By Jaime Nava, Vladik Kreinovich
This publication demonstrates tips to describe and study a system's habit and extract the specified prediction and regulate algorithms from this research. a regular prediction is predicated on looking at related events long ago, understanding the results of those previous occasions, and watching for that the long run end result of the present state of affairs should be just like those previous 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 regulate. purposes hide chemistry, geosciences, clever keep watch over, neural networks, quantum physics, and thermal physics. in particular, it really is proven how the method 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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Additional resources for Algorithmic Aspects of Analysis, Prediction, and Control in Science and Engineering: An Approach Based on Symmetry and Similarity
In general, the action usually the action S = L dt, where L = · 2 dt takes the form S = L dx, where dx means integration over (4-dimensional) spacetime, and the function L is called the Lagrange function. The language of action and Lagrange functions is the standard language of physics: new fundamental theories are very rarely formulated in terms of a differential equation, they are usually formulated in terms of an appropriate Lagrange function – and the corresponding differential equation can then be derived from the resulting minimization principle.
E, we measure the values xi of the parameters and the resulting value y(e) , and then determine the desired coefficients by applying the Least Squares method to the corresponding approximate equations. In the case of linear dependence, we use approximate equations where c0 = f (x1 , . . , xn ), ci = n (e) y(e) ≈ c0 + ∑ ci · Δ xi ; e = 1, 2, . . , E. 5) i=1 In the case of quadratic dependence, we use approximate equations n (e) n y(e) ≈ c0 + ∑ ci · Δ xi + ∑ i=1 n (e) ∑ cii · Δ xi i=1 i =1 (e) · Δ xi .
Kolmogorov, one of the founders of the modern probability theory. , if is a regular sequence like 0101. , we have a reasonable short program for generating this sequence. , unpredictable, then it cannot be described by any law, so the only way to generate this sequence is to simply print it. In other words, for a random sequence s, the length of the shortest program that generates this sequence is approximately equal to the length len(s). , . Once we know the Kolmogorov complexity K(s) of a sequence s, we can tell whether this sequence is random or not: if K(s) ≈ len(s), then the sequence s is random, while if K(s) len(s) the sequence S is not random.