15,179 research outputs found
From "Dirac combs" to Fourier-positivity
Motivated by various problems in physics and applied mathematics, we look for
constraints and properties of real Fourier-positive functions, i.e. with
positive Fourier transforms. Properties of the "Dirac comb" distribution and of
its tensor products in higher dimensions lead to Poisson resummation, allowing
for a useful approximation formula of a Fourier transform in terms of a limited
number of terms. A connection with the Bochner theorem on positive definiteness
of Fourier-positive functions is discussed. As a practical application, we find
simple and rapid analytic algorithms for checking Fourier-positivity in 1- and
(radial) 2-dimensions among a large variety of real positive functions. This
may provide a step towards a classification of positive positive-definite
functions.Comment: 17 pages, 14 eps figures (overall 8 figures in the text
On the positivity of Fourier transforms
Characterizing in a constructive way the set of real functions whose Fourier
transforms are positive appears to be yet an open problem. Some sufficient
conditions are known but they are far from being exhaustive. We propose two
constructive sets of necessary conditions for positivity of the Fourier
transforms and test their ability of constraining the positivity domain. One
uses analytic continuation and Jensen inequalities and the other deals with
Toeplitz determinants and the Bochner theorem. Applications are discussed,
including the extension to the two-dimensional Fourier-Bessel transform and the
problem of positive reciprocity, i.e. positive functions with positive
transforms.Comment: 12 pages, 9 figures (in 4 groups
Are Microwave Induced Zero Resistance States Necessarily Static?
We study the effect of inhomogeneities in Hall conductivity on the nature of
the Zero Resistance States seen in the microwave irradiated two-dimensional
electron systems in weak perpendicular magnetic fields, and we show that
time-dependent domain patterns may emerge in some situations. For an annular
Corbino geometry, with an equilibrium charge density that varies linearly with
radius, we find a time-periodic non-equilibrium solution, which might be
detected by a charge sensor, such as an SET. For a model on a torus, in
addition to static domain patterns seen at high and low values of the
equilibrium charge inhomogeneity, we find that, in the intermediate regime, a
variety of nonstationary states can also exist. We catalog the possibilities we
have seen in our simulations. Within a particular phenomenological model, we
show that linearizing the nonlinear charge continuity equation about a
particularly simple domain wall configuration and analyzing the eigenmodes
allows us to estimate the periods of the solutions to the full nonlinear
equation.Comment: Submitted to PR
Topologically Massive Gauge Theories and their Dual Factorised Gauge Invariant Formulation
There exists a well-known duality between the Maxwell-Chern-Simons theory and
the self-dual massive model in 2+1 dimensions. This dual description has been
extended to topologically massive gauge theories (TMGT) in any dimension. This
Letter introduces an unconventional approach to the construction of this type
of duality through a reparametrisation of the master theory action. The dual
action thereby obtained preserves the same gauge symmetry structure as the
original theory. Furthermore, the dual action is factorised into a propagating
sector of massive gauge invariant variables and a sector with gauge variant
variables defining a pure topological field theory. Combining results obtained
within the Lagrangian and Hamiltonian formulations, a new completed structure
for a gauge invariant dual factorisation of TMGT is thus achieved.Comment: 1+7 pages, no figure
Convergence analysis of the scaled boundary finite element method for the Laplace equation
The scaled boundary finite element method (SBFEM) is a relatively recent
boundary element method that allows the approximation of solutions to PDEs
without the need of a fundamental solution. A theoretical framework for the
convergence analysis of SBFEM is proposed here. This is achieved by defining a
space of semi-discrete functions and constructing an interpolation operator
onto this space. We prove error estimates for this interpolation operator and
show that optimal convergence to the solution can be obtained in SBFEM. These
theoretical results are backed by a numerical example.Comment: 15 pages, 3 figure
Evolutionary dynamics of incubation periods
The incubation period of a disease is the time between an initiating
pathologic event and the onset of symptoms. For typhoid fever, polio, measles,
leukemia and many other diseases, the incubation period is highly variable.
Some affected people take much longer than average to show symptoms, leading to
a distribution of incubation periods that is right skewed and often
approximately lognormal. Although this statistical pattern was discovered more
than sixty years ago, it remains an open question to explain its ubiquity. Here
we propose an explanation based on evolutionary dynamics on graphs. For simple
models of a mutant or pathogen invading a network-structured population of
healthy cells, we show that skewed distributions of incubation periods emerge
for a wide range of assumptions about invader fitness, competition dynamics,
and network structure. The skewness stems from stochastic mechanisms associated
with two classic problems in probability theory: the coupon collector and the
random walk. Unlike previous explanations that rely crucially on heterogeneity,
our results hold even for homogeneous populations. Thus, we predict that two
equally healthy individuals subjected to equal doses of equally pathogenic
agents may, by chance alone, show remarkably different time courses of disease.Comment: 24 pages, 8 figures, 1 tabl
Striped quantum Hall phases
Recent experiments seem to confirm predictions that interactions lead to
charge density wave ground states in higher Landau levels. These new
``correlated'' ground states of the quantum Hall system manifest themselves for
example in a strongly anisotropic resistivity tensor. We give a brief
introduction and overview of this new and emerging field.Comment: 10 pages, 1 figure, updated reference to experimental wor
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