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
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.
- Chaplygin inequality[3]
- Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations.
- Sturm comparison theorem
- Aronson and Weinberger used a comparison theorem to characterize solutions to Fisher's equation, a reaction--diffusion equation.
- Hille-Wintner comparison theorem
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.
- Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart.
- Toponogov's theorem
- Myers's theorem
- Hessian comparison theorem
- Laplacian comparison theorem
- Morse–Schoenberg comparison theorem
- Berger comparison theorem, Rauch–Berger comparison theorem[4]
- Berger–Kazdan comparison theorem[5]
- Warner comparison theorem for lengths of N-Jacobi fields (N being a submanifold of a complete Riemannian manifold)[6]
- Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvatures[7]
- Lichnerowicz comparison theorem
- Eigenvalue comparison theorem
- Cheng's eigenvalue comparison theorem
- See also: Comparison triangle
Other
- Limit comparison theorem, about convergence of series
- Comparison theorem for integrals, about convergence of integrals
- Zeeman's comparison theorem, a technical tool from the theory of spectral sequences
References
- ^ "The Definitive Glossary of Higher Mathematical Jargon — Theorem". Math Vault. 2019-08-01. Retrieved 2019-12-13.
- ^ "Comparison theorem - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-13.
- ^ "Differential inequality - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-13.
- ^ M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700–712
- ^ Weisstein, Eric W. "Berger-Kazdan Comparison Theorem". MathWorld.
- ^ F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356
- ^ R.L. Bishop & R. Crittenden, Geometry of manifolds