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A sparse semi-blind source identification method and its application to Raman spectroscopy for explosives detection
Rapid and reliable detection and identification of unknown chemical substances are critical to homeland security. It is challenging to identify chemical components from a wide range of explosives. There are two key steps involved. One is a non-destructive and informative spectroscopic technique for data acquisition. The other is an associated library of reference features along with a computational method for feature matching and meaningful detection within or beyond the library. In this paper, we develop a new iterative method to identify unknown substances from mixture samples of Raman spectroscopy. In the first step, a constrained least squares method decomposes the data into a sum of linear combination of the known components and a non-negative residual. In the second step, a sparse and convex blind source separation method extracts components geometrically from the residuals. Verification based on the library templates or expert knowledge helps to confirm these components. If necessary, the confirmed meaningful components are fed back into step one to refine the residual and then step two extracts possibly more hidden components. The two steps may be iterated until no more components can be identified. We illustrate the proposed method in processing a set of the so called swept wavelength optical resonant Raman spectroscopy experimental data by a satisfactory blind extraction of a priori unknown chemical explosives from mixture samples. We also test the method on nuclear magnetic resonance (NMR) spectra for chemical compounds identification. © 2013 Published by Elsevier B.V
The regularity of harmonic maps into spheres and applications to Bernstein problems
We show the regularity of, and derive a-priori estimates for (weakly)
harmonic maps from a Riemannian manifold into a Euclidean sphere under the
assumption that the image avoids some neighborhood of a half-equator. The
proofs combine constructions of strictly convex functions and the regularity
theory of quasi-linear elliptic systems.
We apply these results to the spherical and Euclidean Bernstein problems for
minimal hypersurfaces, obtaining new conditions under which compact minimal
hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces
are trivial
Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature
We study minimal hypersurfaces in manifolds of non-negative Ricci curvature,
Euclidean volume growth and quadratic curvature decay at infinity. By
comparison with capped spherical cones, we identify a precise borderline for
the Ricci curvature decay. Above this value, no complete area-minimizing
hypersurfaces exist. Below this value, in contrast, we construct examples.Comment: 31 pages. Comments are welcome
The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension
We identify a region \Bbb{W}_{\f{1}{3}} in a Grassmann manifold
\grs{n}{m}, not covered by a usual matrix coordinate chart, with the
following important property. For a complete submanifold in \ir{n+m} \,
(n\ge 3, m\ge2) with parallel mean curvature whose image under the Gauss map
is contained in a compact subset K\subset\Bbb{W}_{\f{1}{3}}\subset\grs{n}{m},
we can construct strongly subharmonic functions and derive a priori estimates
for the harmonic Gauss map. While we do not know yet how close our region is to
being optimal in this respect, it is substantially larger than what could be
achieved previously with other methods. Consequently, this enables us to obtain
substantially stronger Bernstein type theorems in higher codimension than
previously known.Comment: 36 page
The Gauss image of entire graphs of higher codimension and Bernstein type theorems
Under suitable conditions on the range of the Gauss map of a complete
submanifold of Euclidean space with parallel mean curvature, we construct a
strongly subharmonic function and derive a-priori estimates for the harmonic
Gauss map. The required conditions here are more general than in previous work
and they therefore enable us to improve substantially previous results for the
Lawson-Osseman problem concerning the regularity of minimal submanifolds in
higher codimension and to derive Bernstein type results.Comment: 28 page
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