Comparison theorem

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In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type,[1] and often occur in fields such as calculus, differential equations and Riemannian geometry.

Differential equations[edit]

In the theory of differential equations, comparison theorems assert particular properties of solutions of a differential equation (or of a system thereof), provided that an auxiliary equation/inequality (or a system thereof) possesses a certain property.[2] See also Lyapunov comparison principle.

Riemannian geometry[edit]

In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.

See also: Comparison triangle

Other[edit]

References[edit]

  1. ^ "The Definitive Glossary of Higher Mathematical Jargon — Theorem". Math Vault. 2019-08-01. Retrieved 2019-12-13.
  2. ^ "Comparison theorem - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-13.
  3. ^ "Differential inequality - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-13.
  4. ^ M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700–712
  5. ^ Weisstein, Eric W. "Berger-Kazdan Comparison Theorem". MathWorld.
  6. ^ F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356
  7. ^ R.L. Bishop & R. Crittenden, Geometry of manifolds