10,795 research outputs found
Characterizing the dual mixed volume via additive functionals
Integral representations are obtained of positive additive functionals on
finite products of the space of continuous functions (or of bounded Borel
functions) on a compact Hausdorff space. These are shown to yield
characterizations of the dual mixed volume, the fundamental concept in the dual
Brunn-Minkowski theory. The characterizations are shown to be best possible in
the sense that none of the assumptions can be omitted. The results obtained are
in the spirit of a similar characterization of the mixed volume in the
classical Brunn-Minkowski theory, obtained recently by Milman and Schneider,
but the methods employed are completely different
Fully representable and *-semisimple topological partial *-algebras
We continue our study of topological partial *-algebras, focusing our
attention to *-semisimple partial *-algebras, that is, those that possess a
{multiplication core} and sufficiently many *-representations. We discuss the
respective roles of invariant positive sesquilinear (ips) forms and
representable continuous linear functionals and focus on the case where the two
notions are completely interchangeable (fully representable partial *-algebras)
with the scope of characterizing a *-semisimple partial *-algebra. Finally we
describe various notions of bounded elements in such a partial *-algebra, in
particular, those defined in terms of a positive cone (order bounded elements).
The outcome is that, for an appropriate order relation, one recovers the
\M-bounded elements introduced in previous works.Comment: 26 pages, Studia Mathematica (2012) to appea
Minkowski Functional Description of Microwave Background Gaussianity
A Gaussian distribution of cosmic microwave background temperature
fluctuations is a generic prediction of inflation. Upcoming high-resolution
maps of the microwave background will allow detailed tests of Gaussianity down
to small angular scales, providing a crucial test of inflation. We propose
Minkowski functionals as a calculational tool for testing Gaussianity and
characterizing deviations from it. We review the mathematical formalism of
Minkowski functionals of random fields; for Gaussian fields the functionals can
be calculated exactly. We then apply the results to pixelized maps, giving
explicit expressions for calculating the functionals from maps as well as the
Gaussian predictions, including corrections for map boundaries, pixel noise,
and pixel size and shape. Variances of the functionals for Gaussian
distributions are derived in terms of the map correlation function.
Applications to microwave background maps are discussed.Comment: 24 pages with 2 figures. Submitted to New Astronom
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