5,692 research outputs found
Obervational Model for Microarcsecond Astrometry with the Space Interferometry Mission
The Space Interferometry Mission (SIM) is a space-based long-baseline optical
interferometer for precision astrometry. One of the primary objectives of the
SIM instrument is to accurately determine the directions to a grid of stars,
together with their proper motions and parallaxes, improving a priori knowledge
by nearly three orders of magnitude. The basic astrometric observable of the
instrument is the pathlength delay, a measurement made by a combination of
internal metrology measurements that determine the distance the starlight
travels through the two arms of the interferometer and a measurement of the
white light stellar fringe to find the point of equal pathlength. Because this
operation requires a non--negligible integration time to accurately measure the
stellar fringe position, the interferometer baseline vector is not stationary
over this time period, as its absolute length and orientation are
time--varying. This conflicts with the consistency condition necessary for
extracting the astrometric parameters which requires a stationary baseline
vector. This paper addresses how the time-varying baseline is ``regularized''
so that it may act as a single baseline vector for multiple stars, and thereby
establishing the fundamental operation of the instrument.Comment: 24 pages, 6 figure
Dual Mixed Volumes and the Slicing Problem
We develop a technique using dual mixed-volumes to study the isotropic
constants of some classes of spaces. In particular, we recover, strengthen and
generalize results of Ball and Junge concerning the isotropic constants of
subspaces and quotients of L_p and related spaces. An extension of these
results to negative values of p is also obtained, using generalized
intersection-bodies. In particular, we show that the isotropic constant of a
convex body which is contained in an intersection-body is bounded (up to a
constant) by the ratio between the latter's mean-radius and the former's
volume-radius. We also show how type or cotype 2 may be used to easily prove
inequalities on any isotropic measure.Comment: 38 pages, to appear in Advances in Mathematics. Corrected Remark 4.
Generalized Intersection Bodies
We study the structures of two types of generalizations of
intersection-bodies and the problem of whether they are in fact equivalent.
Intersection-bodies were introduced by Lutwak and played a key role in the
solution of the Busemann-Petty problem. A natural geometric generalization of
this problem considered by Zhang, led him to introduce one type of generalized
intersection-bodies. A second type was introduced by Koldobsky, who studied a
different analytic generalization of this problem. Koldobsky also studied the
connection between these two types of bodies, and noted that an equivalence
between these two notions would completely settle the unresolved cases in the
generalized Busemann-Petty problem. We show that these classes share many
identical structure properties, proving the same results using Integral
Geometry techniques for Zhang's class and Fourier transform techniques for
Koldobsky's class. Using a Functional Analytic approach, we give several
surprising equivalent formulations for the equivalence problem, which reveal a
deep connection to several fundamental problems in the Integral Geometry of the
Grassmann Manifold.Comment: 45 pages, to appear in Journal of Functional Analysis Revised version
after referee's comment
Generalized Intersection Bodies are not Equivalent
In 2000, A. Koldobsky asked whether two types of generalizations of the
notion of an intersection-body, are in fact equivalent. The structures of these
two types of generalized intersection-bodies have been studied by the author in
[http://www.arxiv.org/math.MG/0512058], providing substantial positive evidence
for a positive answer to this question. The purpose of this note is to
construct a counter-example, which provides a surprising negative answer to
this question in a strong sense. This implies the existence of non-trivial
non-negative functions in the range of the spherical Radon transform, and the
existence of non-trivial spaces which embed in L_p for certain negative values
of p.Comment: 18 pages, added a section with equivalent formulations using Fourier
Transforms and Embeddings into L_p for p<
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