5,798 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
Giant Monopole Resonances and nuclear incompressibilities studied for the zero-range and separable pairing interactions
Background: Following the 2007 precise measurements of monopole strengths in
tin isotopes, there has been a continuous theoretical effort to obtain a
precise description of the experimental results. Up to now, there is no
satisfactory explanation of why the tin nuclei appear to be significantly
softer than 208Pb.
Purpose: We determine the influence of finite-range and separable pairing
interactions on monopole strength functions in semi-magic nuclei.
Methods: We employ self-consistently the Quasiparticle Random Phase
Approximation on top of spherical Hartree-Fock-Bogolyubov solutions. We use the
Arnoldi method to solve the linear-response problem with pairing.
Results: We found that the difference between centroids of Giant Monopole
Resonances measured in lead and tin (about 1 MeV) always turns out to be
overestimated by about 100%. We also found that the volume incompressibility,
obtained by adjusting the liquid-drop expression to microscopic results, is
significantly larger than the infinite-matter incompressibility.
Conclusions: The zero-range and separable pairing forces cannot induce
modifications of monopole strength functions in tin to match experimental data.Comment: 11 RevTeX pages, 16 figures, 1 table, extended versio
Correlation studies of fission fragment neutron multiplicities
We calculate neutron multiplicities from fission fragments with specified
mass numbers for events having a specified total fragment kinetic energy. The
shape evolution from the initial compound nucleus to the scission
configurations is obtained with the Metropolis walk method on the
five-dimensional potential-energy landscape, calculated with the
macroscopic-microscopic method for the three-quadratic-surface shape family.
Shape-dependent microscopic level densities are used to guide the random walk,
to partition the intrinsic excitation energy between the two proto-fragments at
scission, and to determine the spectrum of the neutrons evaporated from the
fragments. The contributions to the total excitation energy of the resulting
fragments from statistical excitation and shape distortion at scission is
studied. Good agreement is obtained with available experimental data on neutron
multiplicities in correlation with fission fragments from U(n,f). At higher neutron energies a superlong fission mode appears which
affects the dependence of the observables on the total fragment kinetic energy.Comment: 12 pages, 10 figure
SUMER Observations Confirm the Dynamic Nature of the Quiet Solar Outer Atmosphere: The Inter-network Chromosphere
On 12 March 1996 we obtained observations of the quiet Sun with the SUMER
instrument. The observa- tions were sequences of 15-20 second exposures of
ultraviolet emission line profiles and of the neighboring continua. These data
contain signatures of the dynamics of the solar chromosphere that are uniquely
useful because of wavelength coverage, moderate signal-to-noise ratios, and
image stability. The dominant observed phenomenon is an oscillatory behavior
that is analogous to the 3 minute oscillations seen in Ca II lines. The
oscillations appear to be coherent over 3-8". At any time they occur over
approx. 50 % of the area studied, and they appear as large perturbations in the
intensities of lines and continua. The oscillations are most clearly seen in
intensity variations in the UV (lambda > 912 A) continua, and they are also
seen in the intensities and velocities of chromospheric lines of C I, N I and O
I. Intensity brightenings are accompanied by blueshifts of typically 5 km
s. Phase differences between continuum and line intensities also
indicate the presence of upward propagating waves. Three minute intensity
oscillations are occasionally seen in second spectra (C II 1335), but never in
third spectra (C III and Si III). Third spectra and He I 584 show oscillations
in velocity that are not simply related to the 3 minute oscillations. The
continuum intensity variations are consistent with recent simulations of
chromospheric dynamics (Carlsson & Stein 1994) while the line observations
indicate that important ingredients are missing at higher layers in the
simulations. The data show that time variations are crucial for our
understanding of the chromosphere itself and for the spectral features formed
there.Comment: 8 pages, 3 figs, AASTeX, Accepted for publication in APJ letter
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
Order-N Density-Matrix Electronic-Structure Method for General Potentials
A new order-N method for calculating the electronic structure of general
(non-tight-binding) potentials is presented. The method uses a combination of
the ``purification''-based approaches used by Li, Nunes and Vanderbilt, and
Daw, and a representation of the density matrix based on ``travelling basis
orbitals''. The method is applied to several one-dimensional examples,
including the free electron gas, the ``Morse'' bound-state potential, a
discontinuous potential that mimics an interface, and an oscillatory potential
that mimics a semiconductor. The method is found to contain Friedel
oscillations, quantization of charge in bound states, and band gap formation.
Quantitatively accurate agreement with exact results is found in most cases.
Possible advantages with regard to treating electron-electron interactions and
arbitrary boundary conditions are discussed.Comment: 13 pages, REVTEX, 7 postscript figures (not quite perfect
Collective vibrational states with fast iterative QRPA method
An iterative method we previously proposed to compute nuclear strength
functions is developed to allow it to accurately calculate properties of
individual nuclear states. The approach is based on the
quasi-particle-random-phase approximation (QRPA) and uses an iterative
non-hermitian Arnoldi diagonalization method where the QRPA matrix does not
have to be explicitly calculated and stored. The method gives substantial
advantages over conventional QRPA calculations with regards to the
computational cost. The method is used to calculate excitation energies and
decay rates of the lowest lying 2+ and 3- states in Pb, Sn, Ni and Ca isotopes
using three different Skyrme interactions and a separable gaussian pairing
force.Comment: 10 pages, 11 figure
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