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Theorem in topology
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. Brouwer. It states that for any continuous function f {\displaystyle
Brouwer_fixed-point_theorem
Extension of the Brouwer fixed-point theorem
The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to locally convex topological vector spaces, which may be of infinite
Schauder_fixed-point_theorem
Fixed-point theorem for set-valued functions
a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem
Kakutani_fixed-point_theorem
Mapping theorem in topology
Lefschetz fixed-point theorem are generalizations of other classical results in topology like Brouwer fixed-point theorem or Poincare-Hopf theorem. Lefschetz
Lefschetz_fixed-point_theorem
Condition for a mathematical function to map some value to itself
Banach fixed-point theorem Bekić's theorem Borel fixed-point theorem Bourbaki–Witt theorem Browder fixed-point theorem Brouwer fixed-point theorem Rothe's
Fixed-point_theorem
Theorem in topology
The Jordan curve theorem can be proved from the Brouwer fixed-point theorem (in two dimensions), and the Brouwer fixed-point theorem can be proved from
Jordan_curve_theorem
Theorem about metric spaces
the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important
Banach_fixed-point_theorem
Element mapped to itself by a mathematical function
guaranteeing that, if it is satisfied, fixed-point iteration will always converge to a fixed point. The Brouwer fixed-point theorem (1911) says that any continuous
Fixed_point_(mathematics)
Theorems generalizing the Brouwer fixed-point theorem
In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for
Fixed-point theorems in infinite-dimensional spaces
Fixed-point_theorems_in_infinite-dimensional_spaces
Theorem in topology about homeomorphic subsets of Euclidean space
The theorem and its proof are due to L. E. J. Brouwer, published in 1912. The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem
Invariance_of_domain
Concept in topology
was first defined by Brouwer, who showed that the degree is homotopy invariant and used it to prove the Brouwer fixed point theorem. Less general forms
Degree of a continuous mapping
Degree_of_a_continuous_mapping
Dutch mathematician and logician
in his career, Brouwer proved a number of theorems in the emerging field of topology. The most important were his fixed point theorem, the topological
L._E._J._Brouwer
mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete
Caristi_fixed-point_theorem
Solution concept of a non-cooperative game
Kakutani fixed-point theorem in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler Brouwer fixed-point theorem for the same
Nash_equilibrium
Computing the fixed point of a function
the Brouwer fixed-point theorem: that is, f {\displaystyle f} is continuous and maps the unit d-cube to itself. The Brouwer fixed-point theorem guarantees
Fixed-point_computation
In mathematics, the Earle–Hamilton fixed point theorem is a result in geometric function theory giving sufficient conditions for a holomorphic mapping
Earle–Hamilton fixed-point theorem
Earle–Hamilton_fixed-point_theorem
Mathematical property
has the fixed-point property by the Brouwer fixed-point theorem. A retract A {\displaystyle A} of a space X {\displaystyle X} with the fixed-point property
Fixed-point_property
Continuous function on an interval takes on every value between its values at the ends
topology. The Brouwer fixed-point theorem is a related theorem that, in one dimension, gives a special case of the intermediate value theorem. In constructive
Intermediate_value_theorem
list of PPAD-complete problems. Sperner's lemma Brouwer fixed-point theorem Kakutani fixed-point theorem Nash equilibrium Core of Balanced Games Fisher
List of PPAD-complete problems
List_of_PPAD-complete_problems
be proved from Sperner's lemma and can be used to prove the Brouwer fixed-point theorem. Let Δ n − 1 {\displaystyle \Delta _{n-1}} be an ( n − 1 ) {\displaystyle
Knaster–Kuratowski–Mazurkiewicz lemma
Knaster–Kuratowski–Mazurkiewicz_lemma
Generalisation of the intermediate value theorem
equivalent to the Brouwer fixed-point theorem. It is sometimes called the Miranda theorem or the Bolzano–Poincaré–Miranda theorem. The picture on the right
Poincaré–Miranda_theorem
{\displaystyle f(x)-x} is SGDP, then f has a fixed-point. This is a discrete analogue of the Brouwer fixed-point theorem. [3.9] If X = Z n {\displaystyle \mathbb
Discrete_fixed-point_theorem
Plane figure, bounded by circle
be bijective or even surjective); this is the case n=2 of the Brouwer fixed-point theorem. The statement is false for the open disk: Consider for example
Disk_(mathematics)
to Banach spaces implicit function theorems fixed-point theorems (Brouwer fixed point theorem, Fixed point theorems in infinite-dimensional spaces, topological
Nonlinear_functional_analysis
inequality and its applications", and is closely related to the Brouwer fixed-point theorem, though often more convenient for proving equilibrium results
Ky Fan inequality (game theory)
Ky_Fan_inequality_(game_theory)
Abstract strategy board game
also has profound mathematical underpinnings related to the Brouwer fixed-point theorem, matroids and graph connectivity. Hex is a finite, two-player
Hex_(board_game)
German mathematician (1905–1980)
VI (1928) 265–272. Park, Sehie (1999). "Ninety Years of the Brouwer Fixed Point Theorem" (PDF). Vietnam Journal of Mathematics. 27 (3): 187–222. CiteSeerX 10
Emanuel_Sperner
Space where all functions have fixed points
by the Brouwer fixed-point theorem, every compact bounded convex set in a Euclidean space is a fixed-point space. The definition of a fixed-point space
Fixed-point_space
American mathematician (1878–1932)
In 1922 with George David Birkhoff he generalized the Brouwer fixed point theorem to the theorem of Birkhoff–Kellogg. Among his doctoral students was Arthur
Oliver_Dimon_Kellogg
Provides conditions for a parametric optimization problem to have continuous solutions
to do so. Envelope theorem Brouwer fixed point theorem Kakutani fixed point theorem for correspondences Michael selection theorem Ok, Efe (2007). Real
Maximum_theorem
Theorem on triangulation graph colorings
combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring
Sperner's_lemma
Branch of mathematics
Blakers–Massey theorem Borsuk–Ulam theorem Brouwer fixed point theorem Cellular approximation theorem Dold–Thom theorem Eilenberg–Ganea theorem Eilenberg–Zilber
Algebraic_topology
Mathematical theorem
ISSN 0001-7140. Gale, David (December 1979). "The Game of Hex and the Brouwer Fixed-Point Theorem". The American Mathematical Monthly. 86 (10): 818–827. doi:10
Steinhaus_chessboard_theorem
Economic Model
fulfilling Walras's Law is equivalent to Brouwer fixed-Point theorem. Thus, the use of Brouwer's fixed-point theorem is essential for showing that the equilibrium
Arrow–Debreu_model
Surname list
Brouwer fixed-point theorem, Brouwer–Heyting–Kolmogorov interpretation, Brouwer–Hilbert controversy, Kleene–Brouwer order, Phragmen–Brouwer theorem Leo
Brouwer
Theorem in topology
the Borsuk–Ulam theorem states that every continuous map from the sphere to the plane maps some pair of antipodal points to the same point. Informally, for
Borsuk–Ulam_theorem
Theorem in differential topology
theorem was first proven by Henri Poincaré for the 2-sphere in 1885, and extended to higher even dimensions in 1912 by L. E. J. Brouwer. The theorem has
Hairy_ball_theorem
Brouwer fixed-point theorem is a fixed-point theorem in topology, named after Dutchman Luitzen Brouwer, who proved it in 1911. The hairy ball theorem
List_of_Dutch_discoveries
Topics referred to by the same term
characters (also called the Brauer-Tate theorem). Brauer's main theorems Brauer–Suzuki theorem Brouwer fixed-point theorem This disambiguation page lists mathematics
Brauer's_theorem
On tangency patterns of circles
countably many circles. William Thurston's proof is based on the Brouwer fixed point theorem. Another proof uses a discrete variant of Perron's method of
Circle_packing_theorem
Simplified instance of a general theorem
the theorem. A toy theorem of the Brouwer fixed-point theorem is obtained by restricting the dimension to one. In this case, the Brouwer fixed-point theorem
Toy_theorem
Japanese and American mathematician
critic for The New York Times. The Kakutani fixed-point theorem is a generalization of Brouwer's fixed-point theorem, holding for generalized correspondences
Shizuo_Kakutani
Theorem in mathematical analysis
involves analysis. In topology it is often quoted — as in the Brouwer fixed-point theorem and some applications in Morse theory — in order to prove the
Sard's_theorem
Bolzano–Weierstrass theorem (real analysis, calculus) Borsuk–Ulam theorem (topology) Brouwer fixed-point theorem (topology) Cantor's intersection theorem (real analysis)
List_of_theorems
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
that there are smooth counterexamples to an extension of the Brouwer fixed-point theorem to the unit ball in H. The existence of such counter-examples
Kuiper's_theorem
Theory of equilibrium between supply and demand
traditionally rely on fixed-point theorems such as Brouwer fixed-point theorem for functions (or, more generally, the Kakutani fixed-point theorem for set-valued
General_equilibrium_theory
American mathematician
Charles C. Pugh, Michael Shub: Invariant Manifolds, Springer 1977 Brouwer fixed-point theorem Chern's conjecture (affine geometry) Differential structure Homotopy
Morris_Hirsch
invariant-direction theorem, named after G. D. Birkhoff and O. D. Kellogg, is a generalization of the Brouwer fixed-point theorem. The theorem states that: Let
Birkhoff–Kellogg invariant-direction theorem
Birkhoff–Kellogg_invariant-direction_theorem
diverges Banach fixed-point theorem Banach–Tarski paradox Basel problem Bolzano–Weierstrass theorem Brouwer fixed-point theorem Buckingham π theorem (proof in
List_of_mathematical_proofs
In geometry, set whose intersection with every line is a single line segment
(topological vector space) Brouwer fixed-point theorem Complex convexity Convex cone Convex series Convex metric space Carathéodory's theorem (convex hull) Choquet
Convex_set
French mathematician, physicist and engineer (1854–1912)
3-sphere. Poincaré–Miranda theorem: a generalization of the intermediate value theorem to n dimensions. Brouwer fixed-point theorem Epistemic structural realism
Henri_Poincaré
Theorem in linear algebra
when examined from the point of view of point-set topology. A common thread in many proofs is the Brouwer fixed point theorem. Another popular method
Perron–Frobenius_theorem
Hungarian and American mathematician and physicist (1903–1957)
uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem. Von Neumann's model of an expanding economy considered the matrix
John_von_Neumann
Existence of antipodal pairs in covers of spheres
A_{n+1}} contains a pair of antipodal points. There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant
Lusternik–Schnirelmann theorem
Lusternik–Schnirelmann_theorem
American mathematician (1931–2019)
"Topology in the Complex Plane", which described the Brouwer fixed point theorem, the Jordan curve theorem, and Alexander duality. With Hiroshi Yamaguchi,
Andrew_Browder
Mathematical models of strategic interactions
The book built upon von Neumann's earlier work, which used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, a method that
Game_theory
Function whose values are sets (mathematics)
1016/0022-247X(65)90049-1. Kakutani, Shizuo (1941). "A generalization of Brouwer's fixed point theorem". Duke Mathematical Journal. 8 (3): 457–459. doi:10.1215/S0012-7094-41-00838-4
Set-valued_function
How spheres of various dimensions can wrap around each other
theorem of algebra, which states that every non-constant complex polynomial has a zero. The fact that πn−1(Sn−1) = Z implies the Brouwer fixed point theorem
Homotopy_groups_of_spheres
Abstract strategy game subgenre
tactics and a profound mathematical underpinning related to the Brouwer fixed-point theorem. The game was first marketed as a board game in Denmark under
Connection_game
Combinatorial analog of the Borsuk-Ulam theorem
not too much hope for finding a fast algorithm. There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant
Tucker's_lemma
Properties that homology theories of topological spaces have in common
not a retract of the n-disk. This is used in a proof of the Brouwer fixed point theorem. A "homology-like" theory satisfying all of the Eilenberg–Steenrod
Eilenberg–Steenrod_axioms
Algebraic structure associated with a topological space
Mayer-Vietoris sequences. Notable theorems proved using homology include the following: The Brouwer fixed point theorem: If f is any continuous map from
Homology_(mathematics)
Concept in Nielsen theory
x0) to be the Brouwer degree of the mapping induced by g on some suitably chosen small sphere around x0. The importance of the fixed-point index is largely
Fixed-point_index
Matrix used to describe the transitions of a Markov chain
absolute value of all its eigenvalues is also 1. Finally, the Brouwer Fixed Point Theorem (applied to the compact convex set of all probability distributions
Stochastic_matrix
Japanese economist (1928–2014)
and Uzawa's Theorem, among others. In his 1962 paper, Uzawa proved that the two of Walrasian equilibrium and Brouwer's fixed-point theorem are equivalent
Hirofumi_Uzawa
Every graph has evenly many odd vertices
ISBN 978-1-316-61044-2, MR 3496604 Gale, David (1979), "The game of Hex and the Brouwer fixed-point theorem", The American Mathematical Monthly, 86 (10): 818–827, doi:10
Handshaking_lemma
Polish mathematician (1899–1943)
Schauder is best known for the Schauder fixed-point theorem, an extension of the Brouwer fixed-point theorem to locally convex topological vector spaces
Juliusz_Schauder
Concept in algebraic topology
This is essential to applications of the theory, including the Brouwer fixed point theorem and the topological invariance of simplicial homology. Singular
Simplicial_homology
Continuous, position-preserving mapping from a topological space into a subspace
that is, the (n−1)-sphere, is not a retract of the ball. (See Brouwer fixed-point theorem § A proof using homology or cohomology.) A closed subset X {\textstyle
Retraction_(topology)
identify the Mohorovičić discontinuity. L. E. J. Brouwer makes a proof of the Brouwer fixed-point theorem. August 30 – Discovery of the Burgess Shale Cambrian
1909_in_science
Economic equilibrium concept
\forall i\in 1,\dots ,k} This is a continuous function, so by the Brouwer fixed-point theorem there is a price vector p ∗ {\displaystyle p^{*}} such that:
Competitive_equilibrium
Egbertus Jan Brouwer presents the Brouwer fixed-point theorem. 1912 – Josip Plemelj publishes simplified proof for the Fermat's Last Theorem for exponent
Timeline_of_mathematics
Italian mathematician (1904–1959)
approach. Proceeding in this way, in 1931 he extended the Brouwer fixed point theorem, applying the results obtained both from ordinary differential
Renato_Caccioppoli
Mathematical group of the homotopy classes of loops in a topological space
integers. This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk–Ulam theorem in dimension 2. The fundamental group of the figure
Fundamental_group
Condition in monotone comparative statics
{\partial ^{2}V}{\partial \theta \partial q}}(q,\theta )>0} . Brouwer fixed-point theorem The property need not only relate to continuous functions but
Single-crossing_condition
that can be solved by computing a fixed point of a function that satisfies the conditions of Brouwer's fixed point theorem. More formally, FIXP contains search
FIXP
Branch of mathematical logic
Riemann integrable. The Brouwer fixed point theorem (for continuous functions on an n-simplex). The separable Hahn–Banach theorem in the form: a bounded
Reverse_mathematics
Isomorphism of differentiable manifolds
as this enlarged space was homeomorphic to a closed ball, the Brouwer fixed-point theorem became applicable. Smale conjectured that if M {\displaystyle
Diffeomorphism
Proof assistant program
the Hahn–Banach theorem, Kőnig's lemma, the Brouwer fixed point theorem, Gödel's completeness theorem, and the Jordan curve theorem. This breadth of
Mizar_system
2008 mathematics book
characteristic of Seifert surfaces, the Poincaré–Hopf theorem, the Brouwer fixed point theorem, Betti numbers, and Grigori Perelman's proof of the Poincaré
Euler's_Gem
Branch of applied mathematics
in economic theory, in particular, fixed-point theory through his generalization of Brouwer's fixed-point theorem. Following von Neumann's program, Kenneth
Mathematical_economics
Logical principle
mathematical theorems are often proved by establishing that the negation would involve us in a contradiction, this third possibility which Brouwer suggested
Law_of_excluded_middle
Notable events in the history of geometry
the exterior derivative, 1912 – Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem, 1916 – Einstein's theory of general relativity. 1930
Timeline_of_geometry
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 group decision-making
Arrow's_impossibility_theorem
Latvian mathematician (1865–1921)
three-dimensional case of the Brouwer fixed-point theorem, but his work was not noticed at the time. In 1908, Bohl established a general theorem for locating the roots
Piers_Bohl
Algebraic topology uses abstract algebra to study topological spaces
Applications Jordan curve theorem Brouwer fixed point theorem Invariance of domain Lefschetz fixed-point theorem Hairy ball theorem Degree of a continuous
List of algebraic topology topics
List_of_algebraic_topology_topics
American mathematician (1921–2008)
Mathematical Monthly 81(1974), pp. 876–879. The game of Hex and the Brouwer fixed-point theorem. American Mathematical Monthly 86(1979), pp. 818–827. The strategy
David_Gale
Axiomatic set theories based on the principles of mathematical constructivism
equivalent to the Brouwer fixed point theorem and other theorems regarding values of continuous functions on the reals. The fixed point theorem in turn implies
Constructive_set_theory
Foundational controversy in twentieth-century mathematics
published a number of important papers, in particular the fixed-point theorem. Hilbert admired Brouwer and helped him receive a regular academic appointment
Brouwer–Hilbert_controversy
conjugate Fenchel's inequality Fixed-point theorems in infinite-dimensional spaces, generalise the Brouwer fixed-point theorem. They have applications, for
List_of_convexity_topics
Mathematics glossary
theorem for orthogonal groups say: π q O = π q + 8 O , q ≥ 0 {\displaystyle \pi _{q}O=\pi _{q+8}O,q\geq 0} . Brouwer fixed-point theorem The Brouwer fixed-point
Glossary of algebraic topology
Glossary_of_algebraic_topology
Result of repeatedly applying a mathematical function
guarantee the existence of fixed points in various situations, including the Banach fixed point theorem and the Brouwer fixed point theorem. There are several
Iterated_function
{C} }^{n}} . By Brouwer's fixed point theorem, any holomorphic isometry of the complex hyperbolic space must fix at least one point in D ¯ {\displaystyle
Complex_hyperbolic_space
Basic framework of mathematics
generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical
Foundations_of_mathematics
Method for dividing a simplicial complex
instance in Lefschetz's fixed-point theorem. The Lefschetz number is a useful tool to find out whether a continuous function admits fixed-points. This data
Barycentric_subdivision
Continuous mappings can be approximated by ones that are piecewise simple
mapping by a homotopic one. This theorem was first proved by L.E.J. Brouwer, by use of the Lebesgue covering theorem (a result based on compactness).[citation
Simplicial approximation theorem
Simplicial_approximation_theorem
On the existence of a continuous selection of a multivalued map from a paracompact space
MR 0077107. "proof verification - Reducing Kakutani's fixed-point theorem to Brouwer's using a selection theorem". Mathematics Stack Exchange. Retrieved 2019-10-29
Michael_selection_theorem
Manifold or algebraic variety of dimension n in a space of dimension n+1
into two connected components; this is related to the Jordan–Brouwer separation theorem. An algebraic hypersurface is an algebraic variety that may be
Hypersurface
Theorem which asserts the existence of an object
to Differential Equations in Banach Spaces : An Introduction to Fixed Point Theorems and their Applications. KIT Scientific Publishing. p. 31. ISBN 978-3-7315-0260-9
Existence_theorem
Theorem in game theory
p_{a}(E|\pi _{a})=x_{a}} for some fixed number x a {\displaystyle x_{a}} . In this model, Aumann's agreement theorem claims that if X {\displaystyle X}
Aumann's_agreement_theorem
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BROUWER FIXED-POINT-THEOREM
BROUWER FIXED-POINT-THEOREM
BROUWER FIXED-POINT-THEOREM
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BROUWER FIXED-POINT-THEOREM
BROUWER FIXED-POINT-THEOREM
BROUWER FIXED-POINT-THEOREM
BROUWER FIXED-POINT-THEOREM
BROUWER FIXED-POINT-THEOREM
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