Algorithmic Aspects Of Analysis, Prediction, And Control In Science And Engineering: Symmetry-Based Approach

Algorithms are extremely important in science and engineering. One of the main objectives of science is to predict future events; this usually requires sophisticated algorithms. Once we are able to predict future events, a natural next step is to influence these events, i.e., to control the corresponding systems; control also usually requires complex algorithms. To be able to predict and control a system, we need to have a good description of this system, so that we can use this description to analyze the system\u27s behavior and extract the desired prediction and control algorithms from this analysis. A typical prediction is based on the fact that we observed similar situations in the past; we know the outcomes of these past situations, and we expect that the future outcome of the current situation will be similar to these past observed outcomes. In mathematical terms, similarity corresponds to symmetry, and similarity of outcomes -- to invariance. Because symmetries are ubiquitous and useful, we will show how symmetries can be used in all classes of algorithmic problems of sciences and engineering: from analysis to prediction to control. Specifically, we show how the symmetry-based approach can be used in the analysis of real-life systems, in the algorithmics of prediction, and in the algorithmics of control