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Simplex Tree
In topological data analysis, a simplex tree is a type of trie used to represent efficiently any general simplicial complex. Through its nodes, this data structure notably explicitly represents all the simplices. Its flexible structure allows the implementation of many basic operations useful to computing persistent homology. This data structure was invented by Jean-Daniel Boissonnat and Clément Maria in 2014, in the article ''The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes''. This data structure offers efficient operations on sparse simplicial complexes. For dense or maximal simplices, Skeleton-Blocker representations or Toplex Map representations are used. Definitions Many researchers in topological data analysis consider the simplex tree to be the most compact simplex-based data structure for simplicial complexes, and a data structure allowing an intuitive understanding of simplicial complexes due to integrated usage of their mathematical prope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a -simplex is a -dimensional polytope that is the convex hull of its vertices. More formally, suppose the points u_0, \dots, u_k are affinely independent, which means that the vectors u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points C = \left\. A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vietoris–Rips Complex
In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a 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 a simplex for every finite set of points that has diameter at most δ. That is, it is a family of finite subsets of ''M'', in which we think of a subset of ''k'' points as forming a (''k'' − 1)-dimensional simplex (an edge for two points, a triangle for three points, a tetrahedron for four points, etc.); if a finite set ''S'' has the property that the distance between every pair of points in ''S'' is at most δ, then we include ''S'' as a simplex in the complex. History The Vietoris–Rips complex was originally called the Vietoris complex, for Leopold Vietoris, who introduced it as a means of extending homology theory from simplicial complexes to metric spaces. After Eliyahu Rips applied ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flag Complex
Clique complexes, independence complexes, flag complexes, Whitney complexes and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques (complete subgraphs) of an undirected graph. Clique complex The clique complex of an undirected graph is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the cliques of . Any subset of a clique is itself a clique, so this family of sets meets the requirement of an abstract simplicial complex that every subset of a set in the family should also be in the family. The clique complex can also be viewed as a topological space in which each clique of vertices is represented by a simplex of dimension . The 1-skeleton of (also known as the ''underlying graph'' of the complex) is an undirected graph with a vertex for every 1-element set in the family and an edge for every 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Contraction
In graph theory, an edge contraction is an operation that removes an edge from a graph while simultaneously merging the two vertices that it previously joined. Edge contraction is a fundamental operation in the theory of graph minors. Vertex identification is a less restrictive form of this operation. Definition The edge contraction operation occurs relative to a particular edge, e. The edge e is removed and its two incident vertices, u and v, are merged into a new vertex w, where the edges incident to w each correspond to an edge incident to either u or v. More generally, the operation may be performed on a set of edges by contracting each edge (in any order). The resulting graph is sometimes written as G/e. (Contrast this with G \setminus e, which means simply removing the edge e without merging its incident vertices.) As defined below, an edge contraction operation may result in a graph with multiple edges even if the original graph was a simple graph. However, some a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Betti Number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. The ''n''th Betti number represents the rank of the ''n''th homology group, denoted ''H''''n'', which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. For example, if H_n(X) \cong 0 then b_n(X) = 0, if H_n(X) \cong \mathbb then b_n(X) = 1, if H_n(X) \cong \mathbb \oplus \mathbb then b_n(X) = 2, if H_n(X) \cong \mathbb \oplus \mathbb\oplus \mathbb then b_n(X) = 3, etc. Note that only the ranks of infinite groups are considered, so for example if H_n(X) \cong \mathbb^k \oplus \mathbb/(2), where \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Representation
In mathematics, a complex representation is a representation of a group (or that of Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...) on a complex vector space. Sometimes (for example in physics), the term complex representation is reserved for a representation on a complex vector space that is neither real nor pseudoreal (quaternionic). In other words, the group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation. For compact groups, the Frobenius-Schur indicator can be used to tell whether a representation is real, complex, or pseudo-real. For example, the N-dimensional fundamental representation of SU(N) for N greater than two is a complex representation whose complex co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linked List
In computer science, a linked list is a linear collection of data elements whose order is not given by their physical placement in memory. Instead, each element points to the next. It is a data structure consisting of a collection of nodes which together represent a sequence. In its most basic form, each node contains data, and a reference (in other words, a ''link'') to the next node in the sequence. This structure allows for efficient insertion or removal of elements from any position in the sequence during iteration. More complex variants add additional links, allowing more efficient insertion or removal of nodes at arbitrary positions. A drawback of linked lists is that data access time is linear in respect to the number of nodes in the list. Because nodes are serially linked, accessing any node requires that the prior node be accessed beforehand (which introduces difficulties in pipelining). Faster access, such as random access, is not feasible. Arrays have better cache ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Data Analysis
In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools. The initial motivation is to study the shape of data. TDA has combined algebraic topology and other tools from pure mathematics to allow mathematically rigorous study of "shape". The main tool is persistent homology, an adaptation of homology to point cloud data. Persistent homology has been applied to many types of data across many fields. Moreover, its mathematical foundation is also of theoretica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special case of an ''arc (geometry), arc'', with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline (vinculum (symbol), vinculum) above the symbols for the two endpoints, such as in . Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (geometry), edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point (geometry)
In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical spaces. As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, and higher-dimensional objects consist. In classical Euclidean geometry, a point is a primitive notion, defined as "that which has no part". Points and other primitive notions are not defined in terms of other concepts, but only by certain formal properties, called axioms, that they must satisfy; for example, ''"there is exactly one straight line that passes through two distinct points"''. As physical diagrams, geometric figures are made with tools such as a compass, scriber, or pen, whose pointed tip can mark a small dot or prick a small hole representing a point, or can be drawn across a surface to represent a curve. A po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
