35 research outputs found
Two-divisibility of the coefficients of certain weakly holomorphic modular forms
We study a canonical basis for spaces of weakly holomorphic modular forms of
weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a
relation between the Fourier coefficients of modular forms in this canonical
basis and a generalized Ramanujan tau-function, and use this to prove that
these Fourier coefficients are often highly divisible by 2.Comment: Corrected typos. To appear in the Ramanujan Journa
Self-trapping transition for nonlinear impurities embedded in a Cayley tree
The self-trapping transition due to a single and a dimer nonlinear impurity
embedded in a Cayley tree is studied. In particular, the effect of a perfectly
nonlinear Cayley tree is considered. A sharp self-trapping transition is
observed in each case. It is also observed that the transition is much sharper
compared to the case of one-dimensional lattices. For each system, the critical
values of for the self-trapping transitions are found to obey a
power-law behavior as a function of the connectivity of the Cayley tree.Comment: 6 pages, 7 fig
Harmonic Sums and Mellin Transforms up to two-loop Order
A systematic study is performed on the finite harmonic sums up to level four.
These sums form the general basis for the Mellin transforms of all individual
functions of the momentum fraction emerging in the quantities of
massless QED and QCD up to two--loop order, as the unpolarized and polarized
splitting functions, coefficient functions, and hard scattering cross sections
for space and time-like momentum transfer. The finite harmonic sums are
calculated explicitly in the linear representation. Algebraic relations
connecting these sums are derived to obtain representations based on a reduced
set of basic functions. The Mellin transforms of all the corresponding Nielsen
functions are calculated.Comment: 44 pages Latex, contract number adde
Green function techniques in the treatment of quantum transport at the molecular scale
The theoretical investigation of charge (and spin) transport at nanometer
length scales requires the use of advanced and powerful techniques able to deal
with the dynamical properties of the relevant physical systems, to explicitly
include out-of-equilibrium situations typical for electrical/heat transport as
well as to take into account interaction effects in a systematic way.
Equilibrium Green function techniques and their extension to non-equilibrium
situations via the Keldysh formalism build one of the pillars of current
state-of-the-art approaches to quantum transport which have been implemented in
both model Hamiltonian formulations and first-principle methodologies. We offer
a tutorial overview of the applications of Green functions to deal with some
fundamental aspects of charge transport at the nanoscale, mainly focusing on
applications to model Hamiltonian formulations.Comment: Tutorial review, LaTeX, 129 pages, 41 figures, 300 references,
submitted to Springer series "Lecture Notes in Physics