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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
One of the fundamental concepts in a topological space is the concept of neighborhood. Intuitively, the neighborhood of a point is a set which contains the point and other points arbitrarily defined as similar or close. The notion of similarity or closeness can be defined in terms of a metric or rule.
Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
A neighborhood of a point a in X is any set containing an open set containing a. As we shall see, certain concepts such as continuity and convergence are conveniently phrased in terms of neighborhoods.
Degenerate type fractional evolution hemivariational inequalities and optimal controls via fractional resolvent operators
Published in International Journal of Control, 2020
The multi-valued map is called upper semicontinuous (u.s.c.) on Z if for each , the set is a nonempty, closed subset of Z, and if for each open set of Z containing , there exists an open neighbourhood of such that . Also, G is said to be completely continuous if is relatively compact for every . G has a fixed point if there exists such that .
Slow–fast systems and sliding on codimension 2 switching manifolds
Published in Dynamical Systems, 2019
Paulo Ricardo da Silva, Willian Pereira Nunes
For , S is the slow manifold of system (15). Let be the open subset of S given by The matrix is given by The hypothesis implies that the rank of is 2 and the eigenvalues of have a nonzero real part. In other words, . Since is an open set there exists a neighbourhood such that . Using Theorem 2.1, there exists a family of manifolds with such that and is an invariant manifold , proving that is a sliding point for X.
Extraction of building roof contours from the integration of high-resolution aerial imagery and laser data using Markov random fields
Published in International Journal of Image and Data Fusion, 2018
Vanessa Jordão Marcato Fernandes, Aluir Porfírio Dal Poz
Let be a family of random variables defined over R, where each corresponds to . Thus, is an MRF in with respect to the neighbourhood system if and only if for all realisations of , and . In this context, a clique c is a subset of the primitives of R such that each pair of different primitives in c is a neighbour. The set of all cliques of R with respect to the neighbourhood system is represented as .