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Knowledge Representation and Reasoning for the Design of Resilient Sensor Networks
Published in Fei Hu, Qi Hao, Intelligent Sensor Networks, 2012
David Kelle, Touria El-Mezyani, Sanjeev Srivastava, David Cartes
Simplices may be glued together along their faces. Such a geometric structure is called a simplicial complex Δ if every face of every simplex is also considered to be a simplex in Δ. A simplicial complex that is contained in another is called a sub-complex. However, we are only interested in the combinatorial properties of a simplicial complex. With this in mind, we consider an ASC. Definition 5.2 (Abstract simplicial complexes)An ASC is a collection Δ of subsets of a set of vertices V = {x1, …, xn}, such that if S is an element of Δ then so is every nonempty subset of S [11,14]. The dimension of an ASC is the dimension of its largest facet. An ASC is called pure if all of its facets have the same dimension [6].
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Published in S.P. Bhattacharyya, L.H. Keel, of UNCERTAIN DYNAMIC SYSTEMS, 2020
Edmond A. Jonckheere, Jonathan R. Bar-on
The boundary of a simplicial complex is defined by linearity from the boundary of its simplices. More formally, the boundary appears to be a group homomorphism ∂n:Cn↦Cn−1
Elements of topology and homology
Published in Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama, Computational Topology for Biomedical Image and Data Analysis, 2019
Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama
Simplicial complexes Geometric simplicial complexes are topological spaces comprised of finite sets of points, line segments, triangles, and the triangle generalization into higher dimensions. These constituents are called simplexes. The simplexes of geometric simplicial complexes are embedded in an affine vector space.
Topological machine learning for multivariate time series
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Chengyuan Wu, Carol Anne Hargreaves
The cycle group and boundary group are defined as and respectively. The th homology group is defined to be the quotient group . Informally, the rank of the th homology group (also called the th Betti number) counts the number of -dimensional holes in the simplicial complex . For instance, counts the number of connected components (0-dim holes), counts the number of ‘circular holes’ (1-dim holes), while counts the number of ‘voids’ or ‘cavities’ (2-dim holes). We show an example in Figure 3.
Methods of computational topology and discrete Riemannian geometry for the analysis of arid territories
Published in Cogent Engineering, 2020
Lyailya Karimova, Alexey Terekhov, Nikolai Makarenko, Andrey Rybintsev
In algebraic topology, chain proximity leads to the concept of topological nerve coverage of a point set (Edelsbrunner & Harer, 2009). Let,…N be the sample points on the plane . Decorate each point with a disk with a radius ε centered at. We will simultaneously increase the radii of all the disks. The intersection of 2 disks is replaced by an edge connecting their centers. The intersection of 3 discs is replaced by a triangular face. The resulting structure consisting of the simplest elements of vertices, edges and faces, is called a simplicial complex if its adjacent elements intersect at a point or have a common edge. So, the dilatation of the disks leads to the complex, which is called the Čech complex:
A mosaic of Chu spaces and Channel Theory I: Category-theoretic concepts and tools
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2019
Chris Fields, James F. Glazebrook
We first introduce simplicial complexes as representations of observational data, following Cordier and Porter (1989); Friedman (2012); Goerss and Jardine (1999) and Spanier (1966). Definition 5.1. A simplicial complex consists of a set of objects called the vertices and a set of finite, non-empty subsets of called the simplices. The latter satisfy the condition that if is a simplex, and if (with ), then is also a simplex. Simplicial complexes are objects in a category denoted .