This chapter explains the mathematical background used for polarity index method described in Chapter 3 and its relation with the catastrophic bifurcation points located in the geometric representation of the relative frequencies of a protein group. I discuss the singularities and regularities of this geometric representation and how this metric identifies the main action of a protein with a high level of accuracy. I introduce the concepts of maximum points, minimum points and saddle points, I also calculate the smooth curve from matrix Qi + Σn =1Qi (Sect. 3.3.2), and I justify the exhaustiveness of the metric.
Keywords: Catastrophe theory, Catastrophic bifurcation points, Critical points, Differential calculus, Distributed computing, Function, Polarity index method, Regularity, Serial computing, Singularity, Smooth curves.