Topology, nonlinear

Nonlinear Dynamics and Topological Time Series Analysis Archive http //t13.lanl.gov/ nxt/intro.html... 

Introductory Remarks.—As was mentioned in the introduction to this chapter, the quantitative part of the theory of Poincar6 was first applied in celestial mechanics.1 The two approaches the topological,2 and the analytical are unrelated in the original publications of Poincar6, and the connection between the two appeared nearly 50 years later when the theory of nonlinear oscillations was developed. 

In addition to looking for data trends in physical property space using PCA and PLS, trends in chemical structure space can be delineated by viewing nonlinear maps (NLM) of two-dimensional structure descriptors such as Unity Fingerprints or topological atom pairs using tools such as Benchware DataMiner [42]. Two-dimensional NLM plots provide an overview of chemical structure space and biological activity/molecular properties are mapped in a 3rd and/or 4th dimension to look for trends in the dataset. 

Generally, a chemical plant is composed of several units with several streams and components. The set of material and energy balances constitutes a set of linear/nonlinear equations, which can be represented by an undirected graph. However, when the number of units and streams is large, the graphical representation becomes cumbersome. An alternative representation of the topological structure of the balances in a chemical process is achieved using the occurrence matrix. 

Electronic descriptors were calculated for the ab initio optimized (RHG/STO-3G) structures. In addition, logP as a measure of hydrophobicity and different topological indices were also calculated as additional descriptors. A nonlinear model was constructed using ANN with back propagation. Genetic algorithm (GA) was used as a feature selection method. The best ANN model was utilized to predict the log BB of 23 external molecules. The RMSE of the test set was only... 

More advanced mathematical aspects of aromaticity are given in other references [33, 34]. Some alternative methods beyond the scope of this chapter for the study of aromaticity in deltahedral molecules include tensor surface harmonic theory [35-38] and the related Hirsch 2 N -b 1) electron-counting rule for spherical aromaticity [39]. The topological solitons of nonlinear field theory related to the Skyrmions of nuclear physics have also been used to describe aromatic cluster molecules [40]. 

All topological theories are nonlinear, a feature of both the Sachs and Evans theories, and the whole of quantum theory can be replaced by topology [1], which reduces in some circumstances to the Yang-Mills theory [1], of which 0(3) electrodynamics [3] is an example. 0(3) electrodynamics has been developed into an 0(3) symmetry quantum field theory by Evans and Crowell... 

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