Ellipses numbers and geometric measure representations

• Autorzy: Richter W.-D.
• Języki publikacji: EN

Abstrakty

EN

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) = 2Π (a,b) r of the ellipses E (a,b)(r) with main axes of lengths 2ra and 2rb, 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.

Słowa kluczowe

EN

ellipse number; generalized arc-length; generalized perimeter; generalized method of indivisibles; isoperimetric constant; Minkowski plane; geometric measure representation; intersection-percentage function; generalized uniform distribution on the ellipse; generalized trigonometric functions; generalized elliptical coordinates; disintegration of Lebesgue measure; elliptically contoured Gaussian measure representation; stochastic representation

PL

długość łuku; obwód uogólniony; metoda niepodzielnych; stała izoperymetryczna; płaszczyzna Minkowskiego; uogólniona funkcja trygonometryczna; współrzędne eliptyczne; miara Lebesgue'a; dezintegracja miary; miara Gaussa; reprezentacja stochastyczna

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2011
• Tom: Vol. 17, nr 2
• Strony: 165--179
• Opis fizyczny: Bibliogr. 3 poz.

Twórcy

autor

Richter W.-D.

• University of Rostock, Ulmenstr. 69, Haus 3, 18057 Rostock, Germany,

Bibliografia

• [1] H. Busemann, The isoperimetric problem in the Minkowski plane, Amer. J. Math. 69 (1947), 863-871.
• [2] W.-D. Richter, Generalized spherical and simplicial coordinates, J. Math. Anal. Appl. 336(2007), 1187-1202.
• [3] W.-D. Richter, On l2,p-circle numbers, Lith. Math. J. 48 (2008), no. 2, 228-234.