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Pertinent Properties of Euclidean Space
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
Note that in the example we have A∩B=∅ but A and B are not disconnected. Another way of approaching the topic of connected sets in ℝnor topological spaces in general is by defining the concept of separated sets. More specifically, two subsets A and B of a topological space X are said to be separated if and only if A∩B¯=∅ and A¯∩B=∅. Using this definition of separation results into the following theorem:
Bag of Riemannian Words for Virus Classification
Published in Biju Issac, Nauman Israr, Case Studies in Intelligent Computing, 2014
Masoud Faraki, Mehrtash Harandi
13.3.1 Riemannian Geometry We start the discussion here by rst introducing an analytic manifold. An analytic manifold is a Hausdor, topological space equipped with a complete atlas (Subbarao and Meer 2009). A set X and a family of its subsets T (including the empty set and X ) form a topological space if the union and intersection of any nite number of sets in T also lie in T. As a simple example, one can consider the intervals on a real line. Let X = {(1,3)} be the set of real numbers lying in the open interval (1,3) and T = {(1,3),(1,2),[2,3),{2},{}}. en, T is called a topology of X. A Hausdorff (separated) space is a topological space such that any two points belonging to it can be separated by disjoint neighborhoods. So, in our example, the set X is a Hausdor space. A complete atlas is a set of all smoothly overlapping neighborhood-mapping (U, ϕ) pairs, such that U is an open interval on the m-dimensional manifold and ϕ is a one-to-one and onto mapping (also called homeomorphic) from U to some Euclidean space m. erefore, a manifold can be understood as a (topological) space that resembles Euclidean space at a small neighborhood of each point. An intuitive example of manifold is the surface of a sphere that is not a vector space but can be perceived at (Euclidean) in a small neighborhood of each point of the surface.
Introduction to Learning in Games
Published in Hamidou Tembine, Distributed Strategic Learning for Wireless Engineers, 2018
A function between topological spaces is called continuous if the inverse image of every open set is open. A Hausdorff space or separated space is a topological space in which distinct points can have disjoint neighborhoods.
A long-term field study of the ability to predict thermal cracking of asphalt mixtures tested by the Bending Beam Rheometer
Published in Road Materials and Pavement Design, 2021
Abu Sufian Mohammad Asib, P. Romero
In 2014, researchers showed that, in a manner comparable to binder testing in the BBR, both the creep modulus (or stiffness) and the stress relaxation capacity (m-value) at 60-seconds are needed to evaluate asphalt mixture's thermal cracking potential (Jones et al., 2014). By following previous research (King et al., 2012), Jones et al. (2014) proposed that the data can be presented in a plot analogous to a traditional Black Space diagram in which the stiffness is plotted on the Y-axis against the m-value on the X-axis. They suggested that mixtures with high modulus can still have a good performance if they also have a high m-value. It implies that in their modified Black Space plot, mixtures can be separated by some diagonal lines. Mixtures that fall on the lower right of the line are expected to have good thermal cracking performance, while those in the upper left should not. They defined this line as a ‘thermal stress failure envelope’ (Jones et al., 2014).
A polarization tensor approximation for the Hessian in iterative solvers for non-linear inverse problems
Published in Inverse Problems in Science and Engineering, 2021
F. M. Watson, M. G. Crabb, W. R. B. Lionheart
It is worth noting that for containing perturbations not sufficiently separated, may be a poor approximation to the diagonal of the Hessian. This may limit the effectiveness of our approach, but we could choose instead to initialize with or otherwise some thresholded version.
Surface-patterning generated half-skyrmion lattices in cholesteric blue phase thin films
Published in Liquid Crystals, 2018
Shun Wang, Miha Ravnik, Slobodan Žumer
Figure 4(b) shows the half-skyrmion lattice obtained using Pattern S1 for . Disclinations become very curvy. Each pair of disclinations is separated along the diagonal direction of the midplane of the simulation box. The director field is locally uniform not only between each pair of disclinations but between any two −1/2 disclinations.