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.

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Differential equations

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] [3]

Riemannian geometry

In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. [5]

Other

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References

  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. See also: Lyapunov comparison principle
  4. "Differential inequality - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-13.
  5. Jeff Cheeger and David Gregory Ebin: Comparison theorems in Riemannian Geometry, North Holland 1975.
  6. M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700712
  7. Weisstein, Eric W. "Berger-Kazdan Comparison Theorem". MathWorld .
  8. F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341356
  9. R.L. Bishop & R. Crittenden, Geometry of manifolds