgraphs which contain a single source and a single
sink in arithmetic operations, where is the number of vertices in
the graph. We then demonstrate as an application of these tools a method to
overlay two distinct filtrations of the same underlying space, which allows us
to detect the most significant barcodes using considerably fewer points than
standard persistence.Comment: Revised versio
Mind the Gap: A Study in Global Development through Persistent Homology
The Gapminder project set out to use statistics to dispel simplistic notions
about global development. In the same spirit, we use persistent homology, a
technique from computational algebraic topology, to explore the relationship
between country development and geography. For each country, four indicators,
gross domestic product per capita; average life expectancy; infant mortality;
and gross national income per capita, were used to quantify the development.
Two analyses were performed. The first considers clusters of the countries
based on these indicators, and the second uncovers cycles in the data when
combined with geographic border structure. Our analysis is a multi-scale
approach that reveals similarities and connections among countries at a variety
of levels. We discover localized development patterns that are invisible in
standard statistical methods
Inter-network regions of the Sun at millimetre wavelengths
The continuum intensity at wavelengths around 1 mm provides an excellent way
to probe the solar chromosphere. Future high-resolution millimetre arrays, such
as the Atacama Large Millimeter Array (ALMA), will thus produce valuable input
for the ongoing controversy on the thermal structure and the dynamics of this
layer. Synthetic brightness temperature maps are calculated on basis of
three-dimensional radiation (magneto-)hydrodynamic (MHD) simulations. While the
millimetre continuum at 0.3mm originates mainly from the upper photosphere, the
longer wavelengths considered here map the low and middle chromosphere. The
effective formation height increases generally with wavelength and also from
disk-centre towards the solar limb. The average intensity contribution
functions are usually rather broad and in some cases they are even
double-peaked as there are contributions from hot shock waves and cool
post-shock regions in the model chromosphere. Taking into account the
deviations from ionisation equilibrium for hydrogen gives a less strong
variation of the electron density and with it of the optical depth. The result
is a narrower formation height range. The average brightness temperature
increases with wavelength and towards the limb. The relative contrast depends
on wavelength in the same way as the average intensity but decreases towards
the limb. The dependence of the brightness temperature distribution on
wavelength and disk-position can be explained with the differences in formation
height and the variation of temperature fluctuations with height in the model
atmospheres.Comment: 15 pages, 10 figures, accepted for publication in A&A (15.05.07
Dissecting bombs and bursts: non-LTE inversions of low-atmosphere reconnection in SST and IRIS observations
Ellerman bombs and UV bursts are transient brightenings that are ubiquitously
observed in the lower atmospheres of active and emerging flux regions. Here we
present inversion results of SST/CRISP and CHROMIS, as well as IRIS data of
such transient events. Combining information from the Mg II h & k, Si IV and Ca
II 8542A and Ca II H & K lines, we aim to characterise their temperature and
velocity stratification, as well as their magnetic field configuration. We find
average temperature enhancements of a few thousand kelvin close to the
classical temperature minimum, but localised peak temperatures of up to
10,000-15,000 K from Ca II inversions. Including Mg II generally dampens these
temperature enhancements to below 8000 K, while Si IV requires temperatures in
excess of 10,000 K at low heights, but may also be reproduced with secondary
temperature enhancements of 35,000-60,000 K higher up. However, reproducing Si
IV comes at the expense of overestimating the Mg II emission. The line-of-sight
velocity maps show clear bi-directional jet signatures and strong correlation
with substructure in the intensity images, with slightly larger velocities
towards the observer than away. The magnetic field parameters show an
enhancement of the horizontal field co-located with the brightenings at similar
heights as the temperature increase. We are thus able to largely reproduce the
observational properties of Ellerman bombs with UV burst signature with
temperature stratifications peaking close to the classical temperature minimum.
Correctly modelling the Si IV emission in agreement with all other diagnostics
is, however, an outstanding issue. Accounting for resolution differences,
fitting localised temperature enhancements and/or performing spatially-coupled
inversions is likely necessary to obtain better agreement between all
considered diagnostics.Comment: Accepted for publication in Astronomy & Astrophysics. 24 pages, 17
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