6,150 research outputs found
Diophantine approximation by special primes
We show that whenever , is real and constants
satisfy some necessary conditions, there are infinitely many prime triples
satisfying the inequality and such that, for each
, has at most prime factors
Exact Maps in Density Functional Theory for Lattice Models
In the present work, we employ exact diagonalization for model systems on a
real-space lattice to explicitly construct the exact density-to-potential and
for the first time the exact density-to-wavefunction map that underly the
Hohenberg-Kohn theorem in density functional theory. Having the explicit
wavefunction-to- density map at hand, we are able to construct arbitrary
observables as functionals of the ground-state density. We analyze the
density-to-potential map as the distance between the fragments of a system
increases and the correlation in the system grows. We observe a feature that
gradually develops in the density-to-potential map as well as in the
density-to-wavefunction map. This feature is inherited by arbitrary expectation
values as functional of the ground-state density. We explicitly show the
excited-state energies, the excited-state densities, and the correlation
entropy as functionals of the ground-state density. All of them show this exact
feature that sharpens as the coupling of the fragments decreases and the
correlation grows. We denominate this feature as intra-system steepening. We
show that for fully decoupled subsystems the intra-system steepening transforms
into the well-known inter-system derivative discontinuity. An important
conclusion is that for e.g. charge transfer processes between localized
fragments within the same system it is not the usual inter-system derivative
discontinuity that is missing in common ground-state functionals, but rather
the differentiable intra-system steepening that we illustrate in the present
work
The quaternary Piatetski-Shapiro inequality with one prime of the form
In this paper we show that, for any fixed , every sufficiently
large positive number and a small constant , the diophantine
inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\varepsilon
\end{equation*} has a solution in prime numbers , such
that .Comment: arXiv admin note: substantial text overlap with arXiv:2011.0396
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