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Type of mathematical set
In mathematics, a simplicial complex is a structured set of simplices (for example, points, line segments, triangles, and their n-dimensional counterparts)
Simplicial_complex
Mathematical object
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking
Abstract_simplicial_complex
The link in a simplicial complex is a generalization of the neighborhood of a vertex in a graph. The link of a vertex encodes information about the local
Link_(simplicial_complex)
Mathematical construction used in homotopy theory
mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) and maps between them. Simplicial sets are higher-dimensional
Simplicial_set
Representation of mathematical space
triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism. A space that admits
Triangulation_(topology)
Mathematics glossary
Whitehead torsion vanishes. simplicial approximation See simplicial approximation theorem. simplicial complex See simplicial complex; the basic example is a
Glossary of algebraic topology
Glossary_of_algebraic_topology
refinement) of a simplicial complex is another simplicial complex in which, intuitively, one or more simplices of the original complex have been partitioned
Subdivision (simplicial complex)
Subdivision_(simplicial_complex)
combinatorics, a simplicial (or combinatorial) d-sphere is a simplicial complex homeomorphic to the d-dimensional sphere. Some simplicial spheres arise as
Simplicial_sphere
Roughly, the number of k-dimensional holes on a topological surface
n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the
Betti_number
Mathematical group of the homotopy classes of loops in a topological space
space of a finite connected simplicial complex X {\displaystyle X} can also be described directly as a simplicial complex using edge-paths. Its vertices
Fundamental_group
Computational problem in algebraic topology
The simplicial complex recognition problem is a computational problem in algebraic topology. Given a simplicial complex, the problem is to decide whether
Simplicial complex recognition problem
Simplicial_complex_recognition_problem
Concept in algebraic topology
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of
Simplicial_homology
Multi-dimensional generalization of triangle
simplices to form a simplicial complex. The geometric simplex and simplicial complex should not be confused with the abstract simplicial complex, in which a simplex
Simplex
Topological data
tree is a type of trie used to efficiently represent any general simplicial complex. Through its nodes, this data structure notably represents all the
Simplex_tree
to simplicial complexes and CW complexes in the computation of the homology of topological spaces. Non-positively curved and CAT(0) cube complexes appear
Cubical_complex
Research field in deep learning
interactions among multiple entities and complex hierarchies. This approach leverages structures like simplicial complexes and hypergraphs to capture global
Topological_deep_learning
Topological space
rooms or Bing's house is a particular contractible, 2-dimensional simplicial complex that is not collapsible. The name was given by R. H. Bing. The house
House_with_two_rooms
Type of topological space
dimensions in specific ways. The notion generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially
CW_complex
Simplicial complex
Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the
Coxeter_complex
Branch of mathematics
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory
Algebraic_topology
Method for computing topological features of a space at different spatial resolutions
represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested
Persistent_homology
Algebraic structure associated with a topological space
the task easier. The simplicial homology groups Hn(X) of a simplicial complex X are defined using the simplicial chain complex C(X), with Cn(X) the free
Homology_(mathematics)
Generalized manifold
for each vertex i of X ', a simplicial complex Li' endowed with a rigid simplicial action of a finite group Γi. a simplicial map φi of Li' onto the link
Orbifold
Operation in topology
and B {\displaystyle B} are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows: The vertex set V (
Join_(topology)
collapse reduces a simplicial complex (or more generally, a CW complex) to a homotopy-equivalent subcomplex. Collapses, like CW complexes themselves, were
Collapse_(topology)
Algebraic structure with addition and multiplication
geometry studies maps between the subrings of the function field. Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring
Ring_(mathematics)
About the numbers of faces of different dimensions in an abstract simplicial complex
theorem gives a complete characterization of the f-vectors of abstract simplicial complexes. It includes as a special case the Erdős–Ko–Rado theorem and can
Kruskal–Katona_theorem
Complex in algebraic topology
algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which
Čech_complex
Steinitz is related to the notion of an abstract simplicial complex and it differs from a simplicial complex by the property that its elements are not simplices:
Abstract_cell_complex
Concept in algebraic topology
the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simplices
N-skeleton
Abstraction useful in the construction and triangulation of topological spaces
In mathematics, a Δ-set, often called a Δ-complex or a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation
Delta_set
Continuous mappings can be approximated by ones that are piecewise simple
between spaces that are built up from simplices—that is, finite simplicial complexes. The general continuous mapping between such spaces can be represented
Simplicial approximation theorem
Simplicial_approximation_theorem
≤), ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces σ ⊆ V {\displaystyle
Poset_topology
Abstract simplicial complex describing a graph's cliques
subgraphs) of an undirected graph. The clique complex X(G) of an undirected graph G is an abstract simplicial complex (that is, a family of finite sets closed
Clique_complex
Mathematical ring
Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic
Stanley–Reisner_ring
Branch of geometry that studies combinatorial properties and constructive methods
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory
Discrete_geometry
Branch of mathematics
continuous maps from the standard n-simplex into X; if K is a simplicial complex then the simplicial chains Cn(K) are formal linear combinations of the n-simplices
Homological_algebra
Topics referred to by the same term
loan in California and Nevada, US Simplicial link, a set of simplices "surrounding" a given vertex in a simplicial complex Link (knot theory), a collection
Link
Mathematical structure
conditions on the complexes Δ that can arise in this fashion. By treating these conditions as axioms for a class of simplicial complexes, Tits arrived at
Building_(mathematics)
Generalizations in graph theory
conjecture for r = 3. Let V be a set of vertices. Let C be an abstract simplicial complex on V. Let Vy (for y in Y) be subsets of V. A C-V-transversal is a
Hall-type theorems for hypergraphs
Hall-type_theorems_for_hypergraphs
Point where two or more curves, lines, or edges meet
topological cell complex, as can the faces of a polyhedron or polytope; the vertices of other kinds of complexes such as simplicial complexes are its zero-dimensional
Vertex_(geometry)
Topological space formed from distances
topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and distance δ by forming
Vietoris–Rips_complex
Branch of mathematics
non-negatively graded chain complexes over a fixed base ring. A simplicial set is an abstract generalization of a simplicial complex and can play a role of
Homotopy_theory
Formal linear combination in a cell complex
linear combination of the k-cells in a cell complex. In simplicial complexes (respectively, cubical complexes), k-chains are combinations of k-simplices
Chain_(algebraic_topology)
Method for dividing a simplicial complex
operation on simplicial complexes. In algebraic topology it is sometimes useful to replace the original spaces with simplicial complexes via triangulations:
Barycentric_subdivision
Complex recording the pattern of intersections between a topological family's sets
making N ( C ) {\displaystyle N(C)} an abstract simplicial complex. Hence N(C) is often called the nerve complex of C {\displaystyle C} . Let X be the circle
Nerve_complex
All points in the topological closure not belonging to the interior
slightly different concept from the boundary of a manifold or of a simplicial complex. For example, the boundary of an open disk viewed as a manifold is
Boundary_(topology)
Discrete (i.e., incremental) version of infinitesimal calculus
operator and the chain complex are defined similarly to those for simplicial complexes. More general are cell complexes. A chain complex ( C ∗ , ∂ ∗ ) {\displaystyle
Discrete_calculus
Polytope whose facets are all simplices
cantitruncated alternated cubic honeycomb Simplicial higher polytope families: simplex cross-polytope (Orthoplex) Simplicial complex Delaunay triangulation Cromwell
Simplicial_polytope
Any collection of sets, or subsets of a set
sets as its vertices. An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shape formed by unions of line
Family_of_sets
Tool in homological algebra
finite complex extended to the left and right by 0. An example is the chain complex defining the simplicial homology of a finite simplicial complex. A chain
Chain_complex
Mathematical object in topological graph theory
A chessboard complex is a particular kind of abstract simplicial complex, which has various applications in topological graph theory and algebraic topology
Chessboard_complex
Non-orientable surface with one edge
from an abstract simplicial complex, because all three triangles share the same three vertices, while abstract simplicial complexes require each triangle
Möbius_strip
Mathematical concept
In mathematics, a shelling of a simplicial complex is a way of gluing it together from its maximal simplices (simplices that are not a face of another
Shelling_(topology)
Branch of mathematics
data points with a family of simplicial complexes, indexed by a proximity parameter. Analyse these topological complexes via algebraic topology – specifically
Topology
A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of
Simplicial_map
Fractal curve resembling a blancmange pudding
In mathematics, the blancmange curve is a self-affine fractal curve constructible by midpoint subdivision. It is also known as the Takagi curve, after
Blancmange_curve
Topological invariant in mathematics
CW-complexes. (When only triangular faces are used, they are two-dimensional finite simplicial complexes.) In general, for any finite CW-complex, the
Euler_characteristic
Mathematical space
manifold called the E8 manifold, a manifold not homeomorphic to any simplicial complex. If the form is Z {\displaystyle \mathbb {Z} } , there are two manifolds
4-manifold
In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed
Curve_complex
Mathematical theory
for i ≥ 1 we have Hi(P) = {0}. More generally if X is a simplicial complex or finite CW complex, then the group H0(X) is the free abelian group with the
Reduced_homology
(X):[0,\infty )\to \mathbf {Simp} } valued in the category of simplicial complexes and simplicial maps, where the morphisms (i.e., relations in the poset)
Vietoris–Rips_filtration
complex of an undirected graph G, denoted by I(G), is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking
Independence_complex
How spheres of various dimensions can wrap around each other
certain well known elements, called Hopf elements. If X is any finite simplicial complex with finite fundamental group, in particular if X is a sphere of dimension
Homotopy_groups_of_spheres
of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture stated that for simplicial spheres, all possible h-vectors occur already
H-vector
Mathematical theories
dimension, for extending a continuous mapping defined on a simplicial complex, or CW complex. It is traditionally called Eilenberg obstruction theory,
Obstruction_theory
Analysis of datasets using techniques from topology
simplicial complex to a much smaller cellular complex which is homotopic to the original one. This reduction can in fact be performed as the complex is
Topological_data_analysis
Topological manifold whose homology coincides with that of a sphere
a PL manifold. In other words, this gives an example of a finite simplicial complex that is a topological manifold but not a PL manifold. (It is not a
Homology_sphere
Topics referred to by the same term
sets Chain complex, an algebraic structure Simplicial complex, a kind of topological space CW complex, a kind of topological space Line complex, a 3-dimensional
Complex
a special case of a simplicial homology, as a graph is a special case of a simplicial complex. Since a finite graph is a 1-complex (i.e., its 'faces' are
Graph_homology
Subdivision of space into cells
discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. Mesh cells
Mesh_generation
Topics referred to by the same term
abstract cell complex Cell, a basic unit of a cellular automaton Cell, an element of a CW complex Cell, a k-face of a simplicial complex Cell (journal)
Cell
Mathematical theory
L)&\cong 0\\\end{aligned}}} Let X {\displaystyle X} be an abstract simplicial complex on a vertex set V {\displaystyle V} of size n {\displaystyle n} .
Alexander_duality
Math concept
polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and
Polyhedral_complex
Structure from which the geometry of the universe arises
vertices of an abstract simplicial complex, and a real-valued field associated with every pair of vertices; the abstract simplicial complex is set to correspond
Pregeometry_(physics)
groups on 2-dimensional simplicial complexes that have the structure of CAT(0) spaces. The quotient of the simplicial complex has finite stabilizer groups
Graph_of_groups
the k-th homology group of a simplicial complex depends only on the simplices of dimension at most k+1 (see simplicial homology). Therefore, the above
Homotopical_connectivity
All even-degree subgraphs of a graph
viewed as a simplicial complex with its vertices as zero-dimensional simplices and the edges as one-dimensional simplices. The chain complex of this topological
Cycle_space
Branch of topology
homeomorphic to any simplicial complex. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex was an open problem
Low-dimensional_topology
Feature of a polyhedron, polytope, etc.
facet of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex. For (boundary complexes of) simplicial
Facet_(geometry)
Approximation to shape of a point cloud
α-complex of the given set of points is the simplicial complex formed by the set of edges and triangles whose radii are at most 1/α. The α-complex is
Alpha_shape
Branch of the mathematical field of graph theory
abstract simplicial complex C with a single-element set per vertex and a two-element set per edge. The geometric realization |C| of the complex consists
Topological_graph_theory
Branch of topology
since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from Rn. The Zariski topology is defined
General_topology
Computational problems no algorithm can solve
Determining whether two finite simplicial complexes are homeomorphic. Determining whether a finite simplicial complex is (homeomorphic to) a manifold
List_of_undecidable_problems
Tree graph with one central node and leaves of length 1
subdivision of a star Star (simplicial complex) - a generalization of the concept of a star from a graph to an arbitrary simplicial complex. Wikimedia Commons
Star_(graph_theory)
Topics referred to by the same term
that resembles a star Star (simplicial complex), the set of simplices containing a given vertex in a simplicial complex Kleene star, which can refer
Star_(disambiguation)
Quotient of a weakly contractible space by a free action
the bar construction), that gave concrete descriptions of BG as a simplicial complex for an arbitrary discrete group. Such constructions make evident the
Classifying_space
is a Kan complex. In particular, a category fibered in groupoids over another category is a special case of a right fibration of simplicial sets in the
Fibration_of_simplicial_sets
Type of mathematical function
affine space, as well as on piecewise linear manifolds and simplicial complexes (see simplicial map). In each case, the function may be real-valued, or it
Piecewise_linear_function
Endofunctor on the category of simplicial sets
of simplicial sets (subdivision functor or Sd functor) is an endofunctor on the category of simplicial sets. It refines the structure of simplicial sets
Subdivision_(simplicial_set)
Fourth letter in the Greek alphabet
the position of which is variant between isomeric forms. A simplex, simplicial complex, or convex hull. In chemistry, the addition of heat in a reaction
Delta_(letter)
Subsets whose union equals the whole set
trivial topology). When subdividing simplicial complexes (the first barycentric subdivision of a simplicial complex is a refinement), the situation is
Cover_(topology)
Natural number
The barycentric subdivision of a tetrahedron produces an abstract simplicial complex with exactly 149 simplices. Sloane, N. J. A. (ed.). "Sequence A001632
149_(number)
Subdivision of a planar object into triangles
simplices in T {\displaystyle T} . That is, it is a locally finite simplicial complex that covers the entire space. A point-set triangulation, i.e., a triangulation
Triangulation_(geometry)
refer to such a construction as a simplicial complex. The boundary operator on this triangulation/simplicial complex T is defined in the usual way: for
Discrete_exterior_calculus
topological data analysis (TDA) to associate a sequence of nested simplicial complexes to a finite data set in order to detect the persistence of topological
Degree-Rips_bifiltration
Technique in topological data analysis
analysis, a subdivision bifiltration is a collection of filtered simplicial complexes, typically built upon a set of data points in a metric space, that
Subdivision_bifiltration
Matrix normal form
compute the homology of a finite simplicial complex or CW complex over the integers, because the boundary maps in such a complex are just integer matrices.
Smith_normal_form
Map between simplicial sets with lifting property
In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category
Kan_fibration
American mathematician (born 1937)
work in another monograph with Springer, Boolean Representations of Simplicial Complexes and Matroids. John Rhodes and Benjamin Steinberg (2008), The q-theory
John_Rhodes_(mathematician)
SIMPLICIAL COMPLEX
SIMPLICIAL COMPLEX
Boy/Male
Indian, Punjabi, Sikh
Victory of Simplicity
Girl/Female
Hindu, Indian, Tamil
One with Simplicity; Special Person of All Beings
Girl/Female
Tamil
Hitansi | ஹிதாஂஸீ
Simplicity and purity
Hitansi | ஹிதாஂஸீ
Girl/Female
Indian
Simplicity and purity
Surname or Lastname
English
English : variant of Grice.French (Grisé) : variant spelling of Griset, a nickname for someone with gray hair, a gray complexion, or perhaps one who habitually wore gray, from Old French gris ‘gray’.
Surname or Lastname
Irish
Irish : reduced Anglicized form of Gaelic Ó Duinn, Ó Doinn ‘descendant of Donn’, a byname meaning ‘brown-haired’ or ‘chieftain’.English : nickname for a man with dark hair or a swarthy complexion, from Middle English dunn ‘dark-colored’.Scottish : habitational name from Dun in Angus, named with Gaelic dùn ‘fort’.Scottish : nickname from Gaelic donn ‘brown’. Compare 1.
Surname or Lastname
English (Midlands)
English (Midlands) : nickname for a dark-complexioned man, from Old English earp ‘swarthy’.Americanized spelling of German Erp.
Boy/Male
Hindu, Indian
More Polite; Simplicity
Surname or Lastname
English
English : nickname from Middle English gulle ‘gull’ or gul(le) (Old Norse gulr) ‘yellow’, ‘pale’ (of hair or complexion).Swiss German : nickname for an irascible or unreliable person, from an Alemannic form of Latin gallus ‘rooster’. See also Guell.
Girl/Female
Bengali, Hindu, Indian, Kannada, Marathi, Sanskrit, Tamil, Telugu
Goddess Laxmi; Prosperity; Simplicity; Lovable; Affectionate; Wealthy; Fortunate
Surname or Lastname
English
English : from the popular medieval personal name Hudde, which is of complex origin. It is usually explained as a pet form of Hugh, but there was a pre-existing Old English personal name, Hūda, underlying place names such as Huddington, Worcestershire. This personal name may well still have been in use at the time of the Norman Conquest. If so, it was absorbed by the Norman Hugh and its many diminutives. Reaney adduces evidence that Hudde was also regarded as a pet form of Richard.German : from a short form of a Germanic compound personal name formed with hut ‘guard’ as the first element.Variant spelling of German Hütt (see Huett).Jewish (Ashkenazic) : metonymic occupational name from Yiddish hut, German Hut ‘hat’ (see Huth).
Girl/Female
Greek Latin Spanish
Pastoral simplicity and happiness.
Surname or Lastname
English
English : from Old English dūst ‘dust’, applied as a nickname, possibly for someone with a dusty complexion or hair (as, for example, a miller), or for a worthless person.North German : possibly a Westphalian habitational name from a farm named with dost ‘bush’, ‘brush’. However, the word also means ‘fine dust’, ‘flour’ and may have been applied as an occupational nickname for a miller. Compare 1.
Surname or Lastname
English
English : nickname for a person with a ruddy complexion, from an adjective derivative of Middle English mad(d)er ‘madder’, the dye plant (see Mader 1), here used in a transferred sense.
Surname or Lastname
English
English : nickname for someone with a complexion that was as ‘white as a lily’ (Middle English lilie).
Surname or Lastname
German
German : nickname from the small medieval coin known as the häller or heller because it was first minted (in 1208) at the Swabian town of (Schwäbisch) Hall. Compare Hall.Jewish (Ashkenazic) : habitational name for someone from Schwäbisch Hall.German : topographic name for someone living by a field named as ‘hell’ (see Helle 3).English : topographic name for someone living on a hill, from southeastern Middle English hell + the habitational suffix -er.Dutch : from a Germanic personal name composed of the elements hild ‘strife’ + hari, heri ‘army’.Jewish (Ashkenazic) : nickname for a person with fair hair or a light complexion, from an inflected form, used before a male personal name, of German hell ‘light’, ‘bright’, Yiddish hel.
Girl/Female
Indian
Simplicity and purity
Girl/Female
Tamil
Hitanshi | ஹிதாஂஷீÂ
Simplicity and purity
Hitanshi | ஹிதாஂஷீÂ
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Tamil, Telugu
Virtuous Woman; Simplicity
Boy/Male
Indian, Punjabi, Sikh
Love for Simplicity
SIMPLICIAL COMPLEX
SIMPLICIAL COMPLEX
SIMPLICIAL COMPLEX
SIMPLICIAL COMPLEX
SIMPLICIAL COMPLEX
SIMPLICIAL COMPLEX
SIMPLICIAL COMPLEX