On the Diophantine Equation G n (x) = G m (y) with Q (x, y)=0

Fuchs, Clemens; Pethö, Attila; Tichy, Robert F.

doi:10.1007/978-3-211-74280-8_10

Cited by 24 publications

(30 citation statements)

References 14 publications

(17 reference statements)

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“…See, for instance, the results in [27,48] for related regularity results. The theorems in this paper, while considering the case where (1.6) fails, fall in the realm of (1.39).…”

mentioning

confidence: 99%

Lipschitz Bounds and Nonuniform Ellipticity

Beck

Mingione

2019

Comm Pure Appl Math

150

125

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We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve natural limiting function spaces and reproduce, in this very general context, the classical and optimal ones known in the linear case for the Poisson equation. Moreover, we provide new and natural growth a priori estimates whose validity was an open problem. Finally, we find new results also in the classical uniformly elliptic case. Beyond the specific results, the paper proposes a new approach to nonuniform ellipticity that, in a sense, allows us to reduce nonuniform elliptic problems to uniformly elliptic ones via potential theoretic arguments that are for the first time applied in this setting. © 2019 Wiley Periodicals, Inc.

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“…See, for instance, the results in [27,48] for related regularity results. The theorems in this paper, while considering the case where (1.6) fails, fall in the realm of (1.39).…”

mentioning

confidence: 99%

Lipschitz Bounds and Nonuniform Ellipticity

Beck

Mingione

2019

Comm Pure Appl Math

150

125

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“…RPA typically does a decent job of estimating energy differences between similar species, such as the interactions between two-dimensional layers 4 . RPA yields reasonable structural properties for non-magnetic solids 9 and captures the static correlation in the dissociating H2 10,11 . It does a much poorer job of treating absolute energies, or of predicting energies where the fundamental properties of the different species are different 6,7,12,13 .…”

Section: Introductionmentioning

confidence: 99%

Simple self-interaction correction to random-phase-approximation-like correlation energies

2019

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The random phase approximation (RPA) is exact for the exchange energy of a many-electron ground state, but RPA makes the correlation energy too negative by about 0.5 eV/electron. That large short-range error, which tends to cancel out of iso-electronic energy differences, is largely corrected by an exchange-correlation kernel, or (as in RPA+) by an additive local or semilocal correction. RPA+ is by construction exact for the homogeneous electron gas, and it is also accurate for the jellium surface. RPA+ often gives realistic total energies for atoms or solids in which spinpolarization corrections are absent or small. RPA and RPA+ also yield realistic singlet binding energy curves for H2 and N2, and thus RPA+ yields correct total energies even for spin-unpolarized atoms with fractional spins and strong correlation, as in stretched H2 or N2. However, RPA and RPA+ can be very wrong for spin-polarized one-electron systems (especially for stretched H2 + ), and also for the spin-polarization energies of atoms. The spin-polarization energy is often a small part of the total energy of an atom, but important for ionization energies, electron affinities, and the atomization energies of molecules. Here we propose a computationally efficient generalized RPA+ (gRPA+) that changes RPA+ only for spin-polarized systems by making gRPA+ exact for all oneelectron densities, in the same simple semilocal way that the correlation energy densities of many meta-generalized gradent approximations are made self-correlation free. By construction, gRPA+ does not degrade the exact RPA+ description of jellium. gRPA+ is found to greatly improve upon RPA and RPA+ for the ionization energies and electron affinities of light atoms. Many versions of RPA with an approximate exchange-correlation kernel fail to be exact for all one-electron densities, and they can also be self-interaction corrected in this way.

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“…There is a well-known difficulty to describe long-range correlations with semilocal approximations for the XC energy functional in ground-state calculations [48] . For example, when describing the dissociation of H 2 with a semilocal XC-functional, an artificial transition to a spin-polarized system is in general used to describe total energies at large atom separation.…”

Section: 2mentioning

confidence: 99%

“…For example, when describing the dissociation of H 2 with a semilocal XC-functional, an artificial transition to a spin-polarized system is in general used to describe total energies at large atom separation. [48,49] In time-dependent simulations of H 0 incident on a metal surface, the H 0 atom has been assumed to be electronically spin-polarized at large separations from the surface and loses spin polarization when interacting with the surface. [50,51] We also use an initially spin-polarized H 0 -atom in the present simulations in Section 2.3.…”

Section: 2mentioning

confidence: 99%

“…We therefore suspect that the exchange-correlation effects are not well accounted for by the present spin-unpolarized semilocal approximation to v XC , and more advanced approximations would be desirable. [48] For the model-Hamiltonian calculations, the problem of correlation has been approached using, for example, as approximation for the correlation in multiple-level systems U → ∞. [30] It is argued that no electron-electron relaxation (Auger effect) occurs when the present adiabatic XC-functionals are employed.…”

Section: 2mentioning

confidence: 99%

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Time‐dependent simulation of ion stopping: Charge transfer and electronic excitations

Schlünzen

Balzer

Bönitz

et al. 2019

Contributions to Plasma Physics

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The energy loss of charged particles in matter has been studied for many decades, both, analytically and via computer simulations. While the regime of high projectile energies is well understood, low energy stopping in solids is more challenging due to the importance of non-adiabatic effects and electronic correlations. Here we consider two problems: the charge transfer between substrate and projectile and the role of electronic correlations, specifically formation of doubly occupied lattice sites in the material during the stopping process. The former problem is treated by time-dependent density functional theory simulations and the latter by non-equilibrium Green functions.

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On the Diophantine Equation G n (x) = G m (y) with Q (x, y)=0

Abstract: Dedicated to Wolfgang Schmidt on his 70th birthday.Abstract. Let K be a field of characteristic 0 and let (Gn(X))

Cited by 24 publications

References 14 publications

Lipschitz Bounds and Nonuniform Ellipticity

Lipschitz Bounds and Nonuniform Ellipticity

Simple self-interaction correction to random-phase-approximation-like correlation energies

Time‐dependent simulation of ion stopping: Charge transfer and electronic excitations

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