Linearity requires additivity and scaling, but neural networks need nonlinearity to break free from simple affine transformations. While affine layers only stretch and rotate data without changing its topology, activation functions enable complex reshaping—allowing networks to untangle data manifolds and learn intricate patterns effectively.

Stirling's formula finds a relationship of n!,nⁿ,eⁿ while n trends to infinity.

The Box-Muller transformation is a method to construct a random variable that follows a Gaussian distribution by using a random variable that follows a uniform distribution.

Linear congruential method is a pseudo-random number generation method that generates a random number sequence based on a deterministic mathematical algorithm.

Laplace's theorem in determinants and its applications