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==Random Iterative Systems==

==Random Iterative Systems==

We can extend the above results to linear iterative dynamical systems. For an iterative system of the form:

We can extend the above results to linear iterative dynamical systems<ref name=liu_exact />. For an iterative system of the form:

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</math>

</math>

Among them, [math]W[/math] is a coarse-grained matrix with an order of n * m (m is the dimension of the macroscopic state space), and its function is to map any microscopic state [math]x_t[/math] to a macroscopic state [math]y_t[/math]. [math]W^{\dagger}[/math] is the pseudo inverse operation of W. The first term in the equation is the emergence caused by determinism, abbreviated as Deterministic Emergence, and the second term is the emergence caused by degeneracy, abbreviated as Degenerative Emergence. For more detailed information, please refer to the causal emergence of stochastic iterative systems.

Among them, [math]W[/math] is a coarse-grained matrix with an order of n * m (m is the dimension of the macroscopic state space), and its function is to map any microscopic state [math]x_t[/math] to a macroscopic state [math]y_t[/math]. [math]W^{\dagger}[/math] is the pseudo inverse operation of W. The first term in the equation is the emergence caused by determinism, abbreviated as Deterministic Emergence, and the second term is the emergence caused by degeneracy, abbreviated as Degenerative Emergence. For more detailed information, please refer to [[the causal emergence of stochastic iterative systems]].

==Feedforward Neural Networks==

==Feedforward Neural Networks==