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θ k k th direction of arrival.
Now, let be a vector of length in the th direction for.
where d(x i,y i ) is the distance from the centroid to the contour point (x i,y i ) in the i th direction.
The term u i j is the i th component of the displacement at (x1, x2, x3) due to the j th direction point force of magnitude F at (ξ1, ξ2, ξ3).
If we use ISL d,l (x, y) (l = 1, 2,...,L) to denote the ISL located at (x, y) in the d th direction (d = 1, 2, 3) and l th scale.
where PSL d,l (x, y) denotes the ISL of multiscale products located at (x, y) in l th scale and d th direction; P d,l f x, y) and MPS d,l (x, y) are the corresponding multiscale products and the feature contrast.
Similar(51)
where (overline {n}_{i}) denotes the mean phonon number of the harmonic oscillator in i-th direction and (sigma _{i,0}=sqrt {hbar /(2,m,omega _{i})}).
is the individual best location of particle in the th dimensional direction, is the population best location in the th dimensional direction.
From the known occurrence frequency of the amino acid group i in the ancestral node (the amino acid groups are listed in Table 4),, and in the offspring node,, for the n-th cluster, the direction of change in the amino acid group i on the tree branch j was determined as.
where is the current iteration number; and denote the velocity and location of the particle in the th dimensional direction.
Substituting (4.7) and (4.9) into (4.4), it results the expression of the perturbation in the k th estimated direction of arrival as follows, Δ θ k = Re [ - a ( θ k ) H Δ U p U p H a ( 1 ) ( θ k ) ] ∥ a ( 1 ) ( θ k ) H U p ∥ 2 (4.10).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com