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Geometric theory of differential equations. II. Analytic interpretation of a geometric theorem of Blaschke


Author: H. Guggenheimer
Journal: Proc. Amer. Math. Soc. 29 (1971), 87-90
MSC: Primary 34.40
DOI: https://doi.org/10.1090/S0002-9939-1971-0285761-9
MathSciNet review: 0285761
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Abstract: According to Blaschke, a $ {C^2}$ closed convex plane curve admits at least three pairs of points with parallel tangents and equal curvature radii. We generalize the result to theorems on Hill equations with either first or second intervals of stability collapsed. We also prove a four-extrema theorem for second order linear differential equations without periodicity properties.


References [Enhancements On Off] (What's this?)

  • [1] W. Blaschke, Aufgabe 540, Arch. Math. Phys. (3) 26 (1917), 65.
  • [2] O. Borůvka, Lineare Differentialtransformationen 2. Ordnung, Hochschulbücher für Mathematik, Band 67, VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. MR 38 #4743.
  • [3] Heinrich W. Guggenheimer, Plane geometry and its groups, Holden-Day, Inc., San Francisco, Calif.-Cambridge-Amsterdam, 1967. MR 0213943
  • [4] H. Guggenheimer, Sign changes, extrema, and curves of minimal order, J. Differential Geometry 3 (1969), 511–521. MR 0262999
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  • [6] W. Süss, Lösung der Aufgabe 540, Arch. Math. Phys. 26, Jber. Deutsch. Math.-Verein. 33 (1924), 32-33.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0285761-9
Keywords: Hill equation, dispersion, periodic function
Article copyright: © Copyright 1971 American Mathematical Society