18,336 research outputs found
Harmonic Manifolds and the Volume of Tubes about Curves
H. Hotelling proved that in the n-dimensional Euclidean or spherical space,
the volume of a tube of small radius about a curve depends only on the length
of the curve and the radius. A. Gray and L. Vanhecke extended Hotelling's
theorem to rank one symmetric spaces computing the volumes of the tubes
explicitly in these spaces. In the present paper, we generalize these results
by showing that every harmonic manifold has the above tube property. We compute
the volume of tubes in the Damek-Ricci spaces. We show that if a Riemannian
manifold has the tube property, then it is a 2-stein D'Atri space. We also
prove that a symmetric space has the tube property if and only if it is
harmonic. Our results answer some questions posed by L. Vanhecke, T. J.
Willmore, and G. Thorbergsson.Comment: 17 pages, no figures. This version is different from the journal
versio
Range descriptions for the spherical mean Radon transform
The transform considered in the paper averages a function supported in a ball
in \RR^n over all spheres centered at the boundary of the ball. This Radon
type transform arises in several contemporary applications, e.g. in
thermoacoustic tomography and sonar and radar imaging. Range descriptions for
such transforms are important in all these areas, for instance when dealing
with incomplete data, error correction, and other issues. Four different types
of complete range descriptions are provided, some of which also suggest
inversion procedures. Necessity of three of these (appropriately formulated)
conditions holds also in general domains, while the complete discussion of the
case of general domains would require another publication.Comment: LATEX file, 55 pages, two EPS figure
Generalized Skein Modules of Surfaces
Frobenius extensions play a central role in the link homology theories based
upon the sl(n) link variants, and each of these Frobenius extensions may be
recast geometrically via a category of marked cobordisms in the manner of
Bar-Natan. Here we explore a large family of such marked cobordism categories
that are relevant to generalized sl(n) link homology theories. We also
investigate the skein modules that result from embedding these marked
cobordisms within 3-manifolds, and arrive at an explicit presentation for
several of these generalized skein modules.Comment: 23 pages, multiple figure
Counting generalized Reed-Solomon codes
In this article we count the number of generalized Reed-Solomon (GRS) codes
of dimension k and length n, including the codes coming from a non-degenerate
conic plus nucleus. We compare our results with known formulae for the number
of 3-dimensional MDS codes of length n=6,7,8,9
Orthosymplectically invariant functions in superspace
The notion of spherically symmetric superfunctions as functions invariant
under the orthosymplectic group is introduced. This leads to dimensional
reduction theorems for differentiation and integration in superspace. These
spherically symmetric functions can be used to solve orthosymplectically
invariant Schroedinger equations in superspace, such as the (an)harmonic
oscillator or the Kepler problem. Finally the obtained machinery is used to
prove the Funk-Hecke theorem and Bochner's relations in superspace.Comment: J. Math. Phy
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