Mathematical Preliminaries

This concludes the mathematical preliminaries needed for our discussion of systems with three or more independent concentrations. The different dynamical responses which arise from the various bifurcations listed above will now be exemplified through a number of model schemes, each with three variables. 

In this volume, several key concepts used by practicing computational chemists are brought into focus. The first chapter by Tamar Schlick is dedicated to the mathematics of optimization. After some mathematical preliminaries, approaches to large-scale optimization are described. Basic decent structure of local methods is highlighted, and then nonderivative, gradient, and Newton methods are explained. 

Fluid mechanics is the study of motions of gases and liquids. In the literature the subject is often divided into compressible and incompressible flows. However, the strict mathematical definitions of these two types of flows are not trivial. There is more to this than one may think, as will be discussed in sect. 1.2.5. Meanwhile, considering physical properties rather than strict mathematics, preliminary characterizations of these two classes of flows will provide important ideas why this subdivision of fluid mechanics is useful. 

As a mathematical preliminary, we note the following. To deal with molecules, we are required to transform freely between the laboratory reference frame and the body-fixed frame that rotates with the molecule. The rotation from the lab frame (x,y,z)... 

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