Fractional Calculus

Calculus is the manipulation of one basic operator: the derivative or . This operator operates on functions and by repeatedly applying it, you can get higher order derivatives. It’s inverse operator is known as the integral. Similar to matrix operators which have eigenvalues and eigenvectors, this operator also has eigenvalues and eigenfunctions. The eigenfunction isContinue reading “Fractional Calculus”

Probability Distribution Transformations

Random number generators generate random numbers that tend to follow some kind of distribution. Uniform number generators generate a random number between 0 and 1 with equal likely hood everywhere in the range, Gaussian generators generate numbers that have higher probability to be near zero, etc. These distributions are given functions ( for uniform or  forContinue reading “Probability Distribution Transformations”

On Conservation

Conservation of different properties in nature immensely simplifies calculations to the point where some are impossible without the consideration of them. In some cases, it seems completely intuitive and impossible not to consider. However, not only are there many conservations laws unknown to many but there are also, in some sense, “violations” to these laws.

The Lagrangian

There exists a mathematically different approach to describing classical mechanics than what is usually taught. While it is usually taught using laws like , conservation of energy, and conservation of momentum, there exists a more mathematically elegant and, in some sense, more fundamental way of describing the motion of objects known as Lagrangian mechanics. ItContinue reading “The Lagrangian”