China
Africa
Explore chapters and articles related to this topic
Lattice Algebra
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
Another metric that is computable using only addition and the operations of max and min is the Hausdorff metric. The Hausdorff distance, named after Felix Hausdorff who introduced this distance in 1914 [110], measures the distance between two nonempty subsets of a metric space. More specifically, given a metric space (M,d) and two nonempty subsets X and Y of M, define d(x,Y)=inf{d(x,y):y∈Y}andd(X,Y)=sup{d(x,Y):x∈X}
Graphical Metric Spaces and Fixed Point Theorems
Published in Dhananjay Gopal, Praveen Agarwal, Poom Kumam, Metric Structures and Fixed Point Theory, 2021
The French mathematician Maurice Fréchet [2] introduced the concept of metric spaces, which is at the center of several research activities. According to need and scope, this concept has been generalized by several mathematicians. In ordinary metric spaces, the metric function satisfies the triangular inequality for all points situated in the space. Most of the proofs of fixed point theorems use the triangular inequality, but with limited utility. Shukla et al. [8] pointed out this fact and introduced the notion of graphical metric spaces. They replaced the triangular inequality for ordinary metric functions with a weaker one, which is satisfied by only those points which are situated on some path included in the graphical structure associated with the space. Moreover, they observed that the contraction mappings in this new setting are more competent than those in usual metric spaces and can be applied to obtain the solution of differential equations. This chapter is devoted to the study of fixed point theorems in graphical metric spaces and their application to integral equations.
Set Theory for Concept Modeling
Published in Richard M. Golden, Statistical Machine Learning, 2020
Another example of a metric space is the Hamming space ({0, 1}d, ρ) where the Hamming distance function ρ:{0, 1}d × {0, 1}d → {0, …, d} is defined such that for x = [x1, …, xd] ∈ {0, 1}d and y = [y1, …, yd] ∈ {0, 1}d: ρ(x,y)=∑i=1d|xi−yi|. That is, the Hamming distance function counts the number of unshared binary elements in the vectors x and y.
Human spatial learning strategies in wormhole virtual environments
Published in Spatial Cognition & Computation, 2023
Christopher Widdowson, Ranxiao Frances Wang
Another important theoretical concept needing clarification is the distinction between Euclidean vs. non-Euclidean geometry and metric vs non-metric space (e.g., Montello, 1992). A metric space satisfies the following properties: 1) the distance from A to B is zero if and only if A and B are the same point; 2) the distance between two distinct points is positive (positivity); 3) the distance from A to B is the same as the distance from B to A (symmetry); and 4) the distance from A to B is less than or equal to the distance from A to B via any third point C (triangle inequality). A Euclidean space is a type of metric space that also satisfies the parallel postulate, therefore a space can be Euclidean, non-Euclidean but metric, or non-metric at all. A spatial representation that does not conform to Euclidean geometry can have violations specific to Euclidean metric (e.g., parallel postulate), or violations of general metric principles that are not specific to Euclidean geometry (e.g., symmetry or triangle inequality). Therefore, it is important to distinguish between Euclidean vs non-Euclidean and metric vs non-metric spaces. When the experimental evidence only involves violation of the general metric properties, it is more appropriate to call it “non-metric” than non-Euclidean, and the theoretical distinction should be referred to as metric vs non-metric instead of Euclidean vs non-Euclidean.
A contribution to best proximity point theory and an application to partial differential equation
Published in Optimization, 2023
Sakan Termkaew, Parin Chaipunya, Dhananjay Gopal, Poom Kumam
Let and ζ be a non-triangular metric on X. We will use the following notations The notions of convergent and Cauchy sequences, as well as completeness, are defined similarly as in a metric space. We say that a subset is closed if the limit of a convergent sequence in U is always an element in U. The collection of all closed subsets in is denoted by . Note that closed subsets of a complete non-triangular metric space is again complete.
A comparative study on the application of SIFT, SURF, BRIEF and ORB for 3D surface reconstruction of electron microscopy images
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2018
Ahmad P. Tafti, Ahmadreza Baghaie, Andrew B. Kirkpatrick, Jessica D. Holz, Heather A. Owen, Roshan M. D’Souza, Zeyun Yu
To compare the accuracy and reliability of 3D SEM surface reconstruction technique using different feature detector algorithms, we apply the proposed system on a synthetic data namely “Face” model (Paysan et al. 2009). A set of 12 2D images and the reference 3D face model are presented in Figure 7. We calculated the geometric difference between the reference 3D point cloud of the “Face” model with those estimated by the image feature detector algorithms. To this end, the Hausdorff Distance unit (HDu) (Cignoni et al. 1998; Munkres 1999) is measured to show how close two subsets of a metric space are to each other.