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Fractional Calculus and Taylor Series
Fractional calculus is the attempt to solve equations of the form $\sqrt{\frac{d}{dx}}f(x)$, where $\sqrt{\frac{d}{dx}}$ is some operator that when applied twice is equal to the derivative, and other problems in the same vein. Fractional differentiation is generalized from that idea to raising the derivative operator to an arbitrary exponent and... 
Fluid Mechanics from Particle Motion
In January of 2016 I started taking my first class on fluid mechanics. In the first or second week of that class we started going over the derivation of the partial differential equations of fluid mechanics, but before covering the derivation we covered the concept of a fluid element. In... 
String Dynamics
Since finishing my first class on classical mechanics I have been interested in deriving the equations of motion for strings. Before I begin it should be noted that I do not have formal training in continuum mechanics, so the methods and definitions that I use may be nonstandard. 
PsiPy: Schrodinger's Equation (part 1)
This past fall I was taking my second quantum mechanics class. In that class we learnt how to deal with discreet quantum system using linear algebra and how to extend that notation to solve continuous systems. Midway though the semester I attempted to write a simple program to solve the... 
Memory and Markov
In a previous post I discussed how to generate a Markov Chain with a basic algorithm, but using that algorithm memory usage was a limiting factor. Now we will attempt to find a solution to the memory usage of the algorithm. If you are unfamiliar with how to generate a... 
An Improbable Prime
Over the last couple years security and privacy on the internet has become significant issue due to constant hacking and spying from groups and individuals both foreign and domestic. A common solution to such intrusions is cryptography, one common encryption algorithm is RSA encryption. RSA is a publickey encryption algorithm,... 
The Making of a Markov Chain
Before getting into how to make Markov Chains, lets quickly get a refresh on what a Markov Chain is. A Markov Chain is a set of states and state transition which are selected based on an assigned probability of occurring. The goal of a Markov chain is to model complex... 
What is Momentum?
When you took your first physics class you where introduced to the concept of momentum, and the discussion likely revolved around Newton’s second law of motion. This discussion would then culminate in the definition of momentum: \(\vec{p} = m \vec{v}\). As you progressed in the sciences you learned more about... 
An Introduction to Markov Chains
Markov chains are a statistical tool invented by Andrey Markov to model dependent statistical phenomenon. A Markov Chain is made from two components a discrete set of states and a set of stochastic state transitions. The idea is that if there are two states A and B and the state...
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