371 research outputs found
Local 4/5-Law and Energy Dissipation Anomaly in Turbulence
A strong local form of the ``4/3-law'' in turbulent flow has been proved
recently by Duchon and Robert for a triple moment of velocity increments
averaged over both a bounded spacetime region and separation vector directions,
and for energy dissipation averaged over the same spacetime region. Under
precisely stated hypotheses, the two are proved to be proportional, by a
constant 4/3, and to appear as a nonnegative defect measure in the local energy
balance of singular (distributional) solutions of the incompressible Euler
equations. Here we prove that the energy defect measure can be represented also
by a triple moment of purely longitudinal velocity increments and by a mixed
moment with one longitudinal and two tranverse velocity increments. Thus, we
prove that the traditional 4/5- and 4/15-laws of Kolmogorov hold in the same
local sense as demonstrated for the 4/3-law by Duchon-Robert.Comment: 14 page
Precision Pointing of IBEX-Lo Observations
Post-launch boresight of the IBEX-Lo instrument onboard the Interstellar
Boundary Explorer (IBEX) is determined based on IBEX-Lo Star Sensor
observations. Accurate information on the boresight of the neutral gas camera
is essential for precise determination of interstellar gas flow parameters.
Utilizing spin-phase information from the spacecraft attitude control system
(ACS), positions of stars observed by the Star Sensor during two years of IBEX
measurements were analyzed and compared with positions obtained from a star
catalog. No statistically significant differences were observed beyond those
expected from the pre-launch uncertainty in the Star Sensor mounting. Based on
the star observations and their positions in the spacecraft reference system,
pointing of the IBEX satellite spin axis was determined and compared with the
pointing obtained from the ACS. Again, no statistically significant deviations
were observed. We conclude that no systematic correction for boresight geometry
is needed in the analysis of IBEX-Lo observations to determine neutral
interstellar gas flow properties. A stack-up of uncertainties in attitude
knowledge shows that the instantaneous IBEX-Lo pointing is determined to within
\sim 0.1\degr in both spin angle and elevation using either the Star Sensor
or the ACS. Further, the Star Sensor can be used to independently determine the
spacecraft spin axis. Thus, Star Sensor data can be used reliably to correct
the spin phase when the Star Tracker (used by the ACS) is disabled by bright
objects in its field-of-view. The Star Sensor can also determine the spin axis
during most orbits and thus provides redundancy for the Star Tracker.Comment: 22 pages, 18 figure
Simple Systems with Anomalous Dissipation and Energy Cascade
We analyze a class of linear shell models subject to stochastic forcing in
finitely many degrees of freedom. The unforced systems considered formally
conserve energy. Despite being formally conservative, we show that these
dynamical systems support dissipative solutions (suitably defined) and, as a
result, may admit unique (statistical) steady states when the forcing term is
nonzero. This claim is demonstrated via the complete characterization of the
solutions of the system above for specific choices of the coupling
coefficients. The mechanism of anomalous dissipations is shown to arise via a
cascade of the energy towards the modes () with higher ; this is
responsible for solutions with interesting energy spectra, namely \EE |a_n|^2
scales as as . Here the exponents depend on
the coupling coefficients and \EE denotes expectation with respect to
the equilibrium measure. This is reminiscent of the conjectured properties of
the solutions of the Navier-Stokes equations in the inviscid limit and their
accepted relationship with fully developed turbulence. Hence, these simple
models illustrate some of the heuristic ideas that have been advanced to
characterize turbulence, similar in that respect to the random passive scalar
or random Burgers equation, but even simpler and fully solvable.Comment: 32 Page
Uniform generation in trace monoids
We consider the problem of random uniform generation of traces (the elements
of a free partially commutative monoid) in light of the uniform measure on the
boundary at infinity of the associated monoid. We obtain a product
decomposition of the uniform measure at infinity if the trace monoid has
several irreducible components-a case where other notions such as Parry
measures, are not defined. Random generation algorithms are then examined.Comment: Full version of the paper in MFCS 2015 with the same titl
Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise
We prove that every Markov solution to the three dimensional Navier-Stokes
equation with periodic boundary conditions driven by additive Gaussian noise is
uniquely ergodic. The convergence to the (unique) invariant measure is
exponentially fast.
Moreover, we give a well-posedness criterion for the equations in terms of
invariant measures. We also analyse the energy balance and identify the term
which ensures equality in the balance.Comment: 32 page
On thin plate spline interpolation
We present a simple, PDE-based proof of the result [M. Johnson, 2001] that
the error estimates of [J. Duchon, 1978] for thin plate spline interpolation
can be improved by . We illustrate that -matrix
techniques can successfully be employed to solve very large thin plate spline
interpolation problem
Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators
In this paper we introduce a generalized Sobolev space by defining a
semi-inner product formulated in terms of a vector distributional operator
consisting of finitely or countably many distributional operators
, which are defined on the dual space of the Schwartz space. The types of
operators we consider include not only differential operators, but also more
general distributional operators such as pseudo-differential operators. We
deduce that a certain appropriate full-space Green function with respect to
now becomes a conditionally positive
definite function. In order to support this claim we ensure that the
distributional adjoint operator of is
well-defined in the distributional sense. Under sufficient conditions, the
native space (reproducing-kernel Hilbert space) associated with the Green
function can be isometrically embedded into or even be isometrically
equivalent to a generalized Sobolev space. As an application, we take linear
combinations of translates of the Green function with possibly added polynomial
terms and construct a multivariate minimum-norm interpolant to data
values sampled from an unknown generalized Sobolev function at data sites
located in some set . We provide several examples, such
as Mat\'ern kernels or Gaussian kernels, that illustrate how many
reproducing-kernel Hilbert spaces of well-known reproducing kernels are
isometrically equivalent to a generalized Sobolev space. These examples further
illustrate how we can rescale the Sobolev spaces by the vector distributional
operator . Introducing the notion of scale as part of the
definition of a generalized Sobolev space may help us to choose the "best"
kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D.
thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}
Correction of Electronic Record for Weighing Bucket Precipitation Gauge Measurements
Electronic sensors generate valuable streams of forcing and validation data for hydrologic models but are often subject to noise which must be removed as part of model input and testing database development. We developed an automated precipitation correction program (APCP) for weighing bucket precipitation gauge records, which are subject to several types of mechanical and electronic noise and discontinuities, including gauge maintenance, missing data, wind vibration, and sensor drift. Corrected cumulative water year precipitation from APCP did not exhibit an error bias and matched measured water year total precipitation within 2.1% for 58 station years tested. Removal of low-amplitude periodic noise was especially important for developing accurate instantaneous precipitation records at subdaily time steps. Model flexibility for use with other data types is demonstrated through application to time domain reflectometry soil moisture content data, which are also frequently subject to substantial noise
Area distribution of the planar random loop boundary
We numerically investigate the area statistics of the outer boundary of
planar random loops, on the square and triangular lattices. Our Monte Carlo
simulations suggest that the underlying limit distribution is the Airy
distribution, which was recently found to appear also as area distribution in
the model of self-avoiding loops.Comment: 10 pages, 2 figures. v2: minor changes, version as publishe
Macroscopic models for superconductivity
This paper reviews the derivation of some macroscopic models for superconductivity and also some of the mathematical challenges posed by these models. The paper begins by exploring certain analogies between phase changes in superconductors and those in solidification and melting. However, it is soon found that there are severe limitations on the range of validity of these analogies and outside this range many interesting open questions can be posed about the solutions to the macroscopic models
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