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Theorem in graph theory
In the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number
Menger's_theorem
Formula for the "volume" of an n-simplex
so on. Karl Menger made a further discovery after the development of the Cayley–Menger determinant, which became known as Menger's theorem (unrelated to
Cayley–Menger_determinant
Austrian–American mathematician
credited with Menger's theorem. Outside of mathematics, Menger has substantial contributions to game theory and social sciences. Karl Menger was a student
Karl_Menger
Basic concept of graph theory
analogously. One of the most important facts about connectivity in graphs is Menger's theorem, which characterizes the connectivity and edge-connectivity of a graph
Connectivity_(graph_theory)
Equivalence of optimization problems
case of the duality theorem for linear programs and can be used to derive Menger's theorem and the Kőnig–Egerváry theorem. The theorem equates two quantities:
Max-flow_min-cut_theorem
Limitative results in mathematical logic
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Surname list
Rudolph Menger (1851–1921), American naturalist and physician Menger Hotel, San Antonio Texas Menger sponge, a fractal curve Menger's theorem Menger–Urysohn
Menger
Theorem on extension of bounded linear functionals
In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace
Hahn–Banach_theorem
Lambek–Moser theorem (combinatorics) MacMahon's master theorem (enumerative combinatorics) Menger's theorem (graph theory) Milliken–Taylor theorem (Ramsey
List_of_theorems
Three-dimensional fractal
can also be found within a Menger sponge. The Menger sponge is a closed set; since it is also bounded, the Heine–Borel theorem implies that it is compact
Menger_sponge
MR 2083752. S2CID 119615076.. Aharoni, Ron; Berger, Eli (2009). "Menger's Theorem for infinite graphs". Inventiones Mathematicae. 176 (1): 1–62. arXiv:math/0509397
List of conjectures by Paul Erdős
List_of_conjectures_by_Paul_Erdős
Abstraction of disjoint paths in directed graphs
matroid by Hazel Perfect (1968), based on considerations related to Menger's theorem characterizing the obstacles to the existence of systems of disjoint
Gammoid
Graph which remains connected when k or fewer nodes removed
to find k vertex-independent paths connecting these vertices; see Menger's theorem (Diestel 2005, p. 55). This definition produces the same answer, n − 1
Vertex_connectivity
Theorem on edge-disjoint spanning trees
corollary of the Nash-Williams theorem, every 2k-edge connected graph is k-arboric. Both Nash-Williams' theorem and Menger's theorem characterize when a graph
Nash-Williams_theorem
Invariant of topological spaces
subspaces of the Euclidean spaces, with their usual topology. The Menger–Nöbeling theorem (1932) states that if X {\displaystyle X} is compact metric separable
Inductive_dimension
On forbidden subgraphs in planar graphs
proved by Karl Menger in 1930. Since then, several new proofs of the theorem have been discovered. In the Soviet Union, Kuratowski's theorem was known as
Kuratowski's_theorem
Result in combinatorics and graph theory
mathematics, Hall's marriage theorem, proved by Philip Hall (1935), is a theorem with two equivalent formulations. In each case, the theorem gives a necessary and
Hall's_marriage_theorem
Conjecture on the existence of a cycle in a graph which passes through specified edges
starting with Menger's theorem, guarantee the existence of paths or cycles in a k-connected graph. For 2-connected graphs, Menger's theorem is equivalent
Lovász–Woodall_conjecture
2009) Erdős–Menger conjecture (Ron Aharoni, Eli Berger 2007) Road coloring conjecture (Avraham Trahtman, 2007) Robertson–Seymour theorem (Neil Robertson
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
well-balanced orientations, together with Menger's theorem, immediately implies Robbins' theorem: by Menger's theorem, a 2-edge-connected graph has at least
Strong_orientation
1-measure The Traveling Salesman Theorem was shown to hold in general Euclidean spaces by Kate Okikiolu, that is, the same theorem above holds for sets E ⊆ R
Analyst's traveling salesman theorem
Analyst's_traveling_salesman_theorem
Israeli mathematician
He subsequently proved the appropriate versions of the Kőnig theorem and the Menger theorem for infinite graphs (the latter with Eli Berger). In 2000 he
Ron_Aharoni
number) Lise Meitner, physicist Karl Menger, mathematician (Menger's theorem, Menger sponge); son of Carl Menger) Ronald Micura, chemist Richard von Mises
List_of_Austrian_scientists
This page is a list of network theory topics. Max flow min cut theorem Menger's theorem Metcalfe's law Centrality Betweenness centrality Closeness Bose-Einstein
List_of_network_theory_topics
Lowest number of people removed to disconnect a social group
of a given graph in discrete mathematics. The vertex-cut version of Menger's theorem also proves that the disconnection number is equivalent to a maximally
Structural_cohesion
German mathematician
defining the ends of infinite graphs, for Halin's grid theorem, for extending Menger's theorem to infinite graphs, and for his early research on treewidth
Rudolf_Halin
Any planar graph can be subdivided by removing a few vertices
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split
Planar_separator_theorem
Any individual whose preferences satisfy four axioms has a utility function
In decision theory, the von Neumann–Morgenstern (VNM) utility theorem demonstrates that rational choice under uncertainty involves making decisions that
Von Neumann–Morgenstern utility theorem
Von_Neumann–Morgenstern_utility_theorem
Three-dimensional analog of the Pythagorean theorem
theorem. The generalisation of de Gua's theorem to n-simplices with right-angle corners can also be obtained as a special case from the Cayley–Menger
De_Gua's_theorem
Equation for radii of tangent circles
from the observation that the Cayley–Menger determinant of the four coplanar circle centers is zero. Descartes' theorem is most easily stated in terms of
Descartes'_theorem
Salesman Theorem. Analogous results hold in general metric spaces: Menger-Melnikov curvature of a measure Leymarie, F. (September 2003). "Notes on Menger Curvature"
Menger_curvature
Graph which remains connected when fewer than k edges are removed
disconnects G is a minimum cut in G. The edge connectivity version of Menger's theorem provides an alternative and equivalent characterization, in terms of
Edge_connectivity
four, it does not contain a 5-skein. Using the notion of a skein, Menger's theorem can be formulated as follows: The size of any minimum cut of a given
Skein_(graph_theory)
Austrian economic theory of the purchasing power of money
added to its pre-existing non-monetary demand. The theorem is closely associated with Carl Menger's account of the spontaneous emergence of money, although
Regression_theorem
versions of the Kőnig theorem and the Menger theorem for infinite graphs by Ron Aharoni. Development of the Amitsur–Levitzki theorem by Shimshon Amitsur
List of Israeli inventions and discoveries
List_of_Israeli_inventions_and_discoveries
Mexican mathematician
62: 65–70 (1986) Víctor Neumann-Lara, Luis Montejano "A variation of Menger's theorem for long paths" J. Combin. Theory Ser. B 36: 213–217 (1984) Víctor
Víctor_Neumann-Lara
Directed graph isomorphic to its own transpose graph
101409, hdl:10338.dmlcz/101409. Zelinka, Bohdan (1976b), "Analoga of Menger's theorem for polar and polarized graphs", Czechoslovak Mathematical Journal
Skew-symmetric_graph
\tau (H)} is the size of the smallest edge-cut separating s and t, so Menger's theorem (edge-connectivity version) asserts that ν ( H ) = τ ( H ) {\displaystyle
Sperner_family
proof of Plünnecke's theorem involves a technique known as the "tensor product trick", in addition to an application of Menger's theorem. The Plünnecke–Ruzsa
Plünnecke–Ruzsa_inequality
School of economic thought
The Austrian school originated in 1871 in Vienna with the work of Carl Menger, Eugen von Böhm-Bawerk, Friedrich von Wieser, and others. It was methodologically
Austrian_school_of_economics
Topologically invariant definition of the dimension of a space
Here, S n {\displaystyle S^{n}} is the n-dimensional sphere. Ostrand's theorem on covering dimension. If X is a normal topological space and A {\displaystyle
Lebesgue_covering_dimension
Textbook on the theory of matroids
matching theory and related topics including Hall's marriage theorem, Menger's theorem (an equivalence between minimum cuts and maximum sets disjoint
Independence Theory in Combinatorics
Independence_Theory_in_Combinatorics
Type of metric space in mathematics
mathematics, there are several notions of "convexity" on metric spaces. Karl Menger defined a metric space as convex if any "segment" joining two points in
Convex_metric_space
Branch of mathematics
English for the first time when it was published. Menger proved the following characterization theorem of semimetric spaces: A semimetric space ( R , d
Distance_geometry
In mathematics, invariant of square matrices
multiplication theorem.[clarification needed] The next contributor of importance is Binet (1811, 1812), who formally stated the theorem relating to the
Determinant
Type of metric geometry
x_{i}+\Delta y_{i}=\Delta x_{i}+|f(x_{i})-f(x_{i-1})|.} By the mean value theorem, there exists some point x i ∗ {\displaystyle x_{i}^{*}} between x i {\displaystyle
Taxicab_geometry
Mathematical theorem
Edgeworth's limit theorem is an economic theorem, named after Francis Ysidro Edgeworth, stating that the core of an economy shrinks to the set of Walrasian
Edgeworth's_limit_theorem
Field of economics and game theory
described by Noam Nisan as a way to escape the Gibbard–Satterthwaite theorem. While the theorem is traditionally presented as a result about voting systems, it
Mechanism_design
Proof of social welfare function being pareto efficient given three axioms
In decision theory, the Harsanyi's utilitarian theorem proves mathematically that a rational social welfare function that satisfies the expected utility
Harsanyi's utilitarian theorem
Harsanyi's_utilitarian_theorem
Triangle area in terms of side lengths
incenter and one excircle of the triangle, or as a special case of De Gua's theorem (for the particular case of acute triangles), or as a special case of Brahmagupta's
Heron's_formula
mathematics, a Menger space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Menger space is a
Menger_space
Hungarian and American mathematician and physicist (1903–1957)
the application of this work was instrumental in his mean ergodic theorem. The theorem is about arbitrary one-parameter unitary groups t → V t {\displaystyle
John_von_Neumann
passes through as few of the sensor disks as possible. By applying Menger's theorem to the unit disk graph defined from the barriers, this minimal number
Barrier_resilience
Pathological embedding of the sphere in 3D space
connected, unlike the exterior of the usual round sphere. The Schoenflies theorem in 2D states that any simple closed curve in the plane can be extended
Alexander_horned_sphere
1912 book by Ludwig von Mises
theory. The book also includes the first exposition of Mises's regression theorem, which aimed to explain the purchasing power of money using the subjective
The Theory of Money and Credit
The_Theory_of_Money_and_Credit
Russian mathematician
theory of topological dimension, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which are fundamental results in topology
Pavel_Urysohn
Weakly optimal allocation of resources
per the Greenwald–Stiglitz theorem. The second welfare theorem is essentially the reverse of the first welfare theorem. It states that under similar
Pareto_efficiency
Month in 1902
ancient Babylonian religion. Born: Karl Menger, Austrian-American mathematician who postulated Menger's theorem, in Vienna (d. 1985) Arthur Pue Gorman
January_1902
includes at least one edge from each dicut. According to the Lucchesi-Younger theorem, if the minimum number of edges in a dicut is k {\displaystyle k} , then
Woodall's_conjecture
American mathematician (1925–2020)
of "copulas" into probability theory and proved the theorem that bears his name, Sklar's theorem. That is, that multivariate cumulative distribution functions
Abe_Sklar
Swedish political economist and economic historian (1879–1952)
the Stockholm School of Economics. He is known for the Heckscher–Ohlin theorem, an influential model of international trade that predicts that capital-abundant
Eli_Heckscher
\textstyle \mathbb {U} } . Menger stated and proved the case d = 1 {\displaystyle \textstyle d=1} of the following theorem. The theorem in full generality was
Universal_space
Function that ranks states of society according to their desirability
how much better one choice is compared to another. Arrow's impossibility theorem is a key result of social welfare functions, showing an important difference
Social_welfare_function
Application of mathematical and statistical methods in finance
Financial modeling; Asset pricing. The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes
Mathematical_finance
Economic model of price determination in a market
be determined in equilibrium. However, the Sonnenschein-Mantel-Debreu theorem demonstrates that aggregate demand functions do not necessarily inherit
Supply_and_demand
Subfield of mathematics
mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary
Mathematical_logic
Four-sided polygon
midpoints of the diagonals. This is sometimes known as Euler's quadrilateral theorem and is a generalization of the parallelogram law. The German mathematician
Quadrilateral
and only if a = 1 0 ≤ D(a) ≤ 1 In projective geometry, the Veblen–Young theorem states that a projective geometry of dimension at least 3 is isomorphic
Continuous_geometry
English mathematician (1821–1895)
after him: Cayley's theorem Cayley–Hamilton theorem in linear algebra Cayley–Bacharach theorem Grassmann–Cayley algebra Cayley–Menger determinant Cayley
Arthur_Cayley
Rule in mathematics
spaces having this covering property are called Menger spaces. Hurewicz's reformulation of Menger's property was the first important topological property
Selection_principle
Austrian mathematician (1879–1934)
theorem; the Hahn embedding theorem; the Hahn–Kolmogorov theorem; the Hahn–Mazurkiewicz theorem; the Vitali–Hahn–Saks theorem. Hahn authored the book (Hahn
Hans_Hahn_(mathematician)
table Cayley transform Cayleyan Cayley–Bacharach theorem Cayley–Dickson construction Cayley–Hamilton theorem in linear algebra Cayley-Klein parameters Cayley–Klein
List of things named after Arthur Cayley
List_of_things_named_after_Arthur_Cayley
Croatian-American mathematician
k-dimensional universal Menger compacta. Mathematical Reviews, MR0920964 (89g:54083), 1989 M. Bestvina and M. Feighn, A combination theorem for negatively curved
Mladen_Bestvina
Polish mathematician (1904–1956)
long exact homotopy sequence for fibrations in 1941, and the Hurewicz theorem connecting homotopy and homology groups. His work led to homological algebra
Witold_Hurewicz
In contrast with ordinal utility, in economics
now known as the von Neumann–Morgenstern utility theorem; many similar utility representation theorems exist in other contexts. In 1738, Daniel Bernoulli
Cardinal_utility
Set of points on a line segment with certain topological properties
the discrete two-point space 2 _ {\displaystyle {\underline {2}}} . By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compact
Cantor_set
Property of points all lying on a single line
legs are collinear with the incenter. Pascal's theorem (also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary six points are chosen
Collinearity
American philosopher
of philosophy at the University of Washington. He is known for Fine's theorem in quantum information. Having studied physics, philosophy, and mathematics
Arthur_Fine
Relation between distances of four points
quadrilaterals must obey the triangle inequality. As a special case, Ptolemy's theorem states that the inequality becomes an equality when the four points lie
Ptolemy's_inequality
Mathematical approach
can be viewed from an algebraic point of view (lattice-theoretic). Karl Menger was an early pioneer in the field, and his work on topology without points
Pointless_topology
Field of mathematics and science based on non-linear systems and initial conditions
the system into two open sets. An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit
Chaos_theory
Concept in geometry/topology
metric space is a length metric space (Khamsi & Kirk 2001, Theorem 2.16), a result of Karl Menger. However, the converse does not hold, i.e. there exist length
Intrinsic_metric
German economist (1902–1977)
contributions to decision theory (see von Neumann–Morgenstern utility theorem). He served as a consultant or co-founder for companies including the Market
Oskar_Morgenstern
Type of matrix
ISSN 1860-0980. S2CID 122400126. So (2007), Theorem 2.2.1, p. 10 So (2007), Corollary 3.3.3, p. 42 Menger, Karl (1931). "New Foundation of Euclidean Geometry"
Euclidean_distance_matrix
Solid with six equal square faces
a{\sqrt {3}}} . Both formulas can be determined by using the Pythagorean theorem. The surface area of a cube A {\displaystyle A} is six times the area of
Cube
Austrian philosopher (1879–1944)
the axiom self-evident. The theorem above, however, would then not be demonstrable. The theorem was proven by Karl Menger, the next deontic logician.
Ernst_Mally
German mathematician
Karl Menger and received his PhD in 1931 on a generalization of the embedding theorem, which for one special case can be visualized by the Menger sponge
Georg_Nöbeling
real variables x, as x approaches a point from above or below Squeeze theorem – confirms the limit of a function via comparison with two other functions
List_of_real_analysis_topics
Unforeseen outcomes of an action
Enlightenment, and consequentialism (judging by results). The invisible hand theorem is an example of the unintended consequences of agents acting in their
Unintended_consequences
Approach to economics
uniqueness. However, a result known as the Sonnenschein–Mantel–Debreu theorem suggests that the assumptions that must be made to ensure that equilibrium
Neoclassical_economics
Concept in complex analysis
holomorphic on the set Ω \ K has an analytic extension to all of Ω. By Riemann's theorem for removable singularities, every singleton is removable. This result
Analytic_capacity
Hungarian mathematician
graduated in 1931 with a Ph.D. in mathematics. His advisor there was Karl Menger. Despite Wald's brilliance, he could not obtain a university position because
Abraham_Wald
American mathematician
curve whose solutions characterize the one remaining unsolved case of a theorem of Bilu, Parent & Rebolledo (2013) on the Galois representations of elliptic
Jennifer_Balakrishnan
Fractal named after mathematician Benoit Mandelbrot
{\displaystyle \theta \mapsto } 2 θ {\displaystyle 2\theta } . According to this theorem, when two rays land at the same point, no other rays between them can intersect
Mandelbrot_set
American economist (1847–1938)
same theorem was formulated later independently by John Atkinson Hobson (1891) and Philip Wicksteed (1894). The political message of this theorem is: "[W]hat
John_Bates_Clark
In economics, an imposed cost or benefit
economic decision-making Club good – Type of economic goods Coase theorem – Theorem in economics Externalities of automobiles – Impacts of car usePages
Externality
Atkinson–Stiglitz theorem Where the utility function is separable between labor and all commodities, no indirect taxes need be employed. Aumann's agreement theorem If
Glossary_of_economics
Multi-dimensional generalization of triangle
For a 2-simplex, the theorem is the Pythagorean theorem for triangles with a right angle and for a 3-simplex it is de Gua's theorem for a tetrahedron with
Simplex
English economist and logician (1835–1882)
theory of value. Jevons's work, along with similar discoveries made by Carl Menger in Vienna (1871) and by Léon Walras in Switzerland (1874), marked the opening
William_Stanley_Jevons
NP-hard problem in combinatorial optimization
of the Cambridge Philosophical Society. The Beardwood–Halton–Hammersley theorem provides a practical solution to the travelling salesman problem. The authors
Travelling_salesman_problem
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