By Jaime Nava, Vladik Kreinovich
This ebook demonstrates how one can describe and research a system's habit and extract the specified prediction and keep an eye on algorithms from this research. a customary prediction is predicated on watching related occasions some time past, realizing the results of those previous occasions, and anticipating that the longer term end result of the present state of affairs can be just like those earlier saw results. In mathematical phrases, similarity corresponds to symmetry, and similarity of results to invariance.
This e-book exhibits how symmetries can be utilized in all sessions of algorithmic difficulties of sciences and engineering: from research to prediction to manage. purposes hide chemistry, geosciences, clever keep an eye on, neural networks, quantum physics, and thermal physics. in particular, it's proven how the procedure 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
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 .
For which (s) (s) y(s) ≈ f x1 , . . , xn for all s from 1 to S. (The equality is usually approximate since the measurements are approximate and the value y is often only approximately determined by the values of the variables x1 , . . ) 46 3 Algorithmic Aspects of Prediction: An Approach Based on Symmetry and Similarity First Approximation: Linear Dependence. In many practical situations, the dependence f (x1 , . . , xn ) is smooth: informally, this means that small changes in xi lead to equally small changes in y.
In , we have shown that when we only allow one type of ligand, then the exact same interpolation algorithm can be obtained from a more familiar mathematical technique such as Taylor expansion series. In this section, we showed that the similar equivalence holds in the general case, where we have ligands of different type. 4 Why Feynman Path Integration: An Explanation Based on Symmetry and Similarity Need for Quantization. Since the early 1900s, we know that to describe physics properly, we need to take into account quantum effects.