118 research outputs found
A uniform reconstruction formula in integral geometry
A general method for analytic inversion in integral geometry is proposed. All
classical and some new reconstruction formulas of Radon-John type are obtained
by this method. No harmonic analysis and PDE is used
Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm
Practical applications of thermoacoustic tomography require numerical
inversion of the spherical mean Radon transform with the centers of integration
spheres occupying an open surface. Solution of this problem is needed (both in
2-D and 3-D) because frequently the region of interest cannot be completely
surrounded by the detectors, as it happens, for example, in breast imaging. We
present an efficient numerical algorithm for solving this problem in 2-D
(similar methods are applicable in the 3-D case). Our method is based on the
numerical approximation of plane waves by certain single layer potentials
related to the acquisition geometry. After the densities of these potentials
have been precomputed, each subsequent image reconstruction has the complexity
of the regular filtration backprojection algorithm for the classical Radon
transform. The peformance of the method is demonstrated in several numerical
examples: one can see that the algorithm produces very accurate reconstructions
if the data are accurate and sufficiently well sampled, on the other hand, it
is sufficiently stable with respect to noise in the data
Rarita-Schwinger Type Operators on Spheres and Real Projective Space
In this paper we deal with Rarita-Schwinger type operators on spheres and
real projective space. First we define the spherical Rarita-Schwinger type
operators and construct their fundamental solutions. Then we establish that the
projection operators appearing in the spherical Rarita-Schwinger type operators
and the spherical Rarita-Schwinger type equations are conformally invariant
under the Cayley transformation. Further, we obtain some basic integral
formulas related to the spherical Rarita-Schwinger type operators. Second, we
define the Rarita-Schwinger type operators on the real projective space and
construct their kernels and Cauchy integral formulas.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1106.358
The photometric properties of a vast stellar substructure in the outskirts of M33
We have surveyed sq.degrees surrounding M33 with CFHT MegaCam in the
g and i filters, as part of the Pan-Andromeda Archaeological Survey. Our
observations are deep enough to resolve the top 4mags of the red giant branch
population in this galaxy. We have previously shown that the disk of M33 is
surrounded by a large, irregular, low-surface brightness substructure. Here, we
quantify the stellar populations and structure of this feature using the PAndAS
data. We show that the stellar populations of this feature are consistent with
an old population with dex and an interquartile range in
metallicity of dex. We construct a surface brightness map of M33 that
traces this feature to mags\,arcsec. At these low surface
brightness levels, the structure extends to projected radii of kpc from
the center of M33 in both the north-west and south-east quadrants of the
galaxy. Overall, the structure has an "S-shaped" appearance that broadly aligns
with the orientation of the HI disk warp. We calculate a lower limit to the
integrated luminosity of the structure of mags, comparable to a
bright dwarf galaxy such as Fornax or AndII and slightly less than $1\$ of the
total luminosity of M33. Further, we show that there is tentative evidence for
a distortion in the distribution of young stars near the edge of the HI disk
that occurs at similar azimuth to the warp in HI. The data also hint at a
low-level, extended stellar component at larger radius that may be a M33 halo
component. We revisit studies of M33 and its stellar populations in light of
these new results, and we discuss possible formation scenarios for the vast
stellar structure. Our favored model is that of the tidal disruption of M33 in
its orbit around M31.Comment: Accepted for publication in ApJ. 17 figures. ApJ preprint forma
Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations
We present a method to solve initial-boundary value problems for linear and
integrable nonlinear differential-difference evolution equations. The method is
the discrete version of the one developed by A. S. Fokas to solve
initial-boundary value problems for linear and integrable nonlinear partial
differential equations via an extension of the inverse scattering transform.
The method takes advantage of the Lax pair formulation for both linear and
nonlinear equations, and is based on the simultaneous spectral analysis of both
parts of the Lax pair. A key role is also played by the global algebraic
relation that couples all known and unknown boundary values. Even though
additional technical complications arise in discrete problems compared to
continuum ones, we show that a similar approach can also solve initial-boundary
value problems for linear and integrable nonlinear differential-difference
equations. We demonstrate the method by solving initial-boundary value problems
for the discrete analogue of both the linear and the nonlinear Schrodinger
equations, comparing the solution to those of the corresponding continuum
problems. In the linear case we also explicitly discuss Robin-type boundary
conditions not solvable by Fourier series. In the nonlinear case we also
identify the linearizable boundary conditions, we discuss the elimination of
the unknown boundary datum, we obtain explicitly the linear and continuum limit
of the solution, and we write down the soliton solutions.Comment: 41 pages, 3 figures, to appear in Inverse Problem
Two classes of generalized functions used in nonlocal field theory
We elucidate the relation between the two ways of formulating causality in
nonlocal quantum field theory: using analytic test functions belonging to the
space (which is the Fourier transform of the Schwartz space )
and using test functions in the Gelfand-Shilov spaces . We prove
that every functional defined on has the same carrier cones as its
restrictions to the smaller spaces . As an application of this
result, we derive a Paley-Wiener-Schwartz-type theorem for arbitrarily singular
generalized functions of tempered growth and obtain the corresponding extension
of Vladimirov's algebra of functions holomorphic on a tubular domain.Comment: AMS-LaTeX, 12 pages, no figure
2.11. Necessary conditions for interpolation by entire functions
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41580/1/10958_2005_Article_BF01221572.pd
Twisted convolution and Moyal star product of generalized functions
We consider nuclear function spaces on which the Weyl-Heisenberg group acts
continuously and study the basic properties of the twisted convolution product
of the functions with the dual space elements. The final theorem characterizes
the corresponding algebra of convolution multipliers and shows that it contains
all sufficiently rapidly decreasing functionals in the dual space.
Consequently, we obtain a general description of the Moyal multiplier algebra
of the Fourier-transformed space. The results extend the Weyl symbol calculus
beyond the traditional framework of tempered distributions.Comment: LaTeX, 16 pages, no figure
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