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Array and Array-Handling Intrinsics
Published in Subrata Ray, Fortran 2018 with Parallel Programming, 2019
where m is the number of points, and the range of the summation is 1 to m. The derivation may be found in any standard textbook on numerical analysis. Here, ∑ indicates the summation. Thus, a2=(∑yi ∑xi2 - ∑xi ∑xi yi) / [m ∑xi2 –(∑xi)2] a1=(m ∑xi yi - ∑xi ∑yi) / [m ∑xi2 –(∑xi)2]
Gravitational Search Algorithm
Published in Nazmul Siddique, Hojjat Adeli, Nature-Inspired Computing, 2017
Gravity is the weakest of the four fundamental forces, and appears to have unlimited range (unlike the strong or weak force). The gravitational force is approximately 10−38 times the strength of the strong force (i.e., gravity is 38 orders of magnitude weaker), 10−36 times the strength of the electromagnetic force, and 10−29 times the strength of the weak force. Gravity is responsible for causing the Earth and the other planets to orbit the Sun; and for causing the Moon to orbit the Earth. Gravity is the only force acting on all objects with mass; has an infinite range; is always attractive and never repulsive; and cannot be absorbed, transformed, or shielded against. This means gravity acts between objects without any intermediary and without any delay (Holliday et al., 1993; Schutz, 2003). According to Equation 2.1, there is an attracting gravitational force among all bodies of the universe and the gravitational force is greater for bigger and closer bodies. Increasing the distance between two bodies will decrease the gravitational force between them. This is also illustrated in Figure 2.2. The total gravitational force F on a body of mass m produced by objects with masses mi, i = 1, 2, …, n, is given byF=Gm∑i=1nmiriri3 where ri is the special vector from the mass m to the masses mi. The force on m due to masses mi becomes the sum of the forces due to each mass separately. The individual forces are added together vectorially to yield the direction and magnitude of the force. The symbol ∑ represents summation. The factor ri/ri3 is used to yield the direction, which is numerically equivalent to division by ri2. Equation 2.2 is Newton's gravitational law in its vector form.
Distributed secure estimation for cyber-physical systems with fading measurements and false data injection attacks
Published in International Journal of Systems Science, 2020
Li Li, Jingjing Guo, Yuanqing Xia, Hongjiu Yang
Notation: The notations used in this paper are standard. and represent the n-dimensional Euclidean space and set of all matrices, respectively. For matrices X and Y, the notation X<Y means that X−Y is negative definite. and represent its transpose and inverse, respectively. denotes the trace of a matrix. stands for the Euclidean norm of a vector. describes the mathematical expectation of x. I is an identity matrix with an appropriate dimension. stands for a diagonal matrix. The symbol denotes the summation of a sequence.