Ueber die Bildung der Hagelkörner und über die Ursache des polymorphen Auftretens des festen Wassers

Ohne Zusammenfassung     

Verallgemeinerung eines Satzes von F. Riesz

Let X be a compact Hausdorff space, C(X) the class of all continuous functions f: XR, considered as an ordered vector space over R with respect to its canonical order; furthermore let F be an order complete ordered vector space over R. A linear operator A: C(X)F is called bounded, if it transforms each bounded set of C(X) into a bounded set of F.The purpose of this paper is to represent such a bounded linear operator A: C(X)F as an integral Af= d with respect to some content , defined on the algebraa(X) which is generated by the open sets UX.This representation is unique, if is required to be regular (p. 11).     

Taylor-model-based enclosure of invariant manifolds for planar diffeomorphisms and applications

Fundamental questions in the study of interesting dynamics of planar diffeomorphisms like the Hénon map involve homoclinic phenomena, topological entropy and strange attractors. Inherently, answering these questions requires knowledge about the stable and unstable manifolds, which in the typical case in the plane are smooth curves. We present a method to find highly accurate Taylor Model enclosures of the invariant curves near hyperbolic fixed points. Successive iteration of these local enclosures yields similarly accurate enclosures of pieces of the global manifold tangle. Applications presented include the automatic computation of rigorous enclosures of all homoclinic points up to finite iterates. This allows to find symbolic dynamics in the original system and consequently compute rigorous lower bounds for its topological entropy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Operatoren verallgemeinerter Funktionen

Ohne Zusammenfassung     

Über einen Darstellungssatz von L. Schwartz

Bekanntlich lässt sich jede skalare DistributionT überR lokal als verallgemeinerte Ableitung einer stetigen Funktionf:R C darstellen:T = D ^k f(Schwartz [2], chap. III, §6, Th. XXI). Wir geben in dieser Note einen Beweis für diesen Schwartzschen Darstellungssatz wieder, der die in [2] verwendeten funktionalanalytischen Hilfsmittel vermeidet und sich direkt auf vektorielle Distributionen übertragen lässt, die ihre Werte in einem Banach—Raum oder allgemeiner in einem strikten induktiven Limes einer Folge von Banach—Räumen haben.     

Rigorous and accurate enclosure of invariant manifolds on surfaces

Knowledge about stable and unstable manifolds of hyperbolic fixed points of certain maps is desirable in many fields of research, both in pure mathematics as well as in applications, ranging from forced oscillations to celestial mechanics and space mission design. We present a technique to find highly accurate polynomial approximations of local invariant manifolds for sufficiently smooth planar maps and rigorously enclose them with sharp interval remainder bounds using Taylor model techniques. Iteratively, significant portions of the global manifold tangle can be enclosed with high accuracy. Numerical examples are provided.     

Über die Distributivität von Verbandsgruppen

Ohne Zusammenfassung