Comparison theorem

Comparison theorem

A comparison theorem is any of a variety of theorems that compare properties of various mathematical objects.

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, M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", llinois J. Math., vol. 6 (1962) 700–712
  • Berger–Kazdan comparison theorem [1]
  • Warner comparison theorem for lengths of N-Jacobi fields (N being a submanifold of a complete Riemannian manifold) F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356).
  • Bishop volume comparison theorem / Bishop comparison theorem, conditional on a lower bound for the Ricci curvatures (R.L. Bishop & R. Crittenden, Geometry of manifolds)
  • Lichnerowicz comparison theorem
  • Eigenvalue comparison theorem
See also: Comparison triangle

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. See also Lyapunov comparison principle

  • Sturm comparison theorem [2]

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



Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Cheng's eigenvalue comparison theorem — In Riemannian geometry, Cheng s eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization is not precise, in part… …   Wikipedia

  • Sturm-Picone comparison theorem — In mathematics, in the field of ordinary differential equations, the Sturm Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non… …   Wikipedia

  • Berger-Kazdan comparison theorem — In mathematics, the Berger Kazdan comparison theorem is a result in Riemannian geometry that gives a lower bound on the volume of a Riemannian manifold and also gives a necessary and sufficient condition for the manifold to be isometric to the m… …   Wikipedia

  • Rauch comparison theorem — In Riemannian geometry the Rauch comparison theorem is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, it states that for large curvature, geodesics… …   Wikipedia

  • Comparison of instant runoff voting to other voting systems — This article is a comparison of various voting systems with Instant runoff voting (IRV), also called the Alternative Vote , preferential voting and ranked choice voting. Contents 1 Categories 2 Voting system criteria 3 Voting system results …   Wikipedia

  • Comparison of analog and digital recording — This article compares the two ways in which sound is recorded and stored. Actual sound waves consist of continuous variations in air pressure. Representations of these signals can be recorded using either digital or analog techniques. An analog… …   Wikipedia

  • comparison test — Math. the theorem that a given infinite series converges if the absolute value of each term of the given series is less than or equal to the corresponding term in a known convergent series. * * * …   Universalium

  • comparison test — Math. the theorem that a given infinite series converges if the absolute value of each term of the given series is less than or equal to the corresponding term in a known convergent series …   Useful english dictionary

  • Nachbin's theorem — Exponential type redirects here. For exponential types in type theory and programming languages, see Function type. In mathematics, in the area of complex analysis, Nachbin s theorem (named after Leopoldo Nachbin) is commonly used to establish a… …   Wikipedia

  • Newton's theorem of revolving orbits — Figure 1: An attractive force F(r) causes the blue planet to move on the cyan circle. The green planet moves three times faster and thus requires a stronger centripetal force, which is supplied by adding an attractive inverse cube force. The …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”