Comparison theorem
In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, 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. Differential (or integral) inequalities, derived from differential (respectively, integral) equations by replacing the equality sign with an inequality sign, form a broad class of such auxiliary relations.[1][2]
One instance of such theorem was used by Aronson and Weinberger to characterize solutions of Fisher's equation, a reaction-diffusion equation.[3] Other examples of comparison theorems include:
- Chaplygin's theorem
- Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations
- Lyapunov comparison theorem
- Sturm comparison theorem
- Hille-Wintner comparison theorem
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. [4]
- 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[5]
- Berger–Kazdan comparison theorem[6]
- Warner comparison theorem for lengths of N-Jacobi fields (N being a submanifold of a complete Riemannian manifold)[7]
- Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvatures[8]
- Lichnerowicz comparison theorem
- Eigenvalue comparison theorem
- Comparison triangle
Between algebraic and analytic geometry
[edit]There exist various comparisons between algebraic geometry and analytic geometry often known as GAGA theorems due to the abbreviation of Serre's foundational article Géométrie Algébrique et Géométrie Analytique[9].
- Algebraic and analytic geometry over the complex numbers (GAGA)
- Artin comparison theorem - identifies the Étale cohomology of finite type schemes and the cohomology of their analytic counterparts.
See also
[edit]- 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
[edit]- ↑ Walter, Wolfgang (1970). Differential and Integral Inequalities. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-642-86405-6. ISBN 978-3-642-86407-0.
- ↑ Lakshmikantham, Vangipuram (1969). Differential and integral inequalities: theory and applications. Mathematics in science and engineering. Srinivasa Leela. New York: Academic Press. ISBN 978-0-08-095563-6.
- ↑ Aronson, D. G.; Weinberger, H. F. (1975). Goldstein, Jerome A. (ed.). "Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation". Partial Differential Equations and Related Topics. Berlin, Heidelberg: Springer: 5–49. doi:10.1007/BFb0070595. ISBN 978-3-540-37440-4.
- ↑ Jeff Cheeger and David Gregory Ebin: Comparison theorems in Riemannian Geometry, North Holland 1975.
- ↑ 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
- ↑ Serre, Jean-Pierre (1956), "Géométrie algébrique et géométrie analytique", Annales de l'Institut Fourier, 6: 1–42, doi:10.5802/aif.59, ISSN 0373-0956, MR 0082175
External links
[edit]- "Comparison theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- "Differential inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- nLab - Artin Comparison
If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article.