By Jaime Nava, Vladik Kreinovich
This e-book demonstrates the right way to describe and study a system's habit and extract the specified prediction and regulate algorithms from this research. a customary prediction is predicated on gazing comparable events long ago, understanding the results of those prior events, and watching for that the long run consequence of the present state of affairs should be just like those prior saw results. In mathematical phrases, similarity corresponds to symmetry, and similarity of results to invariance.
This publication exhibits how symmetries can be utilized in all sessions of algorithmic difficulties of sciences and engineering: from research to prediction to manage. purposes conceal chemistry, geosciences, clever keep an eye on, neural networks, quantum physics, and thermal physics. in particular, it's proven how the procedure according to symmetry and similarity can be utilized within the research of real-life structures, within the algorithms of prediction, and within the algorithms of control.
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Extra resources for Algorithmic Aspects of Analysis, Prediction, and Control in Science and Engineering: An Approach Based on Symmetry and Similarity
One can easily check that if a function F(x) satisfies the desired propdef erty, then, for every two real numbers c1 > 0 and c0 , the function F(x) = c1 · F(x) + c0 also satisfies this property. We thus say that the function F(x) = c1 · F(x) + c0 is equivalent to the original function F(x). 1. Every monotonic solution of the above functional equation is equivalent to log(x) or to xα . Conclusion. So, symmetries and similarities do explain the selection of the function F(x) for I-complexity. 1. 1◦ .
For a physical field, we need to know how different field components change with time. Let us assume that we know the state γ at some initial moment of time t (γ (t) = γ ), and we know the state γ at some future moment of time t (γ (t) = γ ). , trajectories γ (t) for which γ (t) = γ and γ (t) = γ . For each fundamental physical theory, we can assign, to each trajectory γ (t), we can assign a value S(γ ) such that among all possible trajectories, the actual one is the one for which the value S(γ ) is the smallest possible.
We can reconstruct the original function p(ξ (ω1 ), . . , ξ (ωn ), . ) as p(ξ (ω1 ), . . , ξ (ωn ), . ) = exp(P(ln(ξ (ω1 )), . . , ln(ξ (ωn )), . ) Since ln(x · y) = ln(x) + ln(y), in terms of P and Zi , the independence requirement takes the form P(Z11 + Z21, . . , Z1n + Z2n , . ) = P(Z11 , . . , Z1n , . ) + P(Z21 , . . , Z2n , . ), def where we denoted Zi j = ln(ξi (ω j )). Let us further simplify this formula by expressing it in terms of real Xi j and imaginary parts Yi j of the real numbers Zi j = Xi j + i ·Yi j .