89,267 research outputs found
Convergence of Adaptive Finite Element Approximations for Nonlinear Eigenvalue Problems
In this paper, we study an adaptive finite element method for a class of a
nonlinear eigenvalue problems that may be of nonconvex energy functional and
consider its applications to quantum chemistry. We prove the convergence of
adaptive finite element approximations and present several numerical examples
of micro-structure of matter calculations that support our theory.Comment: 24 pages, 12 figure
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Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer
Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here we extend the original empirical approach, to the analysis of the Cremona database of quantities relevant to BSD, inspecting more than 2.5 million elliptic curves by means of the latest techniques in data science, machine-learning and topological data analysis. Key quantities such as rank, Weierstrass coefficients, period, conductor, Tamagawa number, regulator and order of the Tate-Shafarevich group give rise to a high-dimensional point-cloud whose statistical properties we investigate. We reveal patterns and distributions in the rank versus Weierstrass coefficients, as well as the Beta distribution of the BSD ratio of the quantities. Via gradient boosted trees, machine learning is applied in finding inter-correlation amongst the various quantities. We anticipate that our approach will spark further research on the statistical properties of large datasets in Number Theory and more in general in pure Mathematics
Radiance and Doppler shift distributions across the network of the quiet Sun
The radiance and Doppler-shift distributions across the solar network provide
observational constraints of two-dimensional modeling of transition-region
emission and flows in coronal funnels. Two different methods, dispersion plots
and average-profile studies, were applied to investigate these distributions.
In the dispersion plots, we divided the entire scanned region into a bright and
a dark part according to an image of Fe xii; we plotted intensities and Doppler
shifts in each bin as determined according to a filtered intensity of Si ii. We
also studied the difference in height variations of the magnetic field as
extrapolated from the MDI magnetogram, in and outside network. For the
average-profile study, we selected 74 individual cases and derived the average
profiles of intensities and Doppler shifts across the network. The dispersion
plots reveal that the intensities of Si ii and C iv increase from network
boundary to network center in both parts. However, the intensity of Ne viii
shows different trends, namely increasing in the bright part and decreasing in
the dark part. In both parts, the Doppler shift of C iv increases steadily from
internetwork to network center. The average-profile study reveals that the
intensities of the three lines all decline from the network center to
internetwork region. The binned intensities of Si ii and Ne viii have a good
correlation. We also find that the large blue shift of Ne viii does not
coincide with large red shift of C iv. Our results suggest that the network
structure is still prominent at the layer where Ne viii is formed in the quiet
Sun, and that the magnetic structures expand more strongly in the dark part
than in the bright part of this quiet Sun region.Comment: 10 pages,9 figure
Observation of Fermi-energy dependent unitary impurity resonances in a strong topological insulator Bi_2Se_3 with scanning tunneling spectroscopy
Scanning tunneling spectroscopic studies of Bi_2Se_3 epitaxial films on Si (111) substrates reveal highly localized unitary impurity resonances associated with non-magnetic quantum impurities. The strength of the resonances depends on the energy difference between the Fermi level (E_F) and the Dirac point (E_D) and diverges as E_F approaches E_D. The Dirac-cone surface state of the host recovers within ~ 2Ã… spatial distance from impurities, suggesting robust topological protection of the surface state of topological insulators against high-density impurities that preserve time reversal symmetry
Making vortices in dipolar spinor condensates via rapid adiabatic passage
We propose to the create vortices in spin-1 condensates via magnetic
dipole-dipole interaction. Starting with a polarized condensate prepared under
large axial magnetic field, we show that by gradually inverting the field,
population transfer among different spin states can be realized in a controlled
manner. Under optimal condition, we generate a doubly quantized vortex state
containing nearly all atoms in the condensate. The resulting vortex state is a
direct manifestation of the dipole-dipole interaction and spin textures in
spinor condensates. We also point out that the whole process can be
qualitatively described by a simple rapid adiabatic passage model.Comment: 4 pages, 4 figure
General Sum Rules for WW Scattering in Higgsless Models: Equivalence Theorem and Deconstruction Identities
We analyze inelastic 2 to 2 scattering amplitudes for gauge bosons and
Nambu-Goldstone bosons in deconstructed Higgsless models. Using the (KK)
Equivalence Theorem in 4D (5D), we derive a set of general sum rules among the
boson masses and multi-boson couplings that are valid for arbitrary
deconstructed models. Taking the continuum limit, our results naturally include
the 5D Higgsless model sum rules for arbitrary 5D geometry and boundary
conditions; they also reduce to the elastic sum rules when applied to the
special case of elastic scattering. For the case of linear deconstructed
Higgsless models, we demonstrate that the sum rules can also be derived from a
set of general deconstruction identities and completeness relations. We apply
these sum rules to the deconstructed 3-site Higgsless model and its extensions;
we show that in 5D ignoring all higher KK modes (n>1) is inconsistent once the
inelastic channels become important. Finally, we discuss how our results
generalize beyond the case of linear Higgsless models.Comment: 36 pages, 2 figure
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