By Jaime Nava, Vladik Kreinovich
This e-book demonstrates how one can describe and study a system's habit and extract the specified prediction and keep watch over algorithms from this research. a standard prediction is predicated on looking at related events long ago, realizing the results of those prior occasions, and looking forward to that the longer term consequence of the present state of affairs could 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 periods 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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Additional resources for Algorithmic Aspects of Analysis, Prediction, and Control in Science and Engineering: An Approach Based on Symmetry and Similarity
In this case, each state γ of the composite system is a pair γ = (γ1 , γ2 ) consisting of a state γ1 of the first subsystem and the state γ2 of the second subsystem. Often, these subsystems are independent. Due to this independence, the probability of going from a state γ = (γ1 , γ2 ) to a state γ = (γ1 , γ2 ) is equal to the product of the probabilities P1 and P2 corresponding to the first and to the second subsystems: P((γ1 , γ2 ) → (γ1 , γ2 )) = P1 (γ1 → γ1 ) · P2(γ2 → γ2 ). In order to describe how to satisfy this property, let us recall that in physics, independence of two subsystems is usually described by assuming that the corresponding action is equal to the sum of the actions corresponding to two subsystems: S((γ1 , γ2 )) = S1 (γ1 ) + S2(γ2 ).
Chemical interactions are, in effect, interaction of electrons. , . In classical (pre-quantum) physics, to describe the state of a particle at any given moment of time, it is sufficient to describe its spatial location and momentum. We may not know the exact values of these quantities, but in principle, we can determine them with an arbitrarily high accuracy. In contrast, in quantum physics, it is impossible to uniquely determine both spatial location and momentum, we can only predict probabilities of different spatial locations (and different momentum values).
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.