While conceptually simple, Bayesian methods can be mathematically and numerically challenging. Computational models require a mathematically and logically formal representation of a problem. Subjective logic gives very efficient computation of mathematically complex models. There are three other mathematically equivalent ways to compute . The Fraunhofer diffraction equation can be expressed in a variety of mathematically equivalent forms. There are many mathematically equivalent formulations of quantum mechanics. HRV is quantified by fitting mathematically defined geometric shapes to the data. Examples exist of mathematically correct results derived by incorrect lines of reasoning. Group theory is an abstract way of mathematically analyzing symmetries. There are several ways to make this intuition mathematically rigorous. Many of the patterns follow as mathematically logical consequences of these assumptions. The fact that the musician meant to pluck it at a mathematically expressible point. A more mathematically rigorous definition is given below. There are several ways to mathematically define quasicrystalline patterns. Battle formations became standardized, based on mathematically calculated ideal models.