43,257 research outputs found
Geometric approach to error correcting codes and reconstruction of signals
We develop an approach through geometric functional analysis to error
correcting codes and to reconstruction of signals from few linear measurements.
An error correcting code encodes an n-letter word x into an m-letter word y in
such a way that x can be decoded correctly when any r letters of y are
corrupted. We prove that most linear orthogonal transformations Q from R^n into
R^m form efficient and robust robust error correcting codes over reals. The
decoder (which corrects the corrupted components of y) is the metric projection
onto the range of Q in the L_1 norm. An equivalent problem arises in signal
processing: how to reconstruct a signal that belongs to a small class from few
linear measurements? We prove that for most sets of Gaussian measurements, all
signals of small support can be exactly reconstructed by the L_1 norm
minimization. This is a substantial improvement of recent results of Donoho and
of Candes and Tao. An equivalent problem in combinatorial geometry is the
existence of a polytope with fixed number of facets and maximal number of
lower-dimensional facets. We prove that most sections of the cube form such
polytopes.Comment: 17 pages, 3 figure
Slepian functions and their use in signal estimation and spectral analysis
It is a well-known fact that mathematical functions that are timelimited (or
spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the
finite precision of measurement and computation unavoidably bandlimits our
observation and modeling scientific data, and we often only have access to, or
are only interested in, a study area that is temporally or spatially bounded.
In the geosciences we may be interested in spectrally modeling a time series
defined only on a certain interval, or we may want to characterize a specific
geographical area observed using an effectively bandlimited measurement device.
It is clear that analyzing and representing scientific data of this kind will
be facilitated if a basis of functions can be found that are "spatiospectrally"
concentrated, i.e. "localized" in both domains at the same time. Here, we give
a theoretical overview of one particular approach to this "concentration"
problem, as originally proposed for time series by Slepian and coworkers, in
the 1960s. We show how this framework leads to practical algorithms and
statistically performant methods for the analysis of signals and their power
spectra in one and two dimensions, and on the surface of a sphere.Comment: Submitted to the Handbook of Geomathematics, edited by Willi Freeden,
Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verla
Scalar and vector Slepian functions, spherical signal estimation and spectral analysis
It is a well-known fact that mathematical functions that are timelimited (or
spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the
finite precision of measurement and computation unavoidably bandlimits our
observation and modeling scientific data, and we often only have access to, or
are only interested in, a study area that is temporally or spatially bounded.
In the geosciences we may be interested in spectrally modeling a time series
defined only on a certain interval, or we may want to characterize a specific
geographical area observed using an effectively bandlimited measurement device.
It is clear that analyzing and representing scientific data of this kind will
be facilitated if a basis of functions can be found that are "spatiospectrally"
concentrated, i.e. "localized" in both domains at the same time. Here, we give
a theoretical overview of one particular approach to this "concentration"
problem, as originally proposed for time series by Slepian and coworkers, in
the 1960s. We show how this framework leads to practical algorithms and
statistically performant methods for the analysis of signals and their power
spectra in one and two dimensions, and, particularly for applications in the
geosciences, for scalar and vectorial signals defined on the surface of a unit
sphere.Comment: Submitted to the 2nd Edition of the Handbook of Geomathematics,
edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be
published by Springer Verlag. This is a slightly modified but expanded
version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the
Handbook, when it was called: Slepian functions and their use in signal
estimation and spectral analysi
Self-similar shear-thickening behavior in CTAB/NaSal surfactant solutions
The effect of salt concentration Cs on the critical shear rate required for
the onset of shear thickening and apparent relaxation time of the
shear-thickened phase, has been investigated systematically for dilute
CTAB/NaSal solutions. Experimental data suggest a self-similar behavior of the
critical shear rate and relaxation time as functions of Cs. Specifically, the
former ~ Cs^(-6) whereas the latter ~ Cs^(6) such that an effective Weissenberg
number for the onset of the shear thickened phase is only weakly dependent on
Cs. A procedure has been developed to collapse the apparent shear viscosity
versus shear rate data obtained for various values of Cs into a single master
curve. The effect of Cs on the elastic modulus and mesh size of the
shear-induced gel phase for different surfactant concentrations is discussed.
Experiments performed using different flow cells (Couette and cone-and-plate)
show that the critical shear rate, relaxation time and the maximum viscosity
attained are geometry-independent. The elastic modulus of the gel phase
inferred indirectly by employing simplified hydrodynamic instability analysis
of a sheared gel-fluid interface is in qualitative agreement with that
predicted for an entangled phase of living polymers. A qualitative mechanism
that combines the effect of Cs on average micelle length and Debye parameter
with shear-induced configurational changes of rod-like micelles is proposed to
rationalize the self-similarity of SIS formation.Comment: 27 pages, 17 figure
Gas cells for tunable diode laser absorption spectroscopy employing optical diffusers. Part 1: single and dual pass cells
New designs for gas cells are presented that incorporate transmissive or
reflective optical diffusers. These components offer simple alignment and can
disrupt the formation of optical etalons. We analyse the performance-limiting
effects in these cells of random laser speckle (both objective and subjective
speckle), interferometric speckle and self-mixing interference, and show how
designs can be optimised. A simple, single pass transmissive gas cell has been
studied using wavelength modulation spectroscopy to measure methane at 1651 nm.
We have demonstrated a short-term noise equivalent absorbance (NEA, 1 sigma) of
2x10(-5), but longer term drift of up to 3x10(-4) over 22 hours
Performance Analysis of Sparse Recovery Based on Constrained Minimal Singular Values
The stability of sparse signal reconstruction is investigated in this paper.
We design efficient algorithms to verify the sufficient condition for unique
sparse recovery. One of our algorithm produces comparable results with
the state-of-the-art technique and performs orders of magnitude faster. We show
that the -constrained minimal singular value (-CMSV) of the
measurement matrix determines, in a very concise manner, the recovery
performance of -based algorithms such as the Basis Pursuit, the Dantzig
selector, and the LASSO estimator. Compared with performance analysis involving
the Restricted Isometry Constant, the arguments in this paper are much less
complicated and provide more intuition on the stability of sparse signal
recovery. We show also that, with high probability, the subgaussian ensemble
generates measurement matrices with -CMSVs bounded away from zero, as
long as the number of measurements is relatively large. To compute the
-CMSV and its lower bound, we design two algorithms based on the
interior point algorithm and the semi-definite relaxation
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