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Ellipses numbers and geometric measure representations

Ellipses numbers and geometric measure representations Abstract Ellipses will be considered as subsets of suitably defined Minkowski planes in such a way that, additionally to the well-known area content property A ( r ) = π ( a,b ) r 2 , the number π ( a,b ) = abπ reflects a generalized circumference property U ( a,b ) ( r ) = 2 π ( a,b ) r of the ellipses E ( a,b ) ( r ) with main axes of lengths 2 ra and 2 rb , respectively. In this sense, the number π ( a,b ) is an ellipse number w.r.t. the Minkowski functional r of the reference set E ( a,b ) (1). This approach is closely connected with a generalization of the method of indivisibles and avoids elliptical integrals. Further, several properties of both a generalized arc-length measure and the ellipses numbers will be discussed, e.g. disintegration of the Lebesgue measure and an elliptically contoured Gaussian measure indivisiblen representation, wherein the ellipses numbers occur in a natural way as norming constants. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Analysis de Gruyter

Ellipses numbers and geometric measure representations

Richter, Wolf-Dieter
Journal of Applied Analysis , Volume 17 (2) – Dec 1, 2011
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References (5)

Publisher
de Gruyter
Copyright
Copyright © 2011 by the
ISSN
1425-6908
eISSN
1869-6082
DOI
10.1515/jaa.2011.011
Publisher site
See Article on Publisher Site

Abstract

Abstract Ellipses will be considered as subsets of suitably defined Minkowski planes in such a way that, additionally to the well-known area content property A ( r ) = π ( a,b ) r 2 , the number π ( a,b ) = abπ reflects a generalized circumference property U ( a,b ) ( r ) = 2 π ( a,b ) r of the ellipses E ( a,b ) ( r ) with main axes of lengths 2 ra and 2 rb , respectively. In this sense, the number π ( a,b ) is an ellipse number w.r.t. the Minkowski functional r of the reference set E ( a,b ) (1). This approach is closely connected with a generalization of the method of indivisibles and avoids elliptical integrals. Further, several properties of both a generalized arc-length measure and the ellipses numbers will be discussed, e.g. disintegration of the Lebesgue measure and an elliptically contoured Gaussian measure indivisiblen representation, wherein the ellipses numbers occur in a natural way as norming constants.

Journal

Journal of Applied Analysis – de Gruyter

Published: Dec 1, 2011

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