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Vectors with given pattern of independence
theory of matroids, a matroid representation is a family of vectors whose linear independence relation is the same as that of a given matroid. Matroid representations
Matroid_representation
Abstraction of linear independence of vectors
In combinatorics, a matroid /ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many
Matroid
Matroid with no linear representation
In mathematics, the Vámos matroid or Vámos cube is a matroid over a set of eight elements that cannot be represented as a matrix over any field. It is
Vámos_matroid
Direct sum of uniform matroids
In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which
Partition_matroid
Abstraction of ordered linear algebra
An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane
Oriented_matroid
Matroid, Inc. is a computer vision company that offers a platform for creating computer vision models, called detectors, to search visual media for objects
Matroid,_Inc.
Abstraction of graph shortest cycles
fixed-parameter tractable for linear matroids when parameterized both by the matroid rank and the field size of a linear representation. The "girth" terminology generalizes
Matroid_girth
Matroid that can be represented over all fields
In mathematics, a regular matroid is a matroid that can be represented over all fields. A matroid is defined to be a family of subsets of a finite set
Regular_matroid
Abstraction of algebraic independence
In mathematics, an algebraic matroid is a matroid, a combinatorial structure, that expresses an abstraction of the relation of algebraic independence.
Algebraic_matroid
Matroid with graph forests as independent sets
In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the
Graphic_matroid
Subroutine for testing independence
mathematics and computer science, a matroid oracle is a subroutine through which an algorithm may access a matroid, an abstract combinatorial structure
Matroid_oracle
Conjecture on forbidden minors of matroids
matroid M {\displaystyle M} ; S {\displaystyle S} is said to be a representation of any matroid isomorphic to M {\displaystyle M} . Not every matroid
Rota's_conjecture
Area of combinatorics
or involve matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group theory and representation theory, lattice
Algebraic_combinatorics
Set without nontrivial polynomial equalities
linearly independent. Every matroid with a linear representation of this type may also be represented as an algebraic matroid, by choosing an indeterminate
Algebraic_independence
Matroid obtained by restrictions and contractions
of matroids, a minor of a matroid M is another matroid N that is obtained from M by a sequence of restriction and contraction operations. Matroid minors
Matroid_minor
Branch of discrete mathematics
nature or involve matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group and representation theory, lattice
Combinatorics
Abstraction of unicyclic subgraphs
In the mathematical subject of matroid theory, the bicircular matroid of a graph G is the matroid B(G) whose points are the edges of G and whose independent
Bicircular_matroid
Number of vectors in any basis of the vector space
|V|=\max(|F|,\dim V).} A vector space can be seen as a particular case of a matroid, and in the latter there is a well-defined notion of dimension. The length
Dimension_(vector_space)
British-Canadian codebreaker and mathematician (1917–2002)
accomplishments, including foundational work in the fields of graph theory and matroid theory. Tutte's research in the field of graph theory proved to be of remarkable
W._T._Tutte
Sequence of locally optimal choices
to solve a class of linear combinatorial optimization problems with a matroid structure. Later Bernhard Korte and László Lovász characterized a broader
Greedy_algorithm
Colombian mathematician
also active as a DJ. His research is in combinatorics, with a focus on matroid theory. Ardila is currently a professor at Queen Mary University of London
Federico_Ardila
Topics referred to by the same term
(graph theory), a symmetric tessellation of a closed surface Regular matroid, a matroid which can be represented over any field Regular paperfolding sequence
Regular
Geometry with 7 points and 7 lines
structure theory of matroids. Excluding the Fano plane as a matroid minor is necessary to characterize several important classes of matroids, such as regular
Fano_plane
American mathematician
oriented matroids; in particular, the Folkman–Lawrence topological representation theorem is "one of the cornerstones of the theory of oriented matroids". In
Jon_Folkman
General concept and operation in mathematics
matroid theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the dual matroid.
Duality_(mathematics)
Convex hull of a finite set of points in a Euclidean space
polytope may have a bit-length which is not polynomial in this representation. Oriented matroid Nef polyhedron Steinitz's theorem for convex polyhedra Branko
Convex_polytope
American/Canadian mathematician and computer scientist
he proved the matroid intersection theorem, a very general combinatorial min-max theorem which, in modern terms, showed that the matroid intersection problem
Jack_Edmonds
Topics referred to by the same term
other repeated vertices than the starting and ending vertices Circuit of a matroid Circuit (neural network), a computational subgraph within an artificial
Circuit
Property of rank functions of matroids
representable matroid. In this sense it is a necessary condition for representability of a matroid over a finite field. For a matroid M and its rank
Ingleton's_inequality
American mathematician
Coxeter matroids; it was published by Serganova and Israel Gelfand in 1987 as part of their research originating the concept of a Coxeter matroid. "Book
Vera_Serganova
Algorithmically defined graph
group, an implicit model for group-theoretic algorithms Matroid oracle, an implicit model for matroid algorithms Korf, Richard E. (2008), "Linear-time disk-based
Implicit_graph
Mathematical system of orderings or sets
defining antimatroids as set systems are very similar to those of matroids, but whereas matroids are defined by an exchange axiom, antimatroids are defined instead
Antimatroid
Representation of symmetric groups
Specht module may be found in Section 1 of "Specht Polytopes and Specht Matroids". The dimension of the Specht module V λ {\displaystyle V_{\lambda }} is
Specht_module
Academic subfield of computer science
languages they are able to recognize. An automaton can be a finite representation of a formal language that may be an infinite set. Automata are used
Theory_of_computation
Fewest dependent columns in a matrix
finds applications in error-correction codes, compressive sensing, and matroid theory, and provides a simple criterion for maximal sparsity of solutions
Spark_(mathematics)
Soviet mathematician (1913–2009)
MR 2000133 Borovik, Alexandre V.; Gelfand, I. M.; White, Neil (2003), Coxeter matroids, Progress in Mathematics, vol. 216, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3764-4
Israel_Gelfand
Indian-American mathematician
relating the schemes defined by Kirchhoff polynomials to the representation spaces of matroids. Moreover, using Mnev's universality theorem, we show that
Prakash_Belkale
of it include enumerative combinatorics, combinatorial design theory, matroid theory, extremal combinatorics and algebraic combinatorics, as well as
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Gluing graphs at complete subgraphs
3-sums of graphic matroids (the matroids representing spanning trees in a graph), cographic matroids, and a certain 10-element matroid. Lovász (2006). As
Clique-sum
a set of points and lines without any two-point lines. Sylvester matroid, a matroid without any two-point lines. Sylvester's determinant identity. Sylvester's
List of things named after James Joseph Sylvester
List_of_things_named_after_James_Joseph_Sylvester
Type of random graph
"The multivariate Tutte polynomial (Alias Potts model) for graphs and matroids". Surveys in Combinatorics 2005. pp. 173–226. arXiv:math/0503607. doi:10
Random_cluster_model
minimums of finite collections of polynomials. Rota's basis conjecture: for matroids of rank n {\displaystyle n} with n {\displaystyle n} disjoint bases B i
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal
Lists_of_mathematics_topics
Sequence of spaces in linear algebra
. Filtration (mathematics) Flag (geometry) Flag manifold Grassmannian Matroid Kostrikin, Alexei I. and Manin, Yuri I. (1997). Linear Algebra and Geometry
Flag_(linear_algebra)
Algorithm for linear programming
(1983, p. 79) There are abstract optimization problems, called oriented matroid programs, on which Bland's rule cycles (incorrectly) while the criss-cross
Simplex_algorithm
English mathematician
geometry. His work on matroids culminated in the paper "Representation of matroids" published in 1969. In his work, Ingleton studied matroids as a generalization
Aubrey_William_Ingleton
of disjoint cycles. Cycle basis Cycle double cover conjecture Eulerian matroid Sabidussi 1964. Euler, L. (1736), "Solutio problematis ad geometriam situs
Veblen's_theorem
British mathematician and computer scientist
research student of Peter Vámos. His doctoral thesis was "Representations of Matroids". Fenton was a postdoctoral fellow in the mathematics department at University
Norman_Fenton
Vertices connected in pairs by edges
they allow for higher-dimensional simplices. Every graph gives rise to a matroid. In model theory, a graph is just a structure. But in that case, there
Graph_(discrete_mathematics)
Analysis of datasets using techniques from topology
reduction can in fact be performed as the complex is constructed by using matroid theory, leading to further performance increases. Another recent algorithm
Topological_data_analysis
-graded sets. Signed sets are fundamental to the definition of oriented matroids. They may also be used to define the faces of a hypercube. If the hypercube
Signed_set
equipped with a further oracle for determining element orders. Implicit graph Matroid oracle Babai, L.; Szemeredi, E. (1984). "On the Complexity of Matrix Group
Black_box_group
Problem in social choice
public goods, with possible constraints on the allocation. They consider matroid constraints, matching constraints, and packing constraints (which correspond
Combinatorial participatory budgeting
Combinatorial_participatory_budgeting
the graphic matroid of a graph, a subset of edges is independent if the corresponding subgraph is a tree or forest. In the bicircular matroid, a subset
Glossary_of_graph_theory
Mathematics award
prize'] for outstanding writings." In the background, there is the representation of Archimedes' tomb, with the carving illustrating his theorem On the
Fields_Medal
mathematician specializing in disease modeling Collette Coullard, American matroid theorist and operations researcher Judith Covington, American mathematics
List_of_women_in_mathematics
Social choice problem
Munagala and Shah focus on three types of constraints: Matroid constraints: there is a fixed matroid M over the items, and the chosen items must form a basis
Multi-issue_voting
Study of Lie groups, Lie algebras and differential equations
essentially algebraic ideas of Killing into the theory of the structure and representation of semisimple Lie algebras that plays such a fundamental role in present-day
Lie_theory
Cycles in a graph that generate all cycles
weight of its longest cycle. In any vector space, and more generally in any matroid, a minimum weight basis may be found by a greedy algorithm that considers
Cycle_basis
Branch of mathematics
commutator of two elements. Burnside, Frobenius, and Molien created the representation theory of finite groups at the end of the nineteenth century. J. A.
Abstract_algebra
Field of knowledge
Coding theory, including error correcting codes and a part of cryptography Matroid theory Discrete geometry Discrete probability distributions Game theory
Mathematics
Mathematical tree of cycles
in any graph may be found in polynomial time using an algorithm for the matroid parity problem. Since triangular cactus graphs are planar graphs, the largest
Cactus_graph
Vectors whose linear combinations are nonzero
independent and M 1 + ⋯ + M d = X . {\displaystyle M_{1}+\cdots +M_{d}=X.} Matroid – Abstraction of linear independence of vectors G. E. Shilov, Linear Algebra
Linear_independence
American mathematician (1907–1989)
paper where he laid the foundations for matroids, a fundamental notion in modern combinatorics and representation theory independently introduced by him
Hassler_Whitney
Economical computational problem
the bases of a matroid over the set of resources, then all best-response sequences converge in polynomial number of steps, and the matroid property is essential
Nash_equilibrium_computation
Branch of elementary mathematics
the ratio of a circle's circumference to its diameter. The decimal representation of an irrational number is infinite without repeating decimals. The
Arithmetic
Area of geometry, about angles and lengths
{\displaystyle x=\cos A} and y = sin A {\displaystyle y=\sin A} . This representation allows for the calculation of commonly found trigonometric values, such
Trigonometry
Computational problems no algorithm can solve
3247570. S2CID 248986512. Kühne, L.; Yashfe, G. (2022). "Representability of Matroids by c-Arrangements is Undecidable". Israel Journal of Mathematics. 252:
List_of_undecidable_problems
Method to solve optimization problems
Method of computing optimal strategies for last-success problems Oriented matroid – Abstraction of ordered linear algebra Quadratic programming – Solving
Linear_programming
Number that is the product of factorials
S2CID 121878844 Golumbic, Martin Charles (1977), "Comparability graphs and a new matroid", Journal of Combinatorial Theory, Series B, 22 (1): 68–90, doi:10
Jordan–Pólya_number
The Avis–Fukuda algorithm adapted the criss-cross algorithm for oriented matroids. A 2025 article by Zelin Dong, Fenglei Fan, Huan Xiong, and Tieyong Zeng
Vertex_enumeration_problem
Smallest convex set containing a given set
convex hulls may also be generalized in a more abstract way, to oriented matroids. It is not obvious that the first definition makes sense: why should there
Convex_hull
Type of functional equation (mathematics)
smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution
Differential_equation
leaks in the pipeline and gender stereotyping contribute to the under-representation of African-American women in the sciences. There are organizations that
List of African-American women in STEM fields
List_of_African-American_women_in_STEM_fields
Geometric system of two mutually inscribed tetrahedra
the two configurations, including the fact that both are self-dual under Matroid duality. In abstract terms, the latter configuration has "points" 0, .
Möbius_configuration
Generalization of graph theory
abstract simplicial complex with the augmentation property is called a matroid. Laminar: for any two hyperedges, either they are disjoint, or one is included
Hypergraph
A. Harvey (MIT) "Algebraic Structures and Algorithms for Matching and Matroid Problems" 2005 Mark Braverman (Toronto) "On the Complexity of Real Functions"
Machtey_Award
Operation combining two oriented knots
125–136. Ramirez Alfonsin, J. L. (1999), "Spatial graphs and oriented matroids: the trefoil", Discrete and Computational Geometry, 22 (1): 149–158, doi:10
Knot_(mathematics)
Flat-sided three-dimensional shape
Bokowski, J.; Guedes de Oliveira, A. (2000), "On the generation of oriented matroids", Discrete and Computational Geometry, 24 (2–3): 197–208, doi:10.1007/s004540010027
Polyhedron
enumeration in matroids. Quasisymmetric functions have been applied in enumerative combinatorics, symmetric function theory, representation theory, and number
Quasisymmetric_function
Embedding a graph in 3D space with no cycles interlinked
D. thesis, University of California, Berkeley. Truemper, Klaus (1992), Matroid Decomposition (PDF), Academic Press, pp. 100–101, archived from the original
Linkless_embedding
Stanley has had a big impact in contemporary combinatorics for his work in matroid theory, for introducing Zeta polynomials, for explicitly defining Eulerian
History_of_combinatorics
Elad; Carmesin, Johannes; Fröhlich, Jan-Oliver (2012-07-09), Infinite matroid union, arXiv:1111.0602 Blossier, Thomas; Bouscaren, Elisabeth (2010). "Finitely
Glossary_of_logic
Canadian geometer (1907–2003)
Robert Frucht, extended LCF notation devised by Joshua Lederberg for the representation of cubic graphs that contain a Hamiltonian cycle.[B81] Since 1978, the
Harold Scott MacDonald Coxeter
Harold_Scott_MacDonald_Coxeter
Branch of mathematics
example of topological spaces. This line of research leads to various representation theorems, often collected under the label of Stone duality. Causal sets
Order_theory
Fair division problem for discrete items
variant is known as the agreeable subset problem. There may be general matroid constraints, matching constraints or knapsack constraints on the chosen
Fair_item_allocation
MATROID REPRESENTATION
MATROID REPRESENTATION
Female
Egyptian
, a wife of Rameses III.
Girl/Female
Arabic
Poetess; Matron
Boy/Male
Arabic, French, Hindu, Indian, Muslim
Rebellious; Ray of Light
Biblical
wand of government
Girl/Female
Muslim
Leadress. Matron.
Boy/Male
Spanish
God's gift.
Girl/Female
Biblical
Rain, prison.
Female
Russian
(Матрона) Russian form of Latin Matrona, MATRYONA means "lady."
Girl/Female
Arabic, German, Hindu, Indian, Kannada, Marathi, Muslim, Telugu
Song; Poetess; Matron
Boy/Male
Biblical
Wand of government.
Boy/Male
Indian
Rebellious
Boy/Male
Muslim
Rebellious
Girl/Female
Australian, British, Chinese, Christian, English, Jamaican, Portuguese
Compound of the Names Polly and Anna; Bitter; Gracious; One who Plays for Real Madrid
Girl/Female
Muslim
Leadress. Matron.
Girl/Female
Hindu, Indian
Representation of Love
Biblical
rain; prison
Girl/Female
Muslim
Poetess. Matron.
Girl/Female
Indian, Sanskrit
Mother
Girl/Female
French Latin
From the Latin Lucretia. Famous bearer: Lucrece, a Roman matron who committed suicide as a public...
MATROID REPRESENTATION
MATROID REPRESENTATION
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : habitational name from any of the numerous places in France so called from the dedication of their churches to St. George (see George).French : secondary surname to the primary surnames De la Porte, Godfroy, Lapointe, and Laporte.
Boy/Male
Gujarati, Hindu, Indian, Kannada
Love to Gods
Boy/Male
Tamil
Madhukesh | மாதà¯à®•à¯‡à®·
Hair of Lord Vishnu
Female
Finnish
 Feminine form of Finnish Simo, SIMONE means "hearkening." Compare with another form of Simone.
Girl/Female
Tamil
Soodnya | ஸூதநà¯à®¯à®¾
One who has achieved her goal, Wise
Girl/Female
Tamil
Mangalavathi | மஂகலாவதீ
Name of a Raga
Surname or Lastname
English
English : perhaps a variant spelling of Whit(t)la, itself a variant of Whitley.
Boy/Male
Tamil
A Ray of light
Boy/Male
American, Anglo, British, English
From the Crane Estate; Settlement of Cranes
Girl/Female
American, Australian, French, German, Latin, Swedish
Dedicated to Mars; The Roman God of War; Longevity; Pillar
MATROID REPRESENTATION
MATROID REPRESENTATION
MATROID REPRESENTATION
MATROID REPRESENTATION
MATROID REPRESENTATION
n. pl.
The maioid crabs.
a.
Like a matron; sedate; grave; matronly.
a.
Like the dartos; dartoic; as, dartoid tissue.
n.
The state of a matron.
a.
Same as Sauroid.
a.
Same as Mastoid.
n.
The collective body of matrons.
n.
A mold; a matrix.
n.
See Matrix.
n.
A housekeeper; esp., a woman who manages the domestic economy of a public instution; a head nurse in a hospital; as, the matron of a school or hospital.
n. pl.
The triangular, or maioid, crabs. See Illust. under Maioid, and Illust. of Spider crab, under Spider.
n.
An old woman or matron.
n.
The point on the side of the skull where the lambdoid, parieto-mastoid and occipito-mastoid sutures.
n.
See Matross.
a.
Resembling a saurian superficially; as, a sauroid fish.
pl.
of Matrix
a.
Pertaining to, or in the region of, the mastoid process; mastoidal.
n.
A native or inhabitant of Madrid.
n.
The state of being a matron.