75,268 research outputs found
Viscous instability of a compressible round jet
The compressible linear stability equations are derived from the Navier Stokes equations in cylindrical polar coordinates. Numerical solutions for locally parallel flow are found using a direct matrix method. Discretization with compact finite difference is found to have better convergence properties than a Chebyshev spectral method for a round jet test case. The method is validated against previous results and convergence is tested for a range of jet profiles. Finally and en method is used to determine the dominant frequency of a Mach 0.9 jet
Calculation of Mutual Information for Partially Coherent Gaussian Channels with Applications to Fiber Optics
The mutual information between a complex-valued channel input and its
complex-valued output is decomposed into four parts based on polar coordinates:
an amplitude term, a phase term, and two mixed terms. Numerical results for the
additive white Gaussian noise (AWGN) channel with various inputs show that, at
high signal-to-noise ratio (SNR), the amplitude and phase terms dominate the
mixed terms. For the AWGN channel with a Gaussian input, analytical expressions
are derived for high SNR. The decomposition method is applied to partially
coherent channels and a property of such channels called "spectral loss" is
developed. Spectral loss occurs in nonlinear fiber-optic channels and it may be
one effect that needs to be taken into account to explain the behavior of the
capacity of nonlinear fiber-optic channels presented in recent studies.Comment: 30 pages, 9 figures, accepted for publication in IEEE Transactions on
Information Theor
Steady advection-diffusion around finite absorbers in two-dimensional potential flows
We perform an exhaustive study of the simplest, nontrivial problem in
advection-diffusion -- a finite absorber of arbitrary cross section in a steady
two-dimensional potential flow of concentrated fluid. This classical problem
has been studied extensively in the theory of solidification from a flowing
melt, and it also arises in Advection-Diffusion-Limited Aggregation. In both
cases, the fundamental object is the flux to a circular disk, obtained by
conformal mapping from more complicated shapes. We construct the first accurate
numerical solution using an efficient new method, which involves mapping to the
interior of the disk and using a spectral method in polar coordinates. Our
method also combines exact asymptotics and an adaptive mesh to handle boundary
layers. Starting from a well-known integral equation in streamline coordinates,
we also derive new, high-order asymptotic expansions for high and low P\'eclet
numbers (\Pe). Remarkably, the `high' \Pe expansion remains accurate even
for such low \Pe as . The two expansions overlap well near \Pe =
0.1, allowing the construction of an analytical connection formula that is
uniformly accurate for all \Pe and angles on the disk with a maximum relative
error of 1.75%. We also obtain an analytical formula for the Nusselt number
() as a function of the P\'eclet number with a maximum relative error of
0.53% for all possible geometries. Because our finite-plate problem can be
conformally mapped to other geometries, the general problem of two-dimensional
advection-diffusion past an arbitrary finite absorber in a potential flow can
be considered effectively solved.Comment: 29 pages, 12 figs (mostly in color
A method to localize gamma-ray bursts using POLAR
The hard X-ray polarimeter POLAR aims to measure the linear polarization of
the 50-500 keV photons arriving from the prompt emission of gamma-ray bursts
(GRBs). The position in the sky of the detected GRBs is needed to determine
their level of polarization. We present here a method by which, despite of the
polarimeter incapability of taking images, GRBs can be roughly localized using
POLAR alone. For this purpose scalers are attached to the output of the 25
multi-anode photomultipliers (MAPMs) that collect the light from the POLAR
scintillator target. Each scaler measures how many GRB photons produce at least
one energy deposition above 50 keV in the corresponding MAPM. Simulations show
that the relative outputs of the 25 scalers depend on the GRB position. A
database of very strong GRBs simulated at 10201 positions has been produced.
When a GRB is detected, its location is calculated searching the minimum of the
chi2 obtained in the comparison between the measured scaler pattern and the
database. This GRB localization technique brings enough accuracy so that the
error transmitted to the 100% modulation factor is kept below 10% for GRBs with
fluence Ftot \geq 10^(-5) erg cm^(-2) . The POLAR localization capability will
be useful for those cases where no other instruments are simultaneously
observing the same field of view.Comment: 13 pages, 10 figure
SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels
We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant
of deep neural networks for irregular structured and geometric input, e.g.,
graphs or meshes. Our main contribution is a novel convolution operator based
on B-splines, that makes the computation time independent from the kernel size
due to the local support property of the B-spline basis functions. As a result,
we obtain a generalization of the traditional CNN convolution operator by using
continuous kernel functions parametrized by a fixed number of trainable
weights. In contrast to related approaches that filter in the spectral domain,
the proposed method aggregates features purely in the spatial domain. In
addition, SplineCNN allows entire end-to-end training of deep architectures,
using only the geometric structure as input, instead of handcrafted feature
descriptors. For validation, we apply our method on tasks from the fields of
image graph classification, shape correspondence and graph node classification,
and show that it outperforms or pars state-of-the-art approaches while being
significantly faster and having favorable properties like domain-independence.Comment: Presented at CVPR 201
Spectral solver for Cauchy problems in polar coordinates using discrete Hankel transforms
We introduce a Fourier-Bessel-based spectral solver for Cauchy problems
featuring Laplacians in polar coordinates under homogeneous Dirichlet boundary
conditions. We use FFTs in the azimuthal direction to isolate angular modes,
then perform discrete Hankel transform (DHT) on each mode along the radial
direction to obtain spectral coefficients. The two transforms are connected via
numerical and cardinal interpolations. We analyze the boundary-dependent error
bound of DHT; the worst case is , which governs the method, and
the best , which then the numerical interpolation governs. The
complexity is . Taking advantage of Bessel functions being the
eigenfunctions of the Laplacian operator, we solve linear equations for all
times. For non-linear equations, we use a time-splitting method to integrate
the solutions. We show examples and validate the method on the two-dimensional
wave equation, which is linear, and on two non-linear problems: a
time-dependent Poiseuille flow and the flow of a Bose-Einstein condensate on a
disk
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