8,342 research outputs found
A Gap Theorem for Willmore Tori and an application to the Willmore Flow
In 1965 Willmore conjectured that the integral of the square of the mean
curvature of a torus immersed in is at least and attains this
minimal value if and only if the torus is a M\"obius transform of the Clifford
torus. This was recently proved by Marques and Neves. In this paper, we show
for tori there is a gap to the next critical point of the Willmore energy and
we discuss an application to the Willmore flow. We also prove an energy gap
from the Clifford torus to surfaces of higher genus.Comment: 9 pages. In this new version we performed some small changes to
improve the exposition. To appear in Nonlinear Analysis: Theory Methods &
Application
Using Classical Music to Increase Productivity in Elementary School Students with Attention Deficit Hyperactivity Disorder
Currently, physicians use prescription medications to treat attention deficit hyperactivity disorder (ADHD) in children known to have low levels of the neurotransmitter dopamine. ADHD children commonly have low adherence to medications used to treat their condition because there is no known dosage to produce the optimal attention span for motivation and focus. The safest method of dopamine production is through natural techniques, which could reduce medication intake and thus side effects. Listening to classical music dominant in consonant sounds has been shown to increase dopamine levels in children with ADHD through the dopamine “reward” pathway in the brain. Because all children exhibit some increase in dopamine while listening to consonance-dominated classical music, educators can introduce this music into the classroom to improve overall academic performance and help treat ADHD children simultaneously. Instructors could rotate music playlists weekly to avoid memorization and monotony. Music could be played in quiet environments when focus is key—not during playtimes or group work. Music treatment is a cost-effective method that—with extant classroom technology—can increase overall focus and work production, proving a worthwhile investment. Physicians should turn to incorporating music in ADHD treatments rather than solely prescribing medications in order to expand the possibilities of discovering more efficient treatment plans. Though consonance dominated classical music is known to increase dopamine, further research should be conducted in order to better define appropriate ADHD treatment plans in the classroom.https://scholarscompass.vcu.edu/uresposters/1056/thumbnail.jp
Koopmans-compliant spectral functionals for extended systems
Koopmans-compliant functionals have been shown to provide accurate spectral
properties for molecular systems; this accuracy is driven by the generalized
linearization condition imposed on each charged excitation - i.e. on changing
the occupation of any orbital in the system, while accounting for screening and
relaxation from all other electrons. In this work we discuss the theoretical
formulation and the practical implementation of this formalism to the case of
extended systems, where a third condition, the localization of Koopmans'
orbitals, proves crucial to reach seamlessly the thermodynamic limit. We
illustrate the formalism by first studying one-dimensional molecular systems of
increasing length. Then, we consider the band gaps of 30 paradigmatic
solid-state test cases, for which accurate experimental and computational
results are available. The results are found to be comparable with the
state-of-the-art in diagrammatic techniques (self-consistent many-body
perturbation theory with vertex corrections), notably using just a functional
formulation for spectral properties and the physics of the generalized-gradient
approximation; when ionization potentials are compared, the results are roughly
twice as accurate.Comment: 9 pages, 3 figures, 1 supporting informatio
Rational Multi-Curve Models with Counterparty-Risk Valuation Adjustments
We develop a multi-curve term structure setup in which the modelling
ingredients are expressed by rational functionals of Markov processes. We
calibrate to LIBOR swaptions data and show that a rational two-factor lognormal
multi-curve model is sufficient to match market data with accuracy. We
elucidate the relationship between the models developed and calibrated under a
risk-neutral measure Q and their consistent equivalence class under the
real-world probability measure P. The consistent P-pricing models are applied
to compute the risk exposures which may be required to comply with regulatory
obligations. In order to compute counterparty-risk valuation adjustments, such
as CVA, we show how positive default intensity processes with rational form can
be derived. We flesh out our study by applying the results to a basis swap
contract.Comment: 34 pages, 9 figure
Mass Destruction of Stars Caused by Black Holes: Scientists might be one step closer to finding out how supermassive black holes form
Screening in orbital-density-dependent functionals
Electronic-structure functionals that include screening effects, such as
Hubbard or Koopmans' functionals, require to describe the response of a system
to the fractional addition or removal of an electron from an orbital or a
manifold. Here, we present a general method to incorporate screening based on
linear-response theory, and we apply it to the case of the orbital-by-orbital
screening of Koopmans' functionals. We illustrate the importance of such
generalization when dealing with challenging systems containing orbitals with
very different chemical character, also highlighting the simple dependence of
the screening on the localization of the orbitals. We choose a set of 46
transition-metal complexes for which experimental data and accurate many-body
perturbation theory calculations are available. When compared to experiment,
results for ionization potentials show a very good performance with a mean
absolute error of eV, comparable to the most accurate many-body
perturbation theory approaches. These results reiterate the role of Koopmans'
compliant functionals as simple and accurate quasiparticle approximations to
the exact spectral functional, bypassing diagrammatic expansions and relying
only on the physics of the local density or generalized-gradient approximation
Koopmans-compliant functionals and their performance against reference molecular data
Koopmans-compliant functionals emerge naturally from extending the constraint
of piecewise linearity of the total energy as a function of the number of
electrons to each fractional orbital occupation. When applied to approximate
density-functional theory, these corrections give rise to
orbital-density-dependent functionals and potentials. We show that the simplest
implementations of Koopmans' compliance provide accurate estimates for the
quasiparticle excitations and leave the total energy functional almost or
exactly intact, i.e., they describe correctly electron removals or additions,
but do not necessarily alter the electronic charge density distribution within
the system. Additional functionals can then be constructed that modify the
potential energy surface, including e.g. Perdew-Zunger corrections. These
functionals become exactly one-electron self-interaction free and, as all
Koopmans-compliant functionals, are approximately many-electron
self-interaction free. We discuss in detail these different formulations, and
provide extensive benchmarks for the 55 molecules in the reference G2-1 set,
using Koopmans-compliant functionals constructed from local-density or
generalized-gradient approximations. In all cases we find excellent performance
in the electronic properties, comparable or improved with respect to that of
many-body perturbation theories, such as GW and self-consistent GW, at
a fraction of the cost and in a variational framework that also delivers energy
derivatives. Structural properties and atomization energies preserve or
slightly improve the accuracy of the underlying density-functional
approximations (Note: Supplemental Material is included in the source)
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