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Tool for working with matrices
Birkhoff's algorithm (also called Birkhoff-von-Neumann algorithm) is an algorithm for decomposing a bistochastic matrix into a convex combination of permutation
Birkhoff_algorithm
American mathematician (1911–1996)
Garrett Birkhoff. Springer Science & Business Media. ISBN 978-0-8176-3114-7. Birkhoff algorithm Birkhoff's condition Birkhoff polytope Birkhoff's representation
Garrett_Birkhoff
Polytope
It is related to the Ford–Fulkerson algorithm that computes the maximum flow in a flow network. Birkhoff algorithm Permutohedron Stable matching polytope
Birkhoff_polytope
Type of square matrix
forward by R. M. Caron et al. Stochastic matrix Unistochastic matrix Birkhoff algorithm Marshal, Olkin (1979). Inequalities: Theory of Majorization and Its
Doubly_stochastic_matrix
Topics referred to by the same term
matrices. It can be found by the Birkhoff algorithm. This disambiguation page lists articles associated with the title Birkhoff decomposition. If an internal
Birkhoff_decomposition
Surname list
different decompositions Birkhoff algorithm Poincaré–Birkhoff–Witt theorem This page lists people with the surname Birkhoff. If an internal link intending
Birkhoff
An algorithm is a fundamental set of rules or defined procedures that are typically designed and used to be a simpler way to solve a specific problem
List_of_algorithms
Statement of spherically symmetric spacetimes
In general relativity, Birkhoff–Jebsen's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically
Birkhoff's theorem (relativity)
Birkhoff's_theorem_(relativity)
Matrix decomposition in mathematics
In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by George David Birkhoff (1909), is a generalization of the LU decomposition
Birkhoff_factorization
Algorithm for allocating divisible objects
convex combination of permutation matrices. This can be done by the Birkhoff algorithm. It is guaranteed to find a combination in which the number of permutation
Simultaneous_eating_algorithm
Theorem in convex and algebraic geometry
the set of non-decomposable multi-hypergraphs is finite. Birkhoff algorithm is an algorithm that, given a bistochastic matrix (a matrix which solves a
Gordan's_lemma
Methodic assignment of colors to elements of a graph
for being the first major computer-aided proof. In 1912, George David Birkhoff introduced the chromatic polynomial to study the coloring problem, which
Graph_coloring
of the things (and topics) listed below. Birkhoff–von Neumann algorithm Birkhoff–von Neumann theorem Birkhoff–von Neumann decomposition Dirac–von Neumann
List of things named after John von Neumann
List_of_things_named_after_John_von_Neumann
Mathematical function defined piecewise by polynomials
at Citroën, Pierre Bézier at Renault, and Birkhoff, Garabedian, and de Boor at General Motors (see Birkhoff and de Boor, 1965), all for work occurring
Spline_(mathematics)
Largest integer that divides given integers
Garrett Birkhoff. A Survey of Modern Algebra, Fourth Edition. MacMillan Publishing Co., 1977. ISBN 0-02-310070-2. 1–7: "The Euclidean Algorithm." gcd(x
Greatest_common_divisor
Problem of fair division
in time O*(2k) by dynamic programming, where k is the size of the Birkhoff algorithm (k ≤ n2). They conjecture that minimizing the largest amount of switches
Rental_harmony
Hungarian and American mathematician and physicist (1903–1957)
paper by von Neumann and Garrett Birkhoff, the first to introduce quantum logics, wherein von Neumann and Birkhoff first proved that quantum mechanics
John_von_Neumann
Visual depiction of a partially ordered set
ISBN 978-3-540-57273-2 Birkhoff, Garrett (1948), Lattice Theory (Revised ed.), American Mathematical Society Chan, Hubert (2004), "A parameterized algorithm for upward
Hasse_diagram
matrix – the sum of elements in each row and each column is one. Birkhoff's algorithm can be used to decompose the matrix into a convex sum of at most
Fractional_matching
Function in algebraic graph theory
function of the number of colors and was originally defined by George David Birkhoff to study the four color problem. It was generalised to the Tutte polynomial
Chromatic_polynomial
colour refinement algorithm also known as the naive vertex classification, or the 1-dimensional version of the Weisfeiler-Leman algorithm, is a routine used
Colour_refinement_algorithm
Fair division problem
In the classic setting, in which m=n, this can be done using the Birkhoff algorithm. It can decompose any n-by-n matrix of agent-object probabilities
Fair_random_assignment
Technique to find exact solutions to Einstein field equations
theories. Physics portal Mathematics portal Astronomy portal Science portal Birkhoff's theorem (relativity) Newman, E. T.; Janis, A. I. (June 1965). "Note on
Newman–Janis_algorithm
American mathematician (1927 to 1992)
acquainted with Garrett Birkhoff. Together they made the Pierce-Birkhoff conjecture, which concerned a minimax approximation algorithm for polynomials. In
Richard_S._Pierce
Polynomial without nontrivial factorization
Quadratic equation § Quadratic factorization Gallian 2012, p. 311 Mac Lane & Birkhoff 1999 do not explicitly define "reducible", but they use it in several places
Irreducible_polynomial
zero matrix Algorithms for matrix multiplication: Strassen algorithm Coppersmith–Winograd algorithm Cannon's algorithm — a distributed algorithm, especially
List of numerical analysis topics
List_of_numerical_analysis_topics
dissection was introduced by George (1973); the name was suggested by Garrett Birkhoff. Nested dissection consists of the following steps: Form an undirected
Nested_dissection
Smallest complete lattice containing a partial order
than or equal to x. The result is a distributive lattice and is used in Birkhoff's representation theorem. However, it may have many more elements than are
Dedekind–MacNeille_completion
Sequence acceleration method in numerical analysis
trapezoid rule, and the Bulirsch–Stoer algorithm for solving ordinary differential equations. In the words of Birkhoff and Rota, "its usefulness for practical
Richardson_extrapolation
Graph with a median for each three vertices
. The concept of median graphs has long been studied, for instance by Birkhoff & Kiss (1947) or (more explicitly) by Avann (1961), but the first paper
Median_graph
Type of mathematical expression
field. See "polynomial" and "binomial", Compact Oxford English Dictionary Birkhoff & Lane 1997, p. 72. Sahai & Bist 2002, p. 20. Young 2022, p. 346. Beauregard
Polynomial
Field of mathematics and science based on non-linear systems and initial conditions
theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies topological transitivity. The Birkhoff Transitivity
Chaos_theory
American mathematician (1935–2020)
graph theory, the Coffman–Graham algorithm for approximate scheduling and graph drawing, and the Graham scan algorithm for convex hulls. He also began
Ronald_Graham
Algebra whose elements are stable matchings
"The NRMP matching algorithm revisited", Academic Medicine, 70 (6): 477–84, doi:10.1097/00001888-199506000-00008, PMID 7786367 Birkhoff, Garrett (1937),
Lattice_of_stable_matchings
Number in {..., –2, –1, 0, 1, 2, ...}
of N is denoted by Z; its elements are called the rational integers.] Birkhoff, Garrett (1948). Lattice Theory (Revised ed.). American Mathematical Society
Integer
Matroid with graph forests as independent sets
co-graphic matroids", pp. 19–20, and section 9, "Graphic matroids", pp. 38–47. Birkhoff, Garrett (1995), Lattice Theory, Colloquium Publications, vol. 25 (3rd ed
Graphic_matroid
Combinitorics of Polyhedra
the Birkhoff polytope lies, and the non-negativity constraints define facets of the Birkhoff polytope within that subspace. However, the Birkhoff polytope
Polyhedral_combinatorics
Property of operations
Basel: Birkhäuser. ISBN 978-3-7643-6996-5. Here: Sect.1.2, p.5. Garrett Birkhoff (1967). Lattice Theory. Colloquium Publications. Vol. 25. Providence: Am
Idempotence
Study of systems of inequalitites
the Pierce–Birkhoff conjecture) are also semialgebraic mappings. Computational real algebraic geometry is concerned with the algorithmic aspects of real
Real_algebraic_geometry
American mathematician (1916–1996)
number theory, was supervised by David Bourgin, a student of George David Birkhoff. In the same year, he joined the RAND Corporation, becoming part of what
Selmer_M._Johnson
Print by M. C. Escher
Patterns in Saracenic Art G. H. Hardy A Mathematician's Apology George David Birkhoff Aesthetic Measure Douglas Hofstadter Gödel, Escher, Bach Nikos Salingaros
Relativity_(M._C._Escher)
Branch of discrete mathematics
polytopes are also considered, such as permutohedra, associahedra and Birkhoff polytopes. Combinatorial geometry is a historical name for discrete geometry
Combinatorics
String that is strictly smaller in lexicographic order than all of its rotations
an explicit construction for the commutators appearing in the Poincaré–Birkhoff–Witt theorem needed for the construction of universal enveloping algebras
Lyndon_word
Distance function
AB, BC, AC do not meet the interior of one of the sides of Ω. Garrett Birkhoff used Hilbert's metric and the Banach contraction principle to rederive
Hilbert_metric
Method of deriving an ontology
mathematical theory of lattices and ordered sets that was developed by Garrett Birkhoff and others in the 1930s. Formal concept analysis finds practical application
Formal_concept_analysis
Matrix equation in control theory
.24.4109W. doi:10.1109/TIP.2015.2458572. PMID 26208345. S2CID 665111. Birkhoff and MacLane. A survey of Modern Algebra. Macmillan. pp. 213, 299. Online
Sylvester_equation
Branch of mathematical statistics
Birkhoff metrized the positive cone using Hilbert's projective metric and proved Jentsch's theorem using the contraction mapping theorem. Birkhoff's results
Algebraic_statistics
Order whose elements are all comparable
a < a {\displaystyle a<a} by asymmetry. References Halmos 1968, Ch.14. Birkhoff 1967, p. 2. Schmidt & Ströhlein 1993, p. 32. Fuchs 1963, p. 2. Davey &
Total_order
American mathematician (1900-1982)
Philosophy (Ph.D.) in mathematics. While he was directed by George David Birkhoff to work on differential equations, his interests continued to shift to
Haskell_Curry
Smallest convex set containing a given set
data", Journal of Algorithms, 31 (1): 1–28, CiteSeerX 10.1.1.134.6921, doi:10.1006/jagm.1998.0988, MR 1670903, S2CID 8013433 Birkhoff, Garrett (1935),
Convex_hull
Mathematical model of the physical space
and to make clear the ramifications of the parallel postulate. Birkhoff's axioms: Birkhoff proposed four postulates for Euclidean geometry that can be confirmed
Euclidean_geometry
German artist and theorist (1471–1528)
Patterns in Saracenic Art G. H. Hardy A Mathematician's Apology George David Birkhoff Aesthetic Measure Douglas Hofstadter Gödel, Escher, Bach Nikos Salingaros
Albrecht_Dürer
matrices contains all permutation matrices and its convex hull is the Birkhoff polytope of all doubly stochastic matrices for n ≥ 3 {\displaystyle n\geq
Unistochastic_matrix
Theory of logic to account for observations from quantum theory
The formal system takes as its starting point an observation of Garrett Birkhoff and John von Neumann, that the structure of experimental tests in classical
Quantum_logic
Subset of incomparable elements
Society, 130 (2): 371–378, doi:10.1090/S0002-9939-01-06058-0, MR 1862115 Birkhoff, Garrett (1937), "Rings of sets", Duke Mathematical Journal, 3 (3): 443–454
Antichain
Gröbner bases for non-commutative algebra
an algorithm for obtaining a non-commutative Gröbner basis of the algebra from its defining relations. However, in contrast to Buchberger's algorithm, in
Bergman's_diamond_lemma
of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures List of data structures List of derivatives
List_of_theorems
Division of New York University, US (founded 1935)
Cathleen Synge Morawetz, 1998 National Medal of Science, Steele Prize, Birkhoff Prize, Noether Lecturer, National Academy of Sciences, American Academy
Courant Institute School of Mathematics, Computing, and Data Science
Courant_Institute_School_of_Mathematics,_Computing,_and_Data_Science
Concept in education theory
set union and set intersection. The name "quasi-ordinal" arises from Birkhoff's representation theorem, which explains that distributive lattices uniquely
Knowledge_space
Boolean algebra and every distributive lattice forms a median algebra. Birkhoff and Kiss showed that a median algebra with elements 0 and 1 satisfying
Median_algebra
Polyhedron whose vertices represent permutations
permutation polytopes, but this terminology is also used for the related Birkhoff polytope, defined as the convex hull of permutation matrices. More generally
Permutohedron
Probability of shared birthdays
reprinted in Frank, P.; Goldstein, S.; Kac, M.; Prager, W.; Szegö, G.; Birkhoff, G., eds. (1964). Selected Papers of Richard von Mises. Vol. 2. Providence
Birthday_problem
Theorem about products in model theory
products of algebraic structures (which is a consequence of one direction of Birkhoff's theorem). Consider a first-order logic signature L. The definition of
Feferman–Vaught_theorem
Mathematical set of all subsets of a set
"Subalgebra lattice". Encyclopedia of Mathematics. Retrieved 2025-11-23. Birkhoff, Garrett; Frink, Orrin, Jr. (1948). "Representations of Lattices by Sets"
Power_set
Equations of motion for viscous fluids
to Navier Stokes Equation and Oceanography, Springer ISBN 3-540-35743-2 Birkhoff, Garrett (1960), Hydrodynamics, Princeton University Press Campos, D.(Editor)
Navier–Stokes_equations
Existence of perfect cuboids and associated cuboid conjectures Pierce–Birkhoff conjecture: every piecewise-polynomial f : R n → R {\displaystyle f:\mathbb
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Axiom set used in first-order logic
modern axiomizations of Euclidean geometry are Hilbert's axioms (1899) and Birkhoff's axioms (1932). Using his axiom system, Tarski was able to show that the
Tarski's_axioms
Mathematical concept
Numbers, W. W. Norton & Company, p. 128, ISBN 9780393040029 Mac Lane and Birkhoff (1999) define "linear", "quadratic", "cubic", "quartic", and "quintic"
Degree_of_a_polynomial
American mathematician
Marquette University in 1959. Formally under the guidance of Professor Garrett Birkhoff, Rockafellar completed his Doctor of Philosophy degree in mathematics from
R._Tyrrell_Rockafellar
Family of graphs based on the Fibonacci sequence
is also the graph of a distributive lattice that may be obtained via Birkhoff's representation theorem from a zigzag poset, a partially ordered set defined
Fibonacci_cube
Number used for counting
Springer-Verlag Berlin Heidelberg. ISBN 978-3-662-02310-5. Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3rd ed.). American Mathematical Society. ISBN 978-0-8218-1646-2
Natural_number
Form of interpolation
may be solved by means of the Chinese remainder theorem for polynomials. Birkhoff interpolation is a further generalization where only derivatives of some
Polynomial_interpolation
Differential equation that is linear with respect to the unknown function
Congress on Mathematical Software (pp. 35-41). Springer, Berlin, Heidelberg. Birkhoff, Garrett & Rota, Gian-Carlo (1978), Ordinary Differential Equations, New
Linear_differential_equation
Rare congenital neuromuscular disorder
Günther, Andreas; Tang, Wakana; Heuberger, Jules; Groeneveld, Geert Jan; Birkhoff, Willem; Mercuri, Eugenio; Lochmüller, Hanns; Wood, Claire; Fischer, Dirk;
Spinal_muscular_atrophy
Cathleen Synge Morawetz, 1998 National Medal of Science, Steele Prize, Birkhoff Prize, Noether Lecturer, National Academy of Sciences, American Academy
List of NYU Courant Institute people
List_of_NYU_Courant_Institute_people
American mathematician
and a Ph.D in 1950, working under the supervision of Professor Garrett Birkhoff. Young began his academic career at the University of Maryland, College
David_M._Young_Jr.
American mathematician (born 1952)
Differential Equations" and was written under the supervision of Garrett Birkhoff. He worked as a L.E. Dickson Instructor in Mathematics at University of
Peter_J._Olver
Square of numbers with equal row, column and diagonal totals
Thus, such matrices constitute a subset of doubly stochastic matrix. The Birkhoff–von Neumann theorem states that for any doubly stochastic matrix A {\displaystyle
Magic_square
Numerical method for solving optimal control problems
Ross, I. M.; Proulx, R. J. (September 2019). "Further Results on Fast Birkhoff Pseudospectral Optimal Control Programming" (PDF). Journal of Guidance
Pseudospectral optimal control
Pseudospectral_optimal_control
be seen as an analogue of the theorem of Garrett Birkhoff that an analogous polytope, the Birkhoff polytope describing the set of all fractional matchings
Stable_matching_polytope
Branch of mathematics
60 Rowen 2006, p. 12 Pratt 2022, § 3.3 Birkhoff's Theorem Grätzer 2008, p. 34 Pratt 2022, § 3.3 Birkhoff's Theorem Rowen 2006, p. 12 Gowers, Barrow-Green
Algebra
Extension of the factorial function
Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. Birkhoff, George D. (1913). "Note on the gamma function". Bull. Amer. Math. Soc
Gamma_function
Branch of physics
Milne-Thomson, L. M. (1996). Theoretical hydrodynamics. Courier Corporation. Birkhoff, G. (2015). Hydrodynamics. Princeton University Press. Batchelor, George
Fluid_mechanics
Algebraic structure modeling logical operations
mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff's 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others
Boolean_algebra_(structure)
Statement that all non empty subsets of positive numbers contains a least element
to Analysis. Jones & Bartlett Learning. p. 18. ISBN 978-0-7637-7492-9. Birkhoff, Garrett; Mac Lane, Saunders (1997). A survey of modern algebra. AKP classics
Well-ordering_principle
1949 by Frederick Ungar Publishing Company. Saunders Mac Lane and Garrett Birkhoff A definitive introductory text for abstract algebra using a category theoretic
List of publications in mathematics
List_of_publications_in_mathematics
Methodology for creation of markets
Milgrom (2013). In doing so, the paper has provided a generalization of the Birkhoff-von Neumann Theorem (a mathematical property about Doubly Stochastic Matrices)
Market_design
Canadian theoretical quantum physicist
Christopher J; Spekkens, Robert W (2015-03-03). "The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations
Robert_Spekkens
American mathematician (1919–1985)
in the arithmetic of rationals." Hilbert's tenth problem asks for an algorithm to determine whether a Diophantine equation has any solutions in integers
Julia_Robinson
Mathematical set with an ordering
to a wide class of partial orders, called distributive lattices; see Birkhoff's representation theorem. Sequence A001035 in OEIS gives the number of partial
Partially_ordered_set
Italian-American computer scientist (1926–2017)
a multiplication algorithm for large numbers, which reduces the multiplication time to as little as 3% of the conventional algorithm. In 2003 he won the
Paul_G._Comba
Theorem in Ramsey theory
equivalent form of the theorem in 1978, using ergodic theory. multiple Birkhoff recurrence theorem (Furstenberg and Weiss, 1978)—If X {\textstyle X} is
Van_der_Waerden's_theorem
Particular correspondence between two partially ordered sets
lecture notes, which also attribute the concept to the article cited. Birkhoff 1940, §32; 3rd edition (1967): Ch. V, §7 and §8. Ganter, B. and Wille,
Galois_connection
American mathematician (1928–2022)
Respective Partial Sums. While at Harvard, Varga also studied with Garrett Birkhoff, who later came to collaborate with Varga in research both on iterative
Richard_S._Varga
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
Mathematics Dictionary (2nd ed.), Van Nostrand Reinhold MacLane, Saunders; Birkhoff, Garrett (1967), Algebra, Macmillan Publishers Michel, Anthony N.; Herget
Jordan_normal_form
Idealised system for theoretical analysis
of integrating the equations of motion to determine its Poincaré map. Birkhoff showed that a billiard system with an elliptic table is integrable. The
Dynamical_billiards
Mathematical operator
uses closure (resp. interior) operation Blyth, p. 10 Blyth, p. 10 Garrett Birkhoff. 1967 (1940). Lattice Theory, 3rd ed. American Mathematical Society. Burris
Closure_operator
massive vector meson of spin-1 as a basis for nuclear forces. 1936 – Garrett Birkhoff and John von Neumann introduce Quantum Logic in an attempt to reconcile
Timeline_of_quantum_mechanics
Basic notion of sameness in mathematics
elements of the set. Stoll 1963, pp. 4–5. Lévy 2002, pp. 13, 358. Mac Lane & Birkhoff 1999, p. 2. Mendelson 1964, p. 5. Cook, Roy T. (2009). A Dictionary Of
Equality_(mathematics)
Awarded every year by the American Mathematical Society
a bequest of Leroy P. Steele and were set up in honor of George David Birkhoff, William Fogg Osgood and William Caspar Graustein. The way the prizes are
Leroy_P._Steele_Prize
BIRKHOFF ALGORITHM
BIRKHOFF ALGORITHM
BIRKHOFF ALGORITHM
BIRKHOFF ALGORITHM
Girl/Female
Hindu, Indian
Intelligent or Name of Lord Lakshmi
Boy/Male
Biblical
All.
Girl/Female
Muslim/Islamic
Success height
Boy/Male
Tamil
Arivumadhi | அரீவà¯à®‚மாஂதீ
Intelligent
Male
Hungarian
Hungarian surname derived from Greek Bartholomaios, BARTOS means "son of Talmai."
Boy/Male
English American French
An English surname based on a French place name, meaning big town. Used both as surname and given...
Boy/Male
Tamil
Neelmadhav | நீல மாதவ
Lord Jagannath
Girl/Female
Australian, Czech, Czechoslovakian
Love; Favour; Grace
Girl/Female
Hindu
Thanking, Adored, Praised, Saluted
Girl/Female
Indian, Punjabi, Sikh
A Winner
BIRKHOFF ALGORITHM
BIRKHOFF ALGORITHM
BIRKHOFF ALGORITHM
BIRKHOFF ALGORITHM
BIRKHOFF ALGORITHM
n.
The art of calculating with any species of notation; as, the algorithms of fractions, proportions, surds, etc.
n.
The art of calculating by nine figures and zero.
n.
Alt. of Algorithm