Algorithmic Aspects of Analysis, Prediction, and Control in by Jaime Nava, Vladik Kreinovich PDF
By Jaime Nava, Vladik Kreinovich
This e-book demonstrates the best way to describe and examine a system's habit and extract the specified prediction and regulate algorithms from this research. a customary prediction is predicated on looking at related events long ago, understanding the results of those previous events, and watching for that the longer term final result of the present state of affairs can 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 regulate. functions disguise chemistry, geosciences, clever keep watch over, neural networks, quantum physics, and thermal physics. in particular, it's proven how the method in response to symmetry and similarity can be utilized within the research of real-life platforms, within the algorithms of prediction, and within the algorithms of control.
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Extra info 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 single ligand of type k at the location i – corresponding to all the places i and types k for which εik = 1; and 28 2 Algorithmic Aspects of Real-Life Systems Analysis def • substances aik,i k = (0, . . , 0, 1, 0, . . , 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 ).
So, here, Bs (5) = 0. Good Properties of I-Complexity. Thus defined I-complexity has many properties which are similar to the properties of the original Kolmogorov complexity K(s): • • • • If a string s starts with a substring s , then I(s) ≤ I(s ). We have I(0s) ≈ I(s) and I(1s) ≈ I(s). We have I(ss ) ≤ I(s) + I(s ). Most strings have high I-complexity. On the other hand, in contrast to non-computable Kolmogorov complexity K(s), I-complexity can be computed feasibly: namely, it can be computed in linear time.
Practical Need to Find Dependencies. In situations where prediction is, in principle, possible, the next questions is: how can we predict? In cases where we know the corresponding equations, we can use these equations for prediction. In practice, it often occurs that we know (or conjecture) that a quantity y depends on quantities x1 , . . , xn , but we do not know the exact form of this dependence. In such situations, we must experimentally determine this dependence y = f (x1 , . . , xn ). For that, in several (S) situations s = 1, .
Algorithmic Aspects of Analysis, Prediction, and Control in Science and Engineering: An Approach Based on Symmetry and Similarity by Jaime Nava, Vladik Kreinovich