91 research outputs found
Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function
This paper continues investigations on the integral transforms of the
Minkowski question mark function. In this work we finally establish the
long-sought formula for the moments, which does not explicitly involve regular
continued fractions, though it has a hidden nice interpretation in terms of
semi-regular continued fractions. The proof is self-contained and does not rely
on previous results by the author.Comment: 8 page
The projective translation equation and unramified 2-dimensional flows with rational vector fields
Let X=(x,y). Previously we have found all rational solutions of the
2-dimensional projective translation equation, or PrTE,
(1-z)f(X)=f(f(Xz)(1-z)/z); here f(X)=(u(x,y),v(x,y)) is a pair of two (real or
complex) functions. Solutions of this functional equation are called projective
flows. A vector field of a rational flow is a pair of 2-homogenic rational
functions. On the other hand, only special pairs of 2-homogenic rational
functions give rise to rational flows. In this paper we are interested in all
non-singular (satisfying the boundary condition) and unramified (without
branching points, i.e. single-valued functions in C^2\{union of curves})
projective flows whose vector field is still rational. We prove that, up to
conjugation with 1-homogenic birational plane transformation, these are of 6
types: 1) the identity flow; 2) one flow for each non-negative integer N -
these flows are rational of level N; 3) the level 1 exponential flow, which is
also conjugate to the level 1 tangent flow; 4) the level 3 flow expressable in
terms of Dixonian (equianharmonic) elliptic functions; 5) the level 4 flow
expressable in terms of lemniscatic elliptic functions; 6) the level 6 flow
expressable in terms of Dixonian elliptic functions again. This reveals another
aspect of the PrTE: in the latter four cases this equation is equivalent and
provides a uniform framework to addition formulas for exponential, tangent, or
special elliptic functions (also addition formulas for polynomials and the
logarithm, though the latter appears only in branched flows). Moreover, the
PrTE turns out to have a connection with Polya-Eggenberger urn models. Another
purpose of this study is expository, and we provide the list of open problems
and directions in the theory of PrTE; for example, we define the notion of
quasi-rational projective flows which includes curves of arbitrary genus.Comment: 34 pages, 2 figure
Asymptotic formula for the moments of Minkowski question mark function in the interval [0,1]
In this paper we prove the asymptotic formula for the moments of Minkowski
question mark function, which describes the distribution of rationals in the
Farey tree. The main idea is to demonstrate that certain a variation of a
Laplace method is applicable in this problem, hence the task reduces to a
number of technical calculations.Comment: 11 pages, 1 figure (final version). Lithuanian Math. J. (to appear
First-principles theory of the luminescence lineshape for the triplet transition in diamond NV centre
In this work we present theoretical calculations and analysis of the vibronic
structure of the spin-triplet optical transition in diamond nitrogen-vacancy
centres. The electronic structure of the defect is described using accurate
first-principles methods based on hybrid functionals. We devise a computational
methodology to determine the coupling between electrons and phonons during an
optical transition in the dilute limit. As a result, our approach yields a
smooth spectral function of electron-phonon coupling and includes both
quasi-localized and bulk phonons on equal footings. The luminescence lineshape
is determined via the generating function approach. We obtain a highly accurate
description of the luminescence band, including all key parameters such as the
Huang-Rhys factor, the Debye-Waller factor, and the frequency of the dominant
phonon mode. More importantly, our work provides insight into the vibrational
structure of nitrogen vacancy centres, in particular the role of local modes
and vibrational resonances. In particular, we find that the pronounced mode at
65 meV is a vibrational resonance, and we quantify localization properties of
this mode. These excellent results for the benchmark diamond nitrogen-vacancy
centre provide confidence that the procedure can be applied to other defects,
including alternative systems that are being considered for applications in
quantum information processing
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Dangling Bonds in Hexagonal Boron Nitride as Single-Photon Emitters.
Hexagonal boron nitride has been found to host color centers that exhibit single-photon emission, but the microscopic origin of these emitters is unknown. We propose boron dangling bonds as the likely source of the observed single-photon emission around 2 eV. An optical transition where an electron is excited from a doubly occupied boron dangling bond to a localized B p_{z} state gives rise to a zero-phonon line of 2.06 eV and emission with a Huang-Rhys factor of 2.3. This transition is linearly polarized with the absorptive and emissive dipole aligned. Because of the energetic position of the states within the band gap, indirect excitation through the conduction band will occur for sufficiently large excitation energies, leading to the misalignment of the absorptive and emissive dipoles seen in experiment. Our calculations predict a singlet ground state and the existence of a metastable triplet state, in agreement with experiment
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