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The integration layer includes the update function for processing the physics engine collisions and callbacks to process expired collisions.
However, in our situation, because of the Markov property of the sight condition, the derivation of the density update function for all the state variables, and the static parameter is not an easy task.
The update function for a reaction depends on its substrates and its contingencies while the update function for a state depends on the reactions connected to it.
So we get an update function for the two possible states and their producing reactions.
We consider network damage in the form of node knockout or constitutive expression; i.e., the update function for a damaged node x i becomes x i (t + τ i ) = 0 or x i (t + τ i ) = 1.
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Similarly, the LLR updating function for u r can be expressed as: L ( u r k ) = ln Pr ( u r k = 0 ) Pr ( u r k = 1 ) = ln ( 1 − p ̂ e ) · exp [ L ( u k ) ] + p ̂ e ( 1 − p ̂ e ) + p ̂ e · exp [ L ( u k ) ]. (12).
Equations (14) provide the updating function for gbest position and gbest value, respectively, as follows: (14) G = { P j, p b e s t j ≥ g b e s t, G, p b e s t j < g b e s t, g b e s t = { p b e s t j, p b e s t j ≥ g b e s t, g b e s t, p b e s t j < g b e s t.
Equations (13) are the updating functions for a particle's pbest position and pbest value, respectively, as follows: (13) P j = { X j, f (X j ) ≥ p b e s t j, P j, f (X j ) < p b e s t j, p b e s t j = { f (X j ), f (X j ) ≥ p b e s t j, p b e s t j, f (X j ) < p b e s t j, where gbest indicates the best value of all pbest values for a particle and its position is denoted by G = (g1, g2, …, g d ).
In the proof we constructed, for sufficiently large N, a suitable updating function f for Boolean systems B = ( 2 N, f ) such that f was cooperative, bi-quadratic, and worked as required.
Double-bottom maps are used to design an updating function to balance the exploration and exploitation for PSO search capability.
The differences in the update functions for other types of contingencies are shown in Fig. 1c.
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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