7,239 research outputs found
Convergence rates for loop-erased random walk and other Loewner curves
We estimate convergence rates for curves generated by Loewner's differential
equation under the basic assumption that a convergence rate for the driving
terms is known. An important tool is what we call the tip structure modulus, a
geometric measure of regularity for Loewner curves parameterized by capacity.
It is analogous to Warschawski's boundary structure modulus and closely related
to annuli crossings. The main application we have in mind is that of a random
discrete-model curve approaching a Schramm-Loewner evolution (SLE) curve in the
lattice size scaling limit. We carry out the approach in the case of
loop-erased random walk (LERW) in a simply connected domain. Under mild
assumptions of boundary regularity, we obtain an explicit power-law rate for
the convergence of the LERW path toward the radial SLE path in the supremum
norm, the curves being parameterized by capacity. On the deterministic side, we
show that the tip structure modulus gives a sufficient geometric condition for
a Loewner curve to be H\"{o}lder continuous in the capacity parameterization,
assuming its driving term is H\"{o}lder continuous. We also briefly discuss the
case when the curves are a priori known to be H\"{o}lder continuous in the
capacity parameterization and we obtain a power-law convergence rate depending
only on the regularity of the curves.Comment: Published in at http://dx.doi.org/10.1214/13-AOP872 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Correlation Functions of Hadron Currents in the QCD Vacuum Calculated in Lattice QCD
Point-to-point vacuum correlation functions for spatially separated hadron
currents are calculated in quenched lattice QCD on a lattice
with . The lattice data are analyzed in terms of dispersion
relations, which enable us to extract physical information from small distances
where asymptotic freedom is apparent to large distances where the hadronic
resonances dominate. In the pseudoscalar, vector, and axial vector channels
where experimental data or phenomenological information are available,
semi-quantitative agreement is obtained. In the nucleon and delta channels,
where no experimental data exist, our lattice data complement experiments.
Comparison with approximations based on sum rules and interacting instantons
are made, and technical details of the lattice calculation are described.Comment: 31 pages in REVTeX (with 10 figures to be added using figures
command), MIT CTP #214
Analytic solutions to the maximum and average exoplanet transit depth for common stellar limb darkening laws
The depth of an exoplanetary transit in the light curve of a distant star is
commonly approximated as the squared planet-to-star radius ratio, (R_p/R_s)^2.
Stellar limb darkening, however, results in significantly deeper transits. Here
we derive analytical solutions to the overshoot of the mid-transit depth caused
by stellar limb darkening compared to the (R_p/R_s)^2 estimate for arbitrary
transit impact parameters. In turn, this allows us to compute the true
planet-to-star radius ratio from the transit depth for a given parameterization
of a limb darkening law and for a known transit impact parameter. We calculate
the maximum emerging specific stellar intensity covered by the planet in
transit and derive analytic solutions for the transit depth overshoot.
Solutions are presented for the linear, quadratic, square-root, logarithmic,
and non-linear stellar limb darkening with arbitrary transit impact parameters.
We also derive formulae to calculate the average intensity along the transit
chord, which allows us to estimate the actual transit depth (and therefore
R_p/R_s) from the mean in-transit flux. The transit depth overshoot of
exoplanets compared to the (R_p/R_s)^2 estimate increases from about 15% for A
main-sequence stars to roughly 20% for sun-like stars and some 30% for K and M
stars. The error in our analytical solutions for R_p/R_s from the small planet
approximation is orders of magnitude smaller than the uncertainties arising
from typical noise in real light curves and from the uncertain limb darkening.
Our equations can be used to predict with high accuracy the expected transit
depth of extrasolar planets. The actual planet radius can be calculated from
the measured transit depth or from the mean in-transit flux if the stellar limb
darkening can be properly parameterized and if the transit impact parameter is
known. Light curve fitting is not required.Comment: 7 pages, 3 figures (2 col, 1 b/w), published in A&
Computing stationary free-surface shapes in microfluidics
A finite-element algorithm for computing free-surface flows driven by
arbitrary body forces is presented. The algorithm is primarily designed for the
microfluidic parameter range where (i) the Reynolds number is small and (ii)
force-driven pressure and flow fields compete with the surface tension for the
shape of a stationary free surface. The free surface shape is represented by
the boundaries of finite elements that move according to the stress applied by
the adjacent fluid. Additionally, the surface tends to minimize its free energy
and by that adapts its curvature to balance the normal stress at the surface.
The numerical approach consists of the iteration of two alternating steps: The
solution of a fluidic problem in a prescribed domain with slip boundary
conditions at the free surface and a consecutive update of the domain driven by
the previously determined pressure and velocity fields. ...Comment: Revised versio
Ising parameterization of QCD Landau free energy and its dynamics
We present a general linear parameterization scheme for the QCD Landau free
energy in the vicinity of the critical point of chiral phase transition in the
- plane. Based on the parametric free energy, we show that, due to the
finite size effects, the regions of fluctuations of the order parameter (i.e.
the field) are broadened, and the discontinuities of the first order
phase transition are smoothed. Meanwhile, the kurtosis of the field is
universally negative around the critical point. Using the Fokker-Plank
equation, we derive the dynamical corrections to the free energy. The dynamical
cumulants of the field on the freeze-out line, record earlier
information in the first order phase transition region as compared to the
crossover region. The typical behavior of dynamical cumulants can be understood
from the equilibrium cumulants by considering the memory effects.Comment: 8 pages, 5 figures. Fig.1 is replaced. More discussions are adde
A geometric approach to phase response curves and its numerical computation through the parameterization method
The final publication is available at link.springer.comThe phase response curve (PRC) is a tool used in neuroscience that measures the phase shift experienced by an oscillator due to a perturbation applied at different phases of the limit cycle. In this paper, we present a new approach to PRCs based on the parameterization method. The underlying idea relies on the construction of a periodic system whose corresponding stroboscopic map has an invariant curve. We demonstrate the relationship between the internal dynamics of this invariant curve and the PRC, which yields a method to numerically compute the PRCs. Moreover, we link the existence properties of this invariant curve as the amplitude of the perturbation is increased with changes in the PRC waveform and with the geometry of isochrons. The invariant curve and its dynamics will be computed by means of the parameterization method consisting of solving an invariance equation. We show that the method to compute the PRC can be extended beyond the breakdown of the curve by means of introducing a modified invariance equation. The method also computes the amplitude response functions (ARCs) which provide information on the displacement away from the oscillator due to the effects of the perturbation. Finally, we apply the method to several classical models in neuroscience to illustrate how the results herein extend the framework of computation and interpretation of the PRC and ARC for perturbations of large amplitude and not necessarily pulsatile.Peer ReviewedPostprint (author's final draft
Approximation by planar elastic curves
We give an algorithm for approximating a given plane curve segment by a
planar elastic curve. The method depends on an analytic representation of the
space of elastic curve segments, together with a geometric method for obtaining
a good initial guess for the approximating curve. A gradient-driven
optimization is then used to find the approximating elastic curve.Comment: 18 pages, 10 figures. Version2: new section 5 added (conclusions and
discussions
- âŠ