# Perturbation Theory

There exists a certain class of “hard” problems that can’t be solved with exact form. Examples include solutions to certain differential equation or higher order polynomials like quintics which can’t be solved with a simple cubic formula or quadratic formula. Perturbation theory is a tool commonly used in mathematical physics and can easily provide solutions to seemingly impossible problems. Read More »

# Heisenberg Uncertainty Derivation

The Heisenberg uncertainty principle seems like a principle that is so fundamentally experimental but it can actually be derived through theory. This requires the consideration of three concepts: the Cauchy-Shwarz inequality, measurement operators, and commutators.Read More »

# Quantum Teleportation

Quantum teleportation is the idea that entangling states can cause very fast information transfer. The name however is misleading as this information transfer is not instantaneous but simply travels at the speed of light. The basis of this idea is based on a very interesting trick that arises out of some of the math of quantum computing.Read More »

# Speed of Light Derivation

The speed of light may seem like an arbitrary constant of nature but, in some sense, it is actually set by other properties of the world. These other properties are the strengths of the electric and magnetic fields which are defined by the constants that are used in determination of them, otherwise known as the permittivity constant ($\inline&space;\epsilon_0=8.85&space;\times&space;10^{-12}&space;\texttt{&space;}\textup{F/m}$) and permeability constant ($\inline&space;\mu_0=4\pi&space;\times&space;10^{-7}\texttt{&space;}\textup{H/m}$). Because light is simply an electromagnetic wave, one can derive its speed using these constants.Read More »

# Complex Impedance

There is a certain luxury of circuit calculations for systems contain direct current that alternating current systems really do not have. It is the idea that voltage and current are “synced.” An increase in voltage will create a corresponding increase in current seemingly instantaneously. However, an alternating current that experiences voltage oscillations experiences a delay. This can mean voltage is at the highest point in its fluctuations while current only reaches such a point a little bit later at which point voltage might already be at its lowest. The ratio of voltage to current is also unclear in these circuits. This makes it hard to describe the system easily.Read More »

# Complex Exponentials

Complex exponentials are used immensely in math and as a result, in many fields of science. It is also used in abundance throughout this site so it is important to understand what they are for future reference. They show the relationship between exponentials and trigonometry on a fundamental level. The following is the relationship.

$e^{ix}=\textup{cos}(x)+i\textup{sin}(x)$Read More »

# Fractional Calculus

Calculus is the manipulation of one basic operator: the derivative or $\inline&space;\frac{\mathrm{d}&space;}{\mathrm{d}&space;x}$. 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 is the function which only goes through some scalar change when acted on by the operator. This scalar that the function is scaled by is called the eigenvalue of the eigenfunction. For the derivative operator, Read More »

# Space Transforms

Functions have the ability to be described in terms of the infinite summation of other functions with a common example being polynomials using Taylor series as shown below.

$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$

However, functions can also be described in terms of trigonometric functions using the Fourier series Read More »

# 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 ($\inline&space;p(x)=1$ for uniform or $\inline&space;p(x)=e^{-x^2/2}$ for Gaussian). In order to calculate the probability that any number generated will fall in the range $\inline&space;[a,b]$, one would take the integral $\inline&space;\int_{a}^{b}p(x)\textup{d}x$. The probability to get a specific number $\inline&space;x$ would be $\inline&space;p(x)\textup{d}x$ which is just 0.Read More »

# 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.Read More »