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Combinatorial representation of a graph on an orientable surface
A combinatorial map is a combinatorial representation of a graph on an orientable surface. A combinatorial map may also be called a combinatorial embedding
Combinatorial_map
Homotopic map of a graph
of here as a 1-dimensional cell complex) a combinatorial map is a continuous map f : Γ → Γ such that: The map f takes vertices to vertices. For every edge
Train_track_map
generalized map is a topological model which allows one to represent and to handle subdivided objects. This model was defined starting from combinatorial maps in
Generalized_map
Graph that can be embedded in the plane
means of stereographic projection. Plane graphs can be encoded with combinatorial maps or rotation systems. An equivalence class of topologically equivalent
Planar_graph
Topics referred to by the same term
overlapping edges Planar graph, a graph drawn on a planar surface Maps of manifolds Combinatorial map, a representation of a topological subdivision of the plane
Map_(disambiguation)
Planar maps require at most four colors
theorem, or the four-color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions
Four_color_theorem
Graph data structure
complexity. Quad-edge data structure Doubly linked face list Winged edge Combinatorial map Muller, D. E.; Preparata, F. P. (1978). "Finding the Intersection
Doubly_connected_edge_list
Method of representing a 3D object by defining the limits of its volume
model constructed by planar surfaces only Bézier curve Bézier surface Combinatorial maps Coons surface Function representation Geometric modeling kernel NURBS
Boundary_representation
Proofs in enumerative combinatorics
the term combinatorial proof is often used to mean either of two types of mathematical proof: A proof by double counting. A combinatorial identity is
Combinatorial_proof
Map distorting size to show another value
A cartogram, also called a value-area map or an anamorphic map, is a thematic map of a set of features (countries, provinces, etc.), in which their geographic
Cartogram
Numbering of combinations of items
In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics
Combinatorial_number_system
Branch of geometry that studies combinatorial properties and constructive methods
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric
Discrete_geometry
Field of mathematics and science based on non-linear systems and initial conditions
on initial conditions property, such as combinatorial chaos (I.e. applying recursively a discrete combinatorial action). This is also comparable and similar
Chaos_theory
MI, Kim S, Weinstein JN, Pommier Y. Depicting combinatorial complexity with the molecular interaction map notation. Mol Syst Biol. 2006;2:51. (2006) doi:10
Molecular_Interaction_Maps
Generalization of linear assignment problem from two to multiple dimensions
The multidimensional assignment problem (MAP) is a fundamental combinatorial optimization problem which was introduced by William Pierskalla. This problem
Multidimensional assignment problem
Multidimensional_assignment_problem
Theory in mathematics
In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for deriving the generating functions of discrete structures
Combinatorial_species
Operation in combinatorial game theory
In the mathematics of combinatorial games, the sum or disjunctive sum of two games is a game in which the two games are played in parallel, with each
Disjunctive_sum
Data structure for representing polygon meshes in computer memory
Edge *edge; } class Face { Edge *edge; } Quad-edge data structure Combinatorial maps Doubly connected edge list Doubly linked face list Half-edge data
Winged_edge
Way to divide polygon into smaller parts
exactly when the subdivision rule is "conformal", as described in the combinatorial Riemann mapping theorem. Applications of subdivision rules. Islamic
Finite_subdivision_rule
Mapping senses to the central nervous system
of topographic maps in the visual cortices, including retinotopic maps, ocular dominance maps and orientation maps. Retinotopic maps are the easiest
Topographic map (neuroanatomy)
Topographic_map_(neuroanatomy)
Symmetric tessellation of a closed surface
regular maps of small genus", Journal of Combinatorial Theory, Series B, 81 (2): 224–242, doi:10.1006/jctb.2000.2008. Nedela, Roman (2007), Maps, Hypermaps
Regular_map_(graph_theory)
Undirected graph with 14 vertices
(2004), "Graphs and digraphs with all 2-factors isomorphic", Journal of Combinatorial Theory, Series B, 92 (2): 395–404, doi:10.1016/j.jctb.2004.09.004, MR 2099150
Heawood_graph
start-left edges until the original edge is reached. Winged edge Combinatorial maps Doubly connected edge list Stolfi, Jorge; Guibas, Leonidas J. (April
Quad-edge
In mathematics, invariant of square matrices
"Division-free algorithms for the determinant and the Pfaffian: algebraic and combinatorial approaches" (PDF), Computational discrete mathematics, Lecture Notes
Determinant
Z-module homomorphism
In algebra, an additive map or additive function is a function f {\displaystyle f} that preserves the addition operation: f ( x + y ) = f ( x ) + f (
Additive_map
NP-hard problem in combinatorial optimization
exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations
Travelling_salesman_problem
Data structure for approximate set membership
from which set elements are taken. It relies on existing non-adaptive combinatorial group testing scheme by Eppstein, Goodrich and Hirschberg. Unlike the
Bloom_filter
Process in digital electronics and integrated circuit design
McGraw-Hill. ISBN 0-07-016333-2. (NB. Chapters 7–9 cover combinatorial two-level, combinatorial multi-level, and respectively sequential circuit optimization
Logic_optimization
Theory in number theory
theory has since grown in varieties (absolute, mono-anabelian, and combinatorial versions) and with multiple interactions with number theory, algebraic
Anabelian_geometry
Motor protein in humans
Tan TC, Oclaman JM, Han JS, Simó S, Niwa S, et al. (April 2020). "A Combinatorial MAP Code Dictates Polarized Microtubule Transport". Developmental Cell
KIF1A
Mathematical models of strategic interactions
game theory include algorithmic game theory, behavioral game theory, combinatorial game theory, evolutionary game theory, and quantum game theory. In 1994
Game_theory
Selection in a particular order
In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set S is a bijection between two specified subsets of
Partial_permutation
Concept in the mathematics of paper folding
stamps", Journal of Combinatorial Theory, 5 (2): 135–152, doi:10.1016/S0021-9800(68)80048-1, MR 0228364 Lunnon, W. F. (1968), "A map-folding problem", Mathematics
Map_folding
Sequence of n-tuples of integers
sequences in higher dimensions" in INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007) [1] S Lettieri, JG Stevens, DM Thomas, "Characteristic
Ducci_sequence
Topics referred to by the same term
algebraic combinatorics Quasisymmetric maps in complex analysis or metric spaces Quasi-symmetric designs in combinatorial design theory Quasisymmetry, a concept
Quasisymmetric
Intelligence in machines
insufficient for solving large reasoning problems because they experienced a "combinatorial explosion", meaning they become exponentially slower as the problems
Artificial_intelligence
Discrete dynamical system on polygons in the projective plane and on their moduli space
In mathematics, the pentagram map is a discrete dynamical system acting on polygons in the projective plane. It defines a new polygon whose vertices are
Pentagram_map
Natural number
That there are maps for which three colors are not sufficient) Molitierno, Jason J. (19 April 2016). Applications of Combinatorial Matrix Theory to
4
Representation of mathematical space
complexes are objects of combinatorial character and therefore one can assign them quantities arising from their combinatorial pattern, for instance, the
Triangulation_(topology)
Intersection graph representing regions on the Euclidean plane
between which the chess king can move. Map graphs can be represented combinatorially as the "half-squares of planar bipartite graphs". That is, let G =
Map_graph
Mathematical construction used in homotopy theory
the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a topological space
Simplicial_set
Concept in mathematics
also referred to as the RSK correspondence or RSK algorithm, is a combinatorial bijection between matrices A with non-negative integer entries and pairs
Robinson–Schensted–Knuth correspondence
Robinson–Schensted–Knuth_correspondence
Overview of and topical guide to algorithms
List of algorithms List of artificial intelligence algorithms List of combinatorial algorithms Al-Khwarizmi — namesake of the term algorithm Charles Babbage
Outline_of_algorithms
Study of discrete mathematical structures
from topology and algebraic topology/combinatorial topology in combinatorics. Design theory is a study of combinatorial designs, which are collections of
Discrete_mathematics
the underlying space) belonging to half or more of the sets Give a combinatorial interpretation of the Kronecker coefficients The size m ( n ) {\displaystyle
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Quadratic polynomial
sets combinatorially, replacing the dynamical plane by the unit circle, rays by angles, and the quadratic polynomial by the doubling modulo one map." Virpi
Complex_quadratic_polynomial
geodesic map, because hyperbolic geodesics in the Klein model are (Euclidean) straight line segments. Ambartzumian, R. V. (1982). Combinatorial integral
Geodesic_map
Abelian group equipped with compatible ring action on both sides
bialgebras. Profunctor Street, Ross (20 Mar 2003). "Categorical and combinatorial aspects of descent theory". arXiv:math/0303175. Jacobson, N. (1989)
Bimodule
Mathematical object
subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. For example
Abstract_simplicial_complex
Any algorithm which solves the search problem
satisfaction, such as: The map coloring problem Filling in a sudoku or crossword In game theory and especially combinatorial game theory, choosing the
Search_algorithm
bound Bruss algorithm: see odds algorithm Chain matrix multiplication Combinatorial optimization: optimization problems where the set of feasible solutions
List_of_algorithms
Decision problem pertaining to equivalence of expressions
word problem takes the form of a normal form theorem and algorithm that maps every element in an equivalence class of expressions to a single encoding
Word_problem_(mathematics)
Smallest convex set containing a given set
functions. Convex hulls have wide applications in mathematics, statistics, combinatorial optimization, economics, geometric modeling, and ethology. Related structures
Convex_hull
Mathematics concept
in a Hadamard matrix represents two perpendicular vectors, while in combinatorial terms, it means that each pair of rows has matching entries in exactly
Hadamard_matrix
Type of plane partition
be represented in a combinatorial way using their vertices, sides, two-dimensional faces, etc. Sometimes the induced combinatorial structure is referred
Voronoi_diagram
Block design in combinatorial mathematics
In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and t = 2 or
Steiner_system
About polynomials in several variables
resulting constraints on the maps between the Picard groups of suitable compactifications as an essentially combinatorial problem and described several
Jacobian_conjecture
Fractal named after mathematician Benoit Mandelbrot
Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle. The boundary of
Mandelbrot_set
Number line and triangular tiling's symmetry mathematical structure
is a simple map (technically, a surjective group homomorphism) π from S~n onto the finite symmetric group Sn. In terms of the combinatorial definition
Affine_symmetric_group
American research software engineer
71).aspx. Applied Combinatorial Mathematics, Ed. E. F. Beckenbach (1964), pp. 27−30; a previous description of a combinatorial representation of integers
James_D._McCaffrey
intersections have certain properties. Combinatorial game theory Combinatorial geometry see discrete geometry Combinatorial group theory the theory of free groups
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Majorcan writer and philosopher (c. 1232 – 1316)
nationalities. The Art consists of a set of general principles and combinatorial operations. It is illustrated with diagrams. A prolific writer, he is
Ramon_Llull
Subpermutation of a longer permutation
In combinatorial mathematics and theoretical computer science, a (classical) permutation pattern is a sub-permutation of a longer permutation. Any permutation
Permutation_pattern
Roli's cube, a skew polytope constructed on the skeleton of the 4-cube. Combinatorial chirality refers to the properties of an abstract polytope regardless
Chiral_polytope
American mathematician
Masur is an American mathematician who works on topology, geometry, and combinatorial group theory. Masur was an invited speaker at the 1994 International
Howard_Masur
Concept in topology
idempotent elements and its ideal structure—can be used to prove deep combinatorial results about the original set S. A prominent example of this utility
Stone–Čech_compactification
Theorem in homotopy theory
is part of the subject of shape theory. Whitehead, J. H. C. (1949). "Combinatorial homotopy. I". Bulletin of the American Mathematical Society. 55 (3):
Whitehead_theorem
Array of numbers
Stevanović, Dragan (2018), Encinas, Andrés M.; Mitjana, Margarida (eds.), Combinatorial Matrix Theory, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer
Matrix_(mathematics)
Counting technique in combinatorics
the most useful principles of enumeration in discrete probability and combinatorial theory is the celebrated principle of inclusion–exclusion. When skillfully
Inclusion–exclusion_principle
Topological space that locally resembles Euclidean space
bracket. A closely related type of manifold is a contact manifold. A combinatorial manifold is a kind of manifold which is discretization of a manifold
Manifold
Class of artificial neural networks
citation networks, molecular biology, chemistry, physics and NP-hard combinatorial optimization problems. Open source libraries implementing GNNs include
Graph_neural_network
Mathematical optimization problem restricted to integers
A Tutorial on Integer Programming Conference Integer Programming and Combinatorial Optimization, IPCO The Aussois Combinatorial Optimization Workshop
Integer_programming
Polygon constructed from another
combinatorially, there is a single family of polygons (given by number of sides); geometric duality of polygons is more varied, as are combinatorial dual
Dual_polygon
Area of discrete mathematics
polynomial on graph connectivity. Geometric graph theory focuses on combinatorial and geometric properties of a graph that is drawn in a plane with straight-line
Graph_theory
If G is a finitely generated group with exponent n, is G necessarily finite?
questions in group theory, and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny
Burnside_problem
Analysis of datasets using techniques from topology
independence configurations analog to homotopical links. Because of its combinatorial complexity, only the simplicial subcase of the cohomology and of information
Topological_data_analysis
Protein-coding gene in the species Homo sapiens
Tan TC, Oclaman JM, Han JS, Simó S, Niwa S, et al. (April 2020). "A Combinatorial MAP Code Dictates Polarized Microtubule Transport". Developmental Cell
Ensconsin
Theorem in topology
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. The theorem
Borsuk–Ulam_theorem
Numbers obtained by adding the two previous ones
memoization). Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that Fn can be interpreted as the number of
Fibonacci_sequence
Sporadic simple group
gave a short proof of the result, describing the 9 classes in terms of combinatorial data on the 24 points: the subgroups fix a point, duad, octad, duum
Mathieu_group_M24
Property of a mathematical space
Riemannian Symplectic Discrete differential Complex Finite Discrete/Combinatorial Digital Convex Computational Fractal Incidence Noncommutative geometry
Dimension
Quantum Computing company in Boston, Massachusetts
from the company to map other problems onto MWIS as well QuEra actively conducts research in Condensed Matter Physics and combinatorial optimization. A QRC
QuEra
Topics referred to by the same term
quick setting cement. Multidimensional knapsack problem, a problem in combinatorial optimization Kilopondmetre, a unit of torque MKP is used to refer to
MKP
Graph describing a topological embedding
ISBN 0-387-94557-1, MR 1367285 Lins, Sóstenes (1982), "Graph-encoded maps", Journal of Combinatorial Theory, Series B, 32 (2): 171–181, doi:10.1016/0095-8956(82)90033-8
Graph-encoded_map
Problem in social choice
Combinatorial participatory budgeting, also called indivisible participatory budgeting or budgeted social choice, is a problem in social choice. There
Combinatorial participatory budgeting
Combinatorial_participatory_budgeting
Branch of mathematics that studies the properties of groups
consisting of generator symbols and their inverses is called a word. Combinatorial group theory studies groups from the perspective of generators and relations
Group_theory
Optimization problem
The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a
Vehicle_routing_problem
Rational number sequence
a combinatorial fact comes to surface which explains the vanishing of the Bernoulli numbers at odd index. Let Sn,m be the number of surjective maps from
Bernoulli_number
Technique for proving sets have equal size
elements, or that the sets in two combinatorial classes have equal size, by finding a bijective function that maps one set one-to-one onto the other.
Bijective_proof
Games in game theory
Several map-coloring games are studied in combinatorial game theory. The general idea is that we are given a map with regions drawn in but with not all
Map-coloring_games
Branch of mathematics
retains a combinatorial nature that allows for computation (often with a much smaller complex). An older name for the subject was combinatorial topology
Algebraic_topology
Award for advancements in discrete mathematics
complexity of combinatorial problems". Networks. 5: 45–68. doi:10.1002/net.1975.5.1.45. Appel, Kenneth; Haken, Wolfgang (1977). "Every planar map is four colorable
Fulkerson_Prize
Type of topological space
categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation (often with a much smaller complex)
CW_complex
Cycle through all length-k sequences
In combinatorial mathematics, a de Bruijn sequence of order n on a size-k alphabet A is a cyclic sequence in which every possible length-n string on A
De_Bruijn_sequence
Branch of mathematics that studies sets
Moore space question was eventually proved to be independent of ZFC. Combinatorial set theory concerns extensions of finite combinatorics to infinite sets
Set_theory
Result in combinatorics and graph theory
gives a necessary and sufficient condition for an object to exist: The combinatorial formulation answers whether a finite collection of sets has a transversal—that
Hall's_marriage_theorem
Topological property
Anders (2003-04-01). "Nerves, fibers and homotopy groups". Journal of Combinatorial Theory. Series A. 102 (1): 88–93. doi:10.1016/S0097-3165(03)00015-3
Homotopical_connectivity
Embedding a graph in a topological space, often Euclidean
embedding or map is an embedding in which every face is homeomorphic to an open disk. A cellular embedding can be described combinatorially by a rotation
Graph_embedding
Algorithm for finding shortest paths
Search or a Case Against Dijkstra's Algorithm. Proc. 4th Int'l Symp. on Combinatorial Search. Archived from the original on 18 February 2020. Retrieved 12
Dijkstra's_algorithm
Theorem in combinatorics
[citation needed] Axiom of choice Blichfeldt's theorem Combinatorial principles Combinatorial proof Dedekind-infinite set Dirichlet's approximation theorem
Pigeonhole_principle
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COMBINATORIAL MAP
COMBINATORIAL MAP
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COMBINATORIAL MAP
COMBINATORIAL MAP
COMBINATORIAL MAP
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