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Subadditive or superadditive integral
A Choquet integral is a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. It was initially used in statistical
Choquet_integral
French mathematician (1915–2006)
topology and measure theory. He is known for creating the Choquet theory, the Choquet integral and the theory of capacities. He did postgraduate work at
Gustave_Choquet
Operation in calculus
Brownian motion. The Choquet integral, a subadditive or superadditive integral created by Gustave Choquet in 1953. The Bochner integral, a generalization
Integral
Area of functional analysis and convex analysis
In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have
Choquet_theory
Surname list
Netherlands Gustave Choquet (1915–2006), French mathematician Yvonne Choquet-Bruhat (1923–2025), French mathematician and physicist Choquet integral, a way of measuring
Choquet
Poisson integrals of homeomorphisms are diffeomorphisms
mathematics, the Radó–Kneser–Choquet theorem, named after Tibor Radó, Hellmuth Kneser and Gustave Choquet, states that the Poisson integral of a homeomorphism of
Radó–Kneser–Choquet_theorem
Preference of known risks to unknown risks
probabilities and the expected utility of an act is defined using a Choquet integral. This representation also rationalizes ambiguity aversion and has the
Ambiguity_aversion
Theory of generalized measures in mathematics
), which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different
Fuzzy_measure_theory
Property of some mathematical functions
of substance in a mixture vs. an ideal solution Choquet integral – Subadditive or superadditive integral Superadditivity – Property of a function Triangle
Subadditivity
Concept in probability theory
Comonotonicity is also related to the comonotonic additivity of the Choquet integral. The concept of comonotonicity has applications in financial risk management
Comonotonicity
Risk measure derived by applying a distortion function to a loss distribution
0} almost surely then ρ g {\displaystyle \rho _{g}} is given by the Choquet integral, i.e. ρ g ( X ) = − ∫ 0 ∞ g ( 1 − F − X ( x ) ) d x . {\displaystyle
Distortion_risk_measure
Generalization of the indicator function for classical sets in fuzzy logic
of an outcome given a certain capacity can be found by taking the Choquet integral over the capacity. Defuzzification Fuzzy measure theory Fuzzy set operations
Membership function (mathematics)
Membership_function_(mathematics)
Mathematical framework to model epistemic uncertainty
evidence). He also introduces a relational integral and compares it to the Choquet integral and Sugeno integral. Any relation m between C and L may be introduced
Dempster–Shafer_theory
Property of a function
exposition of this topic may be found in Steele (1997). Choquet integral – Subadditive or superadditive integral Inner measure Subadditivity – Property of some
Superadditivity
Israeli mathematician (1939–2022)
(not-necessarily-additive set function) and expectation is computed by the Choquet integral. While this approach can be used to explain commonly observed behavior
David_Schmeidler
Spanning tree minimizing sum of distances
Spanjaard study the problem under the criterion of minimizing the Choquet integral. Optimal network design - the problem of finding a spanning set (not
Minimum routing cost spanning tree
Minimum_routing_cost_spanning_tree
Mathematical theorem
simpler proof was given by Boris Korenblum. At around the same time Gustave Choquet studied the much more general concept of monotone functions on semigroups
Bernstein's theorem on monotone functions
Bernstein's_theorem_on_monotone_functions
In Euclidean space, a measure of that set's "size"
"capacitable" set was introduced by Gustave Choquet in 1950: for a detailed account, see reference (Choquet 1986). Let Σ be a closed, smooth, (n − 1)-dimensional
Capacity_of_a_set
Equations describing classical electromagnetism
magnetic field corresponds to the negative curl of an electric field. In integral form, it states that the work per unit charge required to move a charge
Maxwell's_equations
French mathematician (1884–1974)
differential equations. His integral was the first to be able to integrate all derivatives. Among his students is Gustave Choquet. He is also known for the
Arnaud_Denjoy
Membrane enclosed compartments in chloroplasts and cyanobacteria
Bibcode:1998COPB....1..217V. doi:10.1016/S1369-5266(98)80107-6. PMID 10066592. Choquet Y, Wostrikoff K, Rimbault B, et al. (2001). "Assembly-controlled regulation
Thylakoid
American mathematician (1926–2013)
have been republished. His 1966 Lectures on Choquet theory was the first book to explain the theory of integral representations. In these "instant classic"
Robert_Phelps
Conditional independence of exchangeable observations
statistics from the statistics of classical (i.e. independent) particles. Choquet theory Hewitt–Savage zero–one law Krein–Milman theorem Invariant sigma-algebra
De_Finetti's_theorem
French mathematician and physicist (1922–2017)
Y. Choquet-Bruhat and Margaret Dillard-Bleick) Analysis, Manifolds and Physics, (1977) I.T. for Intelligent Grandmothers, (1987) (With Y. Choquet-Bruhat)
Cécile_DeWitt-Morette
Type of random variable
random sets could be found scattered throughout publications before Gustave Choquet formalized the concept of a random set. French mathematician Georges Matheron
Random_closed_set
Enzyme
Biochemistry. New York, NY: Wiley, J. ISBN 978-0-470-57095-1. Stroebel D, Choquet Y, Popot JL, Picot D (Nov 2003). "An atypical haem in the cytochrome b(6)f
Cytochrome_b6f_complex
Configuration space in quantum theory
distributions. The example of a scalar field can be found in the references Y. Choquet-Bruhat, C. Dewitt-Morette, M. Dillard-Bleick, Analysis, Manifold, and Physics
Quantum_configuration_space
German-born theoretical physicist (1879–1955)
was so high I could not follow." Einstein recorded that he had "mastered integral and differential calculus" while still just fourteen. His love of algebra
Albert_Einstein
Vol. 2 (3 ed.). New York: Wiley. ISBN 978-0-471-25709-7. OCLC 279852. Choquet, Gustave (1954). "Theory of capacities". Annales de l'Institut Fourier
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Algebraic structure in linear algebra
Schaefer & Wolff 1999, p. 7. Kreyszig 1989, §4.11-5 Kreyszig 1989, §1.5-5 Choquet 1966, Proposition III.7.2. Treves 1967, p. 34–36. Lang 1983, Cor. 4.1.2
Vector_space
Paradox in decision theory
proposed. These include: Choquet expected utility: Created by French mathematician Gustave Choquet was a subadditive integral used as a way of measuring
Ellsberg_paradox
{E} [X]+\mathbb {E} [Y]\leq \mathbb {E} [X+Y]} Choquet expectation: a subadditive or superadditive integral that is used in image processing and behavioral
Nonlinear_expectation
derivatives and integrals in alternative calculi List of equations List of fundamental theorems List of hypotheses List of inequalities Lists of integrals List of
List_of_theorems
Branch of mathematics that studies dynamical systems
functional analysis Carathéodory's extension theorem Hahn Banach theorem Choquet theory Koopman operator People Ludwig Boltzmann Josiah Willard Gibbs Henri
Ergodic_theory
Description of gravity using discrete values
quantum gravity Integral method Causal dynamical triangulation Causal fermion systems Causal Set Theory Covariant Feynman path integral approach Dilatonic
Quantum_gravity
Influence that can change motion of an object
Computing and the Humanities. Inwit Publishing, Inc. Retrieved 2008-01-04. Choquet-Bruhat, Yvonne (2009). General Relativity and the Einstein Equations. Oxford:
Force
Statistical function that defines the quantiles of a probability distribution
F. (2016). "Of quantiles and expectiles: Consistent scoring functions, Choquet representations, and forecast rankings". J. R. Stat. Soc. B. 78 (3): 505–562
Quantile_function
Physics concept expressed as E = mc²
Erhaltungssätze und die Theorie der räumlich-geschlossenen Welt" [On the integral form of the conservation laws and the theory of the spatially closed world]
Mass–energy_equivalence
Theorem about the dual of a Hilbert space
{\displaystyle A(\cdot )} or A − 1 ( ⋅ ) . {\displaystyle A^{-1}(\cdot ).} Choquet theory – Area of functional analysis and convex analysis Covariance operator –
Riesz_representation_theorem
Debate about credit for general relativity
[scalar] in the Hamiltonian integral' ('insbesondere die Verwendung der Riemannschen Krümmung unter dem Hamiltonschen Integral') was claimed as one of his
General relativity priority dispute
General_relativity_priority_dispute
French mathematician
the Ecole Normale Supérieure in 1965. Her instructors included Gustave Choquet, Henri Cartan, and Laurent Schwartz. After her son was born in 1967, her
Michèle_Artigue
Property of functions which is weaker than continuity
vector space is upper semicontinuous. This fact is used in the proof of the Choquet theorem. Similar ideas applied to subharmonic functions are used in the
Semi-continuity
French-Canadian animated comedy adventure television series
Television and Film. McFarland & Co. p. 290. ISBN 9781476672939. Mégane Choquet (25 March 2020). "Ratz sur Netflix : pourquoi (re)découvrir cette série
Ratz_(TV_series)
Measured time difference as explained by relativity theory
{v(t')}{c}}\right)^{2}}}dt'} In the case where v(0) = v0 = 0 and τ(0) = τ0 = 0 the integral can be expressed as a logarithmic function or, equivalently, as an inverse
Time_dilation
Linear perturbations to solutions of nonlinear Einstein field equations
when the gravitational field is weak. The usage of linearized gravity is integral to the study of gravitational waves and weak-field gravitational lensing
Linearized_gravity
American mathematician
Convexity methods and the Choquet boundary in Banach algebras. Boll. Unione Mat. Ital., V. Ser., A 15, 131–136 (1978). Integral representation theorems
George_Maltese
Theory of gravitation as curved spacetime
Introduction to Special and General Relativity. Springer. ISBN 978-0-387-98641-8. Choquet-Bruhat, Yvonne (2008). General relativity and the Einstein equations. Oxford
General_relativity
Mathematical model combining space and time
proper time between the respective events along the curve (i.e. the path integral) to calculate the total amount of proper time experienced by the traveling
Spacetime
Infinite sum
Topology: Chapters 1–4. Springer. pp. 261–270. ISBN 978-3-540-64241-1. Choquet, Gustave (1966). Topology. Academic Press. pp. 216–231. ISBN 978-0-12-173450-3
Series_(mathematics)
Solution to the Einstein field equations
{r_{\text{s}}}{r}}\right)^{-1}\,dr^{2}+r^{2}\,d\varphi ^{2},} yields an integral expression for w(r): w ( r ) = ∫ d r r r s − 1 = 2 r s r r s − 1 + constant
Schwarzschild_metric
Smallest convex set containing a given set
extreme points. Choquet theory extends this theory from finite convex combinations of extreme points to infinite combinations (integrals) in more general
Convex_hull
Laws in physics about force and motion
doi:10.1088/1751-8113/47/42/424011. ISSN 1751-8113. S2CID 122180759. Choquet-Bruhat, Yvonne (2009). General Relativity and the Einstein Equations. Oxford:
Newton's_laws_of_motion
Probability theory for low quality data
statistics and non-parametric statistics. Included are also concepts based on Choquet integration, and so-called two-monotone and totally monotone capacities
Imprecise_probability
Transmembrane protein family
1016/S0896-6273(02)00693-1. PMID 12062022. S2CID 15936250. Bats C, Groc L, Choquet D (March 2007). "The interaction between Stargazin and PSD-95 regulates
AMPA_receptor
Set of spacetime events, light-connected to a given event
to illustrate this property of Lorentz transformations. Elsewhere, an integral part of light cones is the region of spacetime outside the light cone at
Light_cone
Hungarian-American mathematician (1893–1974)
independently of Théophile De Donder. These were later used by Yvonne Choquet-Bruhat in her proof of the local existence and uniqueness of solutions
Cornelius_Lanczos
French mathematician (born 1952)
1016/j.crma.2006.01.026. ISSN 1631-073X. Talagrand, Michel (1984). Pettis integral and measure theory. Providence, R.I., USA: American Mathematical Society
Michel_Talagrand
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
General Relativity. Addison Wesley. pp. 263, 344. ISBN 9780805387322. Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile (1977), Analysis, Manifolds and
Killing_vector_field
Book by Claude Dellacherie and Paul-André Meyer
Potential as follows: Volume 1 covers integration, analytic sets, and Gustave Choquet's theory of capacities, and includes an introduction to stochastic processes
Probabilities_and_Potential
German physicist (1873–1916)
exposure time, and the resulting contrast on a photographic plate. An integral part of that theory is the Schwarzschild exponent (astrophotography). In
Karl_Schwarzschild
Chinese-American mathematician (born 1949)
1142/9789812779533. ISBN 978-981-277-952-6. MR 2431658. Zbl 1203.58004. Choquet-Bruhat, Yvonne (2009). General relativity and the Einstein equations. Oxford
Shing-Tung_Yau
Constals A, Schulze K, Sobolevsky AI, Rosconi MP, Gouaux E, Tampe R, Choquet D, Cognet L, 2010. Dynamic superresolution imaging of endogenous proteins
Single-particle_trajectory
Exoplanets confirmed or discovered from direct imaging
1051/0004-6361/202244727. ISSN 0004-6361. Milli J, Hibon P, Christiaens V, Choquet É, Bonnefoy M, Kennedy GM, et al. (1 January 2017). "Discovery of a low-mass
List of directly imaged exoplanets
List_of_directly_imaged_exoplanets
American mathematician (1935–2019)
Structure Analysis of Compact Convex Sets (a significant contribution to the Choquet School) [Eff65, Eff08]. Nuclear C*-algebras and related topics in the mid-1970s
Edward_George_Effros
Surface that locally minimizes its area
1016/j.cell.2013.06.031. ISSN 0092-8674. PMC 3767119. PMID 23870120. Yvonne Choquet-Bruhat. General relativity and the Einstein equations. Oxford Mathematical
Minimal_surface
French mathematician, physicist and engineer (1854–1912)
of these equations. He not only faced the question of determining the integral of such equations, but also was the first person to study their general
Henri_Poincaré
Theory of interwoven space and time by Albert Einstein
Relativity Archived 2013-03-21 at the Wayback Machine – An algebraic and integral calculus derivation for E = mc2. MathPages – Reflections on Relativity
Special_relativity
Branch of differential geometry
want to know what these definitions are about. Gauss–Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is
Riemannian_geometry
In geometry, set whose intersection with every line is a single line segment
space Carathéodory's theorem (convex hull) Choquet theory Helly's theorem Holomorphically convex hull Integrally-convex set John ellipsoid Pseudoconvexity
Convex_set
Thought experiment in special relativity
second approach calculates a straightforward but technically complicated integral to determine how the travelling twin measures the elapsed time on the stay-at-home
Twin_paradox
subset of P with d+1 or fewer points such that x lies in its convex hull. Choquet theory - an area of functional analysis and convex analysis concerned with
List_of_convexity_topics
Isomorphism of differentiable manifolds
proof was provided shortly afterwards by Hellmuth Kneser. In 1945, Gustave Choquet, apparently unaware of this result, produced a completely different proof
Diffeomorphism
Generalization of straight line to a curved space time
expression of f into the Euler–Lagrange equation (which makes the value of the integral l stationary), gives d d τ ∂ − g μ ν x ˙ μ x ˙ ν ∂ x ˙ λ = ∂ − g μ ν x
Geodesics in general relativity
Geodesics_in_general_relativity
Mathematics course at the Collège de France
de la dérivation et de la transformation de Fourier 1946–1947 Gustave Choquet Propriétés topologiques des fonctions, applications à la géométrie et à
Peccot_Lectures
Gravitational-wave signal detected in 2017
electromagnetic radiation (EM). Dafermos, Mihalis (4 June 2025). "Yvonne Choquet-Bruhat obituary: mathematician who established that Einstein's equations
GW170817
Astronomical object orbiting Kappa Andromedae
.159...40U. doi:10.3847/1538-3881/ab5afa. S2CID 208248220. Godoy, N.; Choquet, E.; Serabyn, E.; Mâlin, M.; Tremblin, P.; Danielski, C.; Lagage, P. O
Kappa_Andromedae_b
Function in mathematical analysis
theorem for Brownian motion Legendre transform and Lipschitz approximation Choquet, G. (1969). Topologie : espaces topologiques et espaces métriques, fonctions
Modulus_of_continuity
Statistical estimator for ratio of means
almost unbiased estimators for population ratio. Statistics 18: 119-121 Choquet D, L'ecuyer P, Léger C (1999) Bootstrap confidence intervals for ratios
Ratio_estimator
Representation of the symmetry group of spacetime in special relativity
the Weyl group such that if μ is a dominant integral weight, then w0 ⋅ (−μ) is again a dominant integral weight. If π μ 0 {\displaystyle \pi _{\mu _{0}}}
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Partial differential equation
instead. His work is essentially a simpler Riemannian version of Yvonne Choquet-Bruhat's well-known proof and interpretation of well-posedness for the
Ricci_flow
British astrophysicist (1882–1944)
Nature of the Physical World, 276–81. The idealist conclusion was not integral to his epistemology but was based on two main arguments. The first derives
Arthur_Eddington
American theoretical physicist (1911–2008)
connecting the asymptotic behavior of an arbitrary particular solution [of the integral equations] with that of solutions of a standard form". Wheeler did not
John_Archibald_Wheeler
Concept in differential geometry
1007/978-3-540-74311-8. ISBN 3-540-15279-2. MR 0867684. Zbl 0613.53001. Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile; Dillard-Bleick, Margaret (1982)
Exterior_covariant_derivative
administrator YoungJu Choie (born 1959), Korean number theorist Yvonne Choquet-Bruhat (1923–2025), French mathematician and physicist, first woman elected
List_of_women_in_mathematics
multiplication) Gustave Choquet, Existence et unicité des représentations intégrales au moyen des points extrémaux dans les cônes convexes (Choquet theory) Jacques
Séminaire Nicolas Bourbaki (1950–1959)
Séminaire_Nicolas_Bourbaki_(1950–1959)
Reformulation of general relativity
^{3}\mathbf {r} =0\,,} (where d3r is the volume element of the volume integral). So the constructive interference of the matter waves is a maximum. This
Hamilton–Jacobi–Einstein equation
Hamilton–Jacobi–Einstein_equation
General relativity model near spacetime singularities
multiple general integrals, and each of those may contain only a finite subset of all possible initial conditions. Each of those integrals may contain all
BKL_singularity
Intrinsic geometric structures in mathematics
translated from Russian by V. V. Goldberg with a foreword by S. S. Chern. Choquet-Bruhat, Yvonne; Dewitt-Morette, Cécile; Dillard-Bleick, Margaret (1982)
Riemannian connection on a surface
Riemannian_connection_on_a_surface
Mathematics of convex functions and sets
convex set in a locally convex space is generated by its extreme points. Choquet theory refines this idea by representing points of compact convex sets
Convex_analysis
relativistic stars, monograph), Jean Chazy (Chazy-Curzon vacuum), Yvonne Choquet-Bruhat (formerly Yvonne Bruhat; local existence and uniqueness of solutions
List of contributors to general relativity
List_of_contributors_to_general_relativity
Result in general relativity
interpreted as a family or congruence of nonintersecting world lines via the integral curve, not necessarily geodesics), Raychaudhuri's equation in D {\displaystyle
Raychaudhuri_equation
Exact solution in general relativity
{r^{4}(E-1)+2Mr^{3}-(Q^{2}+L^{2})r^{2}+2ML^{2}r-Q^{2}L^{2}}}}.} Multiplying under the integral sign by S 2 {\displaystyle S_{2}} yields the orbital equation c ∫ L r 2
Reissner–Nordström_metric
maximal analytic extension of the Schwarzschild metric. 1952 – Yvonne Choquet-Bruhat proves that the initial-value problem of the Einstein field equations
Timeline of gravitational physics and relativity
Timeline_of_gravitational_physics_and_relativity
Static exact solution in general relativity
and the area A = 4 π r 2 {\displaystyle A=4\pi r^{2}} , the integral for the proper volume is V = ∫ 0 r g A g r r d r = 2 π ( r g 9 / 2 arcsin
Interior_Schwarzschild_metric
Mathematical formulation of special and general relativity
relativistically invariant quantities, take the action as proportional to the integral of the Lorentz invariant line element in spacetime, the length of the particle's
Relativistic Lagrangian mechanics
Relativistic_Lagrangian_mechanics
Teichmüller space of a Fuchsian group. By the Radó–Kneser–Choquet theorem, the Poisson integral F f ( r e i θ ) = 1 2 π ∫ 0 2 π f ( φ ) ⋅ 1 − r 2 1 − 2
Douady–Earle_extension
Relativistic correction
pendulum. The angle of rotation in both cases is determined by the area integral of curvature in agreement with the Gauss–Bonnet theorem. Thomas precession
Thomas_precession
Protein-coding gene in the species Homo sapiens
Poulain-Godefroy O, Bonnefond A, Hara T, Yengo L, Kimura I, Leloire A, Liu N, Iida K, Choquet H, Besnard P, Lecoeur C, Vivequin S, Ayukawa K, Takeuchi M, Ozawa K, Tauber
Free_fatty_acid_receptor_4
derivatives have some common features including that they are derivatives along integral curves of vector fields. The problem in defining derivatives on manifolds
Mathematics of general relativity
Mathematics_of_general_relativity
Protein-coding gene in humans
(3): 791–5. doi:10.2337/db07-0979. PMID 18162508. Cauchi S, Proença C, Choquet H, Gaget S, De Graeve F, Marre M, et al. (March 2008). "Analysis of novel
CDKAL1
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