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Nonlinear Estimation and Filtering
Published in Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala, Nonlinear Filtering, 2017
Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala
In Equation 8.347 the surface area of the unit sphere is An=2πn/Γ(n/2),Γ(n)=∫0∞xn−1e−xdx. By solving Equations 8.347 and 8.348, we obtain w=An2n,u2=1. Thus, the cubature points are located at the intersection of the unit sphere and its axes.
Numerical differentiation and integration
Published in Victor A. Bloomfield, Using R for Numerical Analysis in Science and Engineering, 2018
The formula for the volume of a sphere of radius r is 43πr3, the exact value being V = 4.18879 for radius 1. We see that even for a million points the result is not nearly exact. For good results one needs huge numbers of random points.
Micropolar continua as projective space of Skyrmions
Published in Philosophical Magazine, 2022
In practice, after we identify the order parameter space M of (24), in order to determine the homotopy groups, we will proceed according to the following steps. We identify the dimension m of the manifold M where the medium is defined. This can be different from the dimension of physical space where the medium is placed.We take account of the dimensionality d of the physically possible defect.We identify the n-sphere which surrounds the region of defects.
Observation of string defects in liquid crystal
Published in Liquid Crystals, 2021
Kirandeep Kaur Matharu, Samriti Khosla, Sapna Sethi, Nitin Sood
Phase transitions associated with change of symmetry are described through Spontaneous symmetry breaking [14]. In this case the higher symmetry phase (less ordered) is associated with a unique true vacuum whereas in the ordered phase, the vacuum has degenerated structure and the system can choose any one of the degenerate vacua randomly after the phase transition. Consequently, a set of minimum energy configurations forms which represent vacuum manifold (M). For topological excitations manifold defines the order parameter [11]. The topology of ‘M’ represents the kind of defect to be formed while approaching the more ordered state [6,7]. These defects are represented by different homotopy groups ᴨn(M). The homotopy groups characterise mapping from the n-sphere Sn enclosing the topological excitation in real space into the vacuum manifold M [11]. In general, πn (Mo) gives the number of topologically distinct noncontractible n-spheres and π1(Mo) gives the number of topologically distinct noncontractible loops. Strings occur in theories with nontrivial π1(Mo) [2,6,7].
The wild Fox–Artin arc in invariant sets of dynamical systems
Published in Dynamical Systems, 2018
T. V. Medvedev, O. V. Pochinka
Now we project this dynamics to the n-sphere. Denote by the North Pole of the sphere . For each point there is the unique line through N and x in and this line intersects the plane Ox1…xn in exactly one point ϑ+(x) (see Figure 3). The stereographic projection of the point x is defined to be this point ϑ+(x). One can easily find that Thus, stereographic projection is a diffeomorphism.