# Algorithmic Aspects of Analysis, Prediction, and Control in by Jaime Nava, Vladik Kreinovich

By Jaime Nava, Vladik Kreinovich

This e-book demonstrates the way to describe and study a system's habit and extract the specified prediction and keep an eye on algorithms from this research. a customary prediction is predicated on gazing related occasions long ago, understanding the results of those previous events, and anticipating that the longer term end result of the present scenario should be just like those prior saw results. In mathematical phrases, similarity corresponds to symmetry, and similarity of results to invariance.

This publication indicates how symmetries can be utilized in all sessions of algorithmic difficulties of sciences and engineering: from research to prediction to regulate. functions hide chemistry, geosciences, clever keep watch over, neural networks, quantum physics, and thermal physics. in particular, it's proven how the strategy in response 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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**Example text**

As a result, we obtain the following expression: n N n N n y = y0 + ∑ ∑ yi j · Δ xi j + ∑ ∑ i=1 j=1 N ∑ ∑ yi j,i j · Δ xi j · Δ xi j , i=1 j=1 i =1 j =1 def where Δ xi j = xi j − di0 j , and y0 , yi j , and yi j,i j are appropriate coefficients. 3 Properties of Molecules with Variant Ligands 25 These formulas can be applied to all possible situations, in which at each location i, different parameters xi1 , . . , xiN can change independently. Situations in which we are interested are characterized by describing, for each location, whether there is a ligand or not, and if yes, exactly which ligand.

I=1 Example. For example, for aaaab, the corresponding values of Bs (i) are 01230. 2 Is Prediction Possible? 43 • For i = 1, the sequence s1 . . si−1 is empty, so Bs (1) = 0. • For i = 2, with s1 s2 = aa, a string s2 = a is a substring of length 1 of the sequence s1 . . si−1 = s1 = a. So, here, Bs (2) = 1. • For i = 3, with s1 s2 s3 = aaa, a string s2 s3 = aa is a substring of length 2 of the sequence s1 . . si−1 = s1 s2 = aa. So, here, Bs (3) = 2. • For i = 4, with s1 s2 s3 s4 = aaaa, a string s2 s3 s4 = aaa is a substring of length 3 of the sequence s1 .

Situations in which we are interested are characterized by describing, for each location, whether there is a ligand or not, and if yes, exactly which ligand. Let εik denote the discrete variable that describes the presence of a ligand of type k at the location i: • when there is no ligand of type k at the location i, we take εik = 0, and • when there is a ligand of type k at the location i, we take εik = 1. , if εik = 1 for some k, then εik = 0 for all k = k. 8), the value y of the desired physical quantity depends on the differences Δ xi j corresponding to different i and j.