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Index of articles associated with the same name
In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often
Comparison_theorem
Theorem in Riemannian geometry
In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its
Cheng's eigenvalue comparison theorem
Cheng's_eigenvalue_comparison_theorem
Relates sectional curvature of a Riemannian manifold to the rate geodesics spread apart
In Riemannian geometry, the Rauch comparison theorem, named after Harry Rauch, who proved it in 1951, is a fundamental result which relates the sectional
Rauch_comparison_theorem
equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria
Sturm–Picone comparison theorem
Sturm–Picone_comparison_theorem
On when a morphism of spectral sequences in homological algebra is an isomorphism
comparison theorem, introduced by Christopher Zeeman, gives conditions for a morphism of spectral sequences to be an isomorphism. Comparison theorem—Let
Zeeman's_comparison_theorem
Triangle comparison theorem in Riemannian geometry
Toponogov's theorem (named after Victor Andreevich Toponogov) is a triangle comparison theorem. It is one of a family of comparison theorems that quantify
Toponogov's_theorem
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
On volumes in complete Riemannian n-manifolds whose Ricci curvature has a lower bound
inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the
Bishop–Gromov_inequality
Differential form in commutative algebra
\Omega _{X/k}} . The Riemann–Roch theorem and its far-reaching generalization, the Grothendieck–Riemann–Roch theorem, contain as a crucial ingredient the
Kähler_differential
Method for finding limits in calculus
functions. The squeeze theorem is used in calculus and mathematical analysis, typically to confirm the limit of a function via comparison with two other functions
Squeeze_theorem
Mathematical theorem
provides a comparison theorem that can be used to prove uniqueness of a solution to the initial value problem; see the Picard–Lindelöf theorem. It is named
Grönwall's_inequality
Cheng's eigenvalue comparison theorem Clifford's theorem on special divisors Cohn-Vossen's inequality Erdős–Mordell inequality Euler's theorem in geometry Gromov's
List_of_inequalities
Describes the objects of a given type, up to some equivalence
group Representation theorem – Proof that every structure with certain properties is isomorphic to another structure Comparison theorem Moduli space – Geometric
Classification_theorem
Theorem in vector calculus
theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem,
Stokes'_theorem
Elliptic curves
d-torsion points. It is called a 'comparison theorem' as it is an analogue for Arakelov theory of comparison theorems in cohomology relating de Rham cohomology
Hodge–Arakelov_theory
Complete manifolds of non-negative sectional curvature largely reduce to the compact case
In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature
Soul_theorem
French mathematician (1927-2016)
Arthur Besse Berger's inequality for Einstein manifolds Berger–Kazdan comparison theorem Musical isomorphism Parametrix Quaternion-Kähler manifold Spin(7)-manifold
Marcel_Berger
Group of mathematical theorems
specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients
Isomorphism_theorems
17th-century conjecture proved by Andrew Wiles in 1994
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that there are no positive integers a
Fermat's_Last_Theorem
Conditions for switching order of integration in calculus
Fubini's theorem gives the conditions under which a double integral can be computed as an iterated integral, i.e. by integrating in one variable at a
Fubini's_theorem
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
Restricts the possible topology of a negatively curved compact Riemannian manifold
MR 1138207. Zbl 0752.53001. Cheeger, Jeff; Ebin, David G. (2008). Comparison theorems in Riemannian geometry (Revised reprint of the 1975 original ed.)
Preissmann's_theorem
Bounds the length of geodetic segments in Riemannian manifolds based in Ricci curvature
Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was
Myers's_theorem
Relation between sides of a right triangle
In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle
Pythagorean_theorem
Class of ordinary differential equations
{\textstyle \{x\in (a,b):u(x)=0\}} is infinite. The Bolzano-Weierstrass Theorem tells us that this set has some limit point c ∈ [ a , b ] {\textstyle c\in
Sturm–Liouville_theory
Set of points where the shortest paths from a specific starting point cease to be unique
Laplacian comparison theorem and the local Hessian comparison theorem. These are used in the proof of the local version of the Toponogov theorem, and many
Cut_locus
Vector field in Riemannian geometry
Riemannian manifold. Conjugate points Geodesic deviation equation Rauch comparison theorem N-Jacobi field Manfredo Perdigão do Carmo. Riemannian geometry. Translated
Jacobi_field
Theorem in mathematics
In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating
Mean_value_theorem
Italian-born American mathematician (1923–2023)
special case of Calabi's construction. Calabi found the Laplacian comparison theorem in Riemannian geometry, which relates the Laplace–Beltrami operator
Eugenio_Calabi
ISSN 0002-9947. JSTOR 1989854. Kleiner, Bruce (1992). "An isoperimetric comparison theorem". Inventiones Mathematicae. 108 (1): 37–47. Bibcode:1992InMat.108
Cartan–Hadamard_conjecture
Relationship between derivatives and integrals
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at every
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Gives a lower bound on the volume of a Riemannian manifold
Society. doi:10.1090/mmono/149. ISBN 0-8218-0284-4. MR 1390760. Zbl 0886.53002. Weisstein, Eric W. "Berger-Kazdan comparison theorem". MathWorld. v t e
Berger's isoembolic inequality
Berger's_isoembolic_inequality
Well-quasi-ordering of finite trees
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under
Kruskal's_tree_theorem
Statement relating differentiable symmetries to conserved quantities
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law
Noether's_theorem
in association with an almost immediate proof of the Sturm comparison theorem, a theorem whose proof took up many pages in Sturm's original memoir of
Picone_identity
Theorem in mathematics
In mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that
Inverse_function_theorem
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree
Taylor's_theorem
Theorem in differential geometry
to the two endpoints of the line. From the fundamental Laplacian comparison theorem proved earlier by Eugenio Calabi, these functions are both superharmonic
Splitting_theorem
Subfield of automated reasoning and mathematical logic
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving
Automated_theorem_proving
Polish mathematician
Deligne cohomology for algebraic varieties over the real numbers. A comparison theorem, via syntomic methods, for p {\displaystyle p} -adic analytic varieties
Wiesława_Nizioł
Mathematical theorem
open interval (x0, x1). It is a special case of the Sturm-Picone comparison theorem. Since u {\displaystyle \displaystyle u} and v {\displaystyle \displaystyle
Sturm_separation_theorem
Theorem in calculus relating line and double integrals
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in R
Green's_theorem
On when a set of compact Riemannian manifolds of a given dimension is relatively compact
Gromov's metric compactness theorem is automatically satisfied as a corollary of the Bishop–Gromov volume comparison theorem. As such, it follows that:
Gromov's compactness theorem (geometry)
Gromov's_compactness_theorem_(geometry)
Italian mathematician (1885–1977)
mathematician. He is known for the Picone identity, the Sturm-Picone comparison theorem and being the founder of the Istituto per le Applicazioni del Calcolo
Mauro_Picone
Proof all ranked voting rules have spoilers
Arrow's impossibility theorem is a key result in social choice theory, proved by American economist Kenneth Arrow. It shows that no procedure for group
Arrow's_impossibility_theorem
Uniqueness of countable dense linear orders
isomorphism theorem is stated using the following concepts: A linear order or total order is defined by a set of elements and a comparison operation that
Cantor's_isomorphism_theorem
Grothendieck's axioms Grothendieck category Grothendieck's comparison theorem Grothendieck's connectedness theorem Grothendieck connection Grothendieck construction
List of things named after Alexander Grothendieck
List_of_things_named_after_Alexander_Grothendieck
Israeli mathematician
BC. Benjamini, Itai; Cao, Jianguo (1996), "A new isoperimetric comparison theorem for surfaces of variable curvature", Duke Mathematical Journal, 85 (2):
Itai_Benjamini
Statement about integration on manifolds
generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about
Generalized_Stokes_theorem
Mathematical problem in spectral theory
sphere (for instance) is spectrally rigid, by Cheng's eigenvalue comparison theorem. It is also known, by a result of Osgood, Phillips, and Sarnak that
Hearing_the_shape_of_a_drum
Branch of differential geometry
including hundreds of references.) Cheeger, Jeff; Ebin, David G. (2008), Comparison theorems in Riemannian geometry, Providence, RI: AMS Chelsea Publishing; Revised
Riemannian_geometry
Theorem in Mathematics
Halanay inequality is a comparison theorem for differential equations with delay. This inequality and its generalizations have been applied to analyze
Halanay_inequality
Hong Kong mathematician
differential equations, including Cheng's eigenvalue comparison theorem, Cheng's maximal diameter theorem, and a number of works with Shing-Tung Yau. Many
Shiu-Yuen_Cheng
Tensor in differential geometry
geometric and topological consequences, as in Myers's theorem and related comparison theorems. In dimension three, the Ricci tensor determines the full
Ricci_curvature
On the structure of complete Riemannian manifolds of non-positive sectional curvature
the CAT(0) condition is an abstract form of Toponogov's triangle comparison theorem. The assumption of non-positive curvature can be weakened (Alexander
Cartan–Hadamard_theorem
Certain vector fields are the sum of an irrotational and a solenoidal vector field
In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector
Helmholtz_decomposition
Mathematical theorem
Sturm–Picone comparison theorem. There are many extensions to this result, such as the Gesztesy–Ünal criterion. While Peano's existence theorem guarantees
Kneser's theorem (differential equations)
Kneser's_theorem_(differential_equations)
On converting relations to functions of several real variables
In multivariable calculus, the implicit function theorem is a theorem that provides sufficient conditions under which a planar curve specified by F ( x
Implicit_function_theorem
Shape with three sides
area enclosed by the triangle. In more general spaces, there are comparison theorems relating the properties of a triangle in the space to properties
Triangle
Perron–Frobenius theorem Z-matrix L-matrix M-matrix H-matrix (iterative method) Varga, Richard S. (2006). "Basic Iterative Methods and Comparison Theorems". Matrix
Comparison_matrix
Gives general conditions under which sheaf cohomology groups with indices > 0 are zero
and the GAGA comparison theorems. However, in 1987 Pierre Deligne and Luc Illusie gave a purely algebraic proof of the vanishing theorem in (Deligne &
Kodaira_vanishing_theorem
French mathematician (1803–1855)
made a significant addition to equation theory with his work, Sturm's theorem. Sturm was born in Geneva, France in 1803. The family of his father, Jean-Henri
Jacques Charles François Sturm
Jacques_Charles_François_Sturm
American mathematician
Finiteness theorems for Riemannian manifolds. American Journal of Mathematics. 92 (1970) 61–74. Cheeger, Jeff; Ebin, David G. Comparison theorems in Riemannian
Jeff_Cheeger
Introduction and Reference on Differential Geometry
manifolds, Jacobi fields, the Morse index, the Rauch comparison theorems, and the Cartan–Hadamard theorem. Then it ascends to complex manifolds, Kähler manifolds
Foundations of Differential Geometry
Foundations_of_Differential_Geometry
Theorem in political science
In political science and social choice, Black's median voter theorem says that if voters and candidates are distributed along a one-dimensional political
Median_voter_theorem
French mathematician (born 1958)
Band 110, Springer Verlag, 2004, S. 263–346. with Persi Diaconis Comparison theorems for random walks on finite groups, Annals of Probability, Band 21
Laurent_Saloff-Coste
British mathematician and logician
contributions include the Spector–Gandy theorem, the Gandy Stage Comparison theorem, and the Gandy Selection theorem. He also made a significant contribution
Robin_Gandy
Branch of mathematics
curves. These two branches are related to each other by the fundamental theorem of calculus. Calculus uses convergence of infinite sequences and infinite
Calculus
Fundamental theorem in probability theory and statistics
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample
Central_limit_theorem
Theorem in classical statistical mechanics
mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of
Equipartition_theorem
Mathematical theorem
for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem. In the context of partial differential equations
Symmetry of second derivatives
Symmetry_of_second_derivatives
British mathematician (1925–2016)
piecewise linear topology and is credited with working out the engulfing theorem (independently also worked out by John Stallings), which can be used to
Christopher_Zeeman
oscillation theory are: Kneser's theorem (differential equations) Sturm–Picone comparison theorem Sturm separation theorem Atkinson, F.V. (1964). Discrete
Oscillation_theory
Circulation density in a vector field
vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector
Curl_(mathematics)
Differentiation under the integral sign formula
integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above
Leibniz_integral_rule
Matrix of partial derivatives of a vector-valued function
generalization includes generalizations of the inverse function theorem and the implicit function theorem, where the non-nullity of the derivative is replaced by
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Motion of a curve based on its curvature
"Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem", Journal für die Reine und Angewandte Mathematik
Curve-shortening_flow
Description in Riemannian geometry
to give a comparison theorem between geodesic triangles in M and those in a suitable simply connected space form; see Toponogov's theorem. Simple consequences
Sectional_curvature
Differential calculus on function spaces
L}{\partial x}}=0} implies that the Lagrangian is time-independent. By Noether's theorem, there is an associated conserved quantity. In this case, this quantity
Calculus_of_variations
Differentiable manifold
as comparison spaces in studying geodesics and volume comparison theorems on CR manifolds with zero Webster torsion akin to the H.E. Rauch comparison theorem
CR_manifold
Indefinite integral
Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval
Antiderivative
3D generalization of the Leibniz integral rule
calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or simply the Reynolds theorem, named after Osborne Reynolds
Reynolds_transport_theorem
Study of rates of change
Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation and integration are inverse
Differential_calculus
Largest distance between two points
1007/978-3-319-91755-9, ISBN 978-3-319-91755-9 Cheng, Shiu Yuen (1975), "Eigenvalue comparison theorems and its geometric applications", Mathematische Zeitschrift, 143 (3):
Diameter_of_a_set
Theorem that tells the maximum rate at which information can be transmitted
In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified
Shannon–Hartley_theorem
Relates rational elliptic curves to modular forms
In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way
Modularity_theorem
Method of evaluating certain integrals along paths in the complex plane
application of the Cauchy integral formula or residue theorem is possible application of Cauchy's integral theorem The integral is reduced to only an integration
Contour_integration
Mathematical method in calculus
The discrete analogue for sequences is called summation by parts. The theorem can be derived as follows. For two continuously differentiable functions
Integration_by_parts
Method of mathematical integration
under the integral sign (via the monotone convergence theorem and dominated convergence theorem). While the Riemann integral considers the area under
Lebesgue_integral
Algebraic expansion of powers of a binomial
algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power ( x
Binomial_theorem
Length of a line segment
calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. These names
Euclidean_distance
Study of rational collective decision-making
Arrow had penned his theorem with the assumption of ordinality, Sen introduced the assumption of cardinality, or interpersonal comparison, into the social-choice
Social_choice_theory
ordinary differential equation. This theorem was stated by Sergey Chaplygin in 1919. It is one of many comparison theorems. The Chaplygin Method is a method
Chaplygin's Theorem and Method for Solving ODE
Chaplygin's_Theorem_and_Method_for_Solving_ODE
Topics referred to by the same term
linear second order differential equations Sturm–Picone comparison theorem, a classical theorem which provides criteria for the oscillation and non-oscillation
Picone
Technique in integral evaluation
theorem. Alternatively, the requirement that det(Dφ) ≠ 0 can be eliminated by applying Sard's theorem. For Lebesgue measurable functions, the theorem
Integration_by_substitution
Operation on differential forms
natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential k {\displaystyle
Exterior_derivative
Mathematical identities
\varphi )} in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials). This result is a special
Vector_calculus_identities
American mathematician
Retrieved 10 January 2016. Kleiner, Bruce (1992). "An isoperimetric comparison theorem". Inventiones Mathematicae. 108 (1): 37–47. Bibcode:1992InMat.108
Bruce_Kleiner
Evaluates a line integral through a gradient field using the original scalar field
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated
Gradient_theorem
Gauss–Bonnet theorem Hopf–Rinow theorem Cartan–Hadamard theorem Myers theorem Rauch comparison theorem Morse index theorem Synge theorem Weinstein theorem Toponogov
List of differential geometry topics
List_of_differential_geometry_topics
COMPARISON THEOREM
COMPARISON THEOREM
Boy/Male
Arabic American Muslim
Companion.
Girl/Female
Hindu
Comparison
Boy/Male
Bengali, Celebrity, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Sindhi, Telugu
Without Comparison; Incomparable; Unique
Boy/Male
Tamil
Without comparison
Girl/Female
Tamil
Comparison
Boy/Male
Hindu, Indian
Compassion
Girl/Female
African, Indian, Swahili
Compassion
Girl/Female
Indian
Compassion.
Boy/Male
Tamil
Compassion
Boy/Male
Arabic American
Companion.
Boy/Male
Hindu, Indian, Marathi, Punjabi, Sikh, Telugu
Compassion
Girl/Female
American, British, English, Latin
Compassion
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Without Comparison
Girl/Female
Muslim
Companion
Girl/Female
Indian, Telugu
Compassion
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Telugu
Example; Comparison; Respectful; Brilliant
Girl/Female
Arabic
Compassion
Boy/Male
Arabic American
Companion.
Girl/Female
Gujarati, Indian, Kannada, Tamil
Compassion
Boy/Male
Gujarati, Indian
Compassion
COMPARISON THEOREM
COMPARISON THEOREM
Girl/Female
Christian & English(British/American/Australian)
Symbol of Peace
Girl/Female
Arabic, Muslim
Beautiful
Girl/Female
Indian
A musical instrument, Wise, Far-sighted
Girl/Female
Tamil
Goddess Parvati
Girl/Female
Muslim
Consolation, Comfort
Boy/Male
Gujarati, Hindu, Indian, Kannada, Kashmiri, Malayalam, Marathi, Sanskrit, Telugu, Traditional
Lord Krishna
Girl/Female
Latin
Full of sorrows.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Son of Dhritrashtra
Girl/Female
Muslim
Friendly, Affable
Boy/Male
Hindu, Indian, Marathi
Saint; Lord
COMPARISON THEOREM
COMPARISON THEOREM
COMPARISON THEOREM
COMPARISON THEOREM
COMPARISON THEOREM
n.
That to which, or with which, a thing is compared, as being equal or like; illustration; similitude.
n.
The modification, by inflection or otherwise, which the adjective and adverb undergo to denote degrees of quality or quantity; as, little, less, least, are examples of comparison.
n.
The faculty of the reflective group which is supposed to perceive resemblances and contrasts.
v. t.
To be a companion to; to attend on; to accompany.
n.
Similitude; comparison.
n.
A comparison; a simile.
n.
Mutual comparison of corresponding parts.
n.
A knight of the lowest rank in certain orders; as, a companion of the Bath.
n.
Comparison or illustration by contraries.
v. t.
To compare.
n.
Illustration by comparison; simile.
n.
Comparison.
n.
The act of comparing; an examination of two or more objects with the view of discovering the resemblances or differences; relative estimate.
v. t.
To qualify as a companion; to make equal.
a.
Of or pertaining to comparison.
n.
A wooden hood or penthouse covering the companion way; a companion hatch.
n.
The state of being compared; a relative estimate; also, a state, quality, or relation, admitting of being compared; as, to bring a thing into comparison with another; there is no comparison between them.
n.
A comparison; parable; proverb.
n.
A figure by which one person or thing is compared to another, or the two are considered with regard to some property or quality, which is common to them both; e.g., the lake sparkled like a jewel.
v. i.
To be equal; to hold comparison.