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General properties of fractals
Published in Xie Heping, Fractals in Rock Mechanics, 2020
One of the common ways to form a topological space is through the use of a distance function, also called a metric. For example, on Rn the standard distance, d(x,y)=(∑i=1n(xi−yi))1/2, between x = (xl,…,xn) and y = (yx,…,yn) can be used to construct the open disks and from them the topology. The abstraction of this proceeds as follows.
Perception
Published in Hanky Sjafrie, Introduction to Self-Driving Vehicle Technology, 2019
Applying the above standard non-linear least-square solvers directly to the SLAM problem may lead to a sub-optimal result because the parameters, i.e., the configurations X, are assumed to be in Euclidean space [26]. Therefore, modern SLAM back-ends typically perform on-manifold least-squares optimization, as the space of the rotational parameter is not Euclidian. A manifold is a topological space that locally resembles Euclidean space, but globally might not be Euclidean [33]. On-manifold optimization has basically the same structure as the Euclidian counterpart. For each iteration, a new step is computed in the local Euclidian approximation space. The accumulated increments are projected in the global non-Euclidian space and the process is repeated until convergence [26].
Topological and Metric Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Hausdorff Spaces. In what follows, we restrict ourselves to a class of topological spaces called Hausdorff spaces. A topological space X is said to be Hausdorff if for every two distinct points x and y there exist neighborhoods B of x and C of y such that B∩C=∅. $ B \cap C = \emptyset . $ In other words, every two distinct points can be separated by disjoint neighborhoods. We will see a fundamental consequence of this definition in the next section when we define the limit of a sequence.
A categorical approach to graded fuzzy topological system and fuzzy geometric logic with graded consequence
Published in Journal of Applied Non-Classical Logics, 2022
Purbita Jana, Mihir K. Chakraborty
The motivation of this work mostly comes from the main topic of Vickers' book ‘Topology via Logic’ (Vickers, 1989), where he introduced the notion of topological system and indicated its connection with geometric logic. The relationships among topological space, topological system, frame and geometric logic play an important role in the study of topology through logic (geometric logic). Naturally the question ‘from which logic fuzzy topology can be studied?’ comes to mind. If such a logic is obtained what could be its significance?