Download e-book for iPad: Algorithmic Aspects of Analysis, Prediction, and Control in by Jaime Nava, Vladik Kreinovich
By Jaime Nava, Vladik Kreinovich
This ebook demonstrates the right way to describe and study a system's habit and extract the specified prediction and regulate algorithms from this research. a standard prediction is predicated on looking at comparable occasions long ago, figuring out the results of those earlier occasions, and anticipating that the long run final result of the present scenario should be just like those previous saw results. In mathematical phrases, similarity corresponds to symmetry, and similarity of results to invariance.
This booklet indicates how symmetries can be utilized in all sessions of algorithmic difficulties of sciences and engineering: from research to prediction to manage. functions hide chemistry, geosciences, clever regulate, neural networks, quantum physics, and thermal physics. in particular, it's proven how the strategy in accordance with 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
0, 1, 0, . . , 0) with a ligand of type k at the locations i and a ligand of type k at a location i – corresponding to all possible pairs (i, k) and (i , k ), i < i , for which εik = εi k = 1. 17) takes the form ∑ y = V (a0 ) + ∑ V (aik ) + V (aik,i k ). 18) i
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. This value S(γ ) is called action, and the principle that action is minimized along the actual trajectory is called the least action principle.
As we have mentioned earlier, we show that the poset-related approaches are, in effect, equivalent to the use of a much simpler (and much more familiar) tool of (discrete) Taylor series. Discrete Taylor Series: Reminder. In many practical situations, we have a physical variable y that depends on the discrete parameters εik which take two possible values: 0 and 1, and for which, for every i, at most one value εik can be equal to 1. 3 Properties of Molecules with Variant Ligands 27 Then, in the first approximation, the dependence of y on εik can be described by the following linear formula n m y = v0 + ∑ ∑ vik · εik .
Algorithmic Aspects of Analysis, Prediction, and Control in Science and Engineering: An Approach Based on Symmetry and Similarity by Jaime Nava, Vladik Kreinovich