In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
,
differential equations and
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
.
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. 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.
One instance of such theorem was used by Aronson and Weinberger to characterize solutions of
Fisher's equation, a reaction-diffusion equation. Other examples of comparison theorems include:
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Chaplygin's theorem
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Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations
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Lyapunov comparison theorem
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Sturm comparison theorem
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Hille-Wintner comparison theorem
Riemannian geometry
In
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.
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Rauch comparison theorem relates the
sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a po ...
of a
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
to the rate at which its
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
s spread apart
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Toponogov's theorem
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Myers's theorem
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Hessian comparison theorem
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Laplacian comparison theorem
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Morse–Schoenberg comparison theorem
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Berger comparison theorem,
Rauch–Berger comparison theorem
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Berger–Kazdan comparison theorem
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Warner comparison theorem for
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
s of
N-Jacobi fields (''N'' being a submanifold of a complete Riemannian manifold)
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Bishop–Gromov inequality, conditional on a lower bound for the
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
s
[ R.L. Bishop & R. Crittenden, ''Geometry of manifolds'']
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Lichnerowicz comparison theorem
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Eigenvalue comparison theorem
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Cheng's eigenvalue comparison theorem
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Comparison triangle
See also
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Limit comparison theorem, about convergence of series
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Comparison theorem for integrals, about convergence of integrals
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Zeeman's comparison theorem, a technical tool from the theory of
spectral sequences
References
External links
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{{sia, mathematics
Mathematical theorems