A Cantilever beam is a rigid structure which is extended horizontally and supported on one end.

Working with Cantilever Beams

curvature

Let’s first define a function:

\begin{equation} w(x) \end{equation}

this represents the deflection of the beam at point \(x\). We will begin by taking its derivative by location:

\begin{equation} \Delta w = \pdv{w}{x} \end{equation}

is the change in deflection over location. “How much deviation of the beam from the resting axi is there as you run along it?”

The capacitance is the amount of change something can hold; this scales based on how much electric potential is being applied.

Parallel plates

\begin{equation} C = \frac{\epsilon_{0} A}{d} \end{equation}

where, \(e_0\) is the permittivity of free space, \(A\) the area of the plates, and \(d\) their distance.

A capacitor changes, then resists being charged further. Their rules work opposite to resistors.

capacitor in series

\begin{equation} \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \end{equation}

and yet,

capacitor in parallel

\begin{equation} C_{eq} = C_1 + C_2 + C_3 \end{equation}

energy stored by a capacitor

\begin{equation} E = \frac{1}{2} CV^{2} \end{equation}

where, \(E\) is the energy stored, \(C\) the capacitance, and \(V\) the voltage across the capacitor.

Which, subbing the formula below:

A cancer drug to synthesize Fluoropyrimidine.

CAPM is a method of portfolio selection analysis which focuses on maximizing return given some fixed variance.

It deals with optimal Capital Market Line, given here:

\begin{equation} E[R_{p}] = r_{f}+\frac{\sigma_{p}}{\sigma_{T}}\qty(E[R_{T}]-r_{f}) \end{equation}

Which describes \(E[R_{p}]\), the expected return of an optimal portfolio in a market, given, \(R_{T}\) is the market return, \(r_{f}\) is the risk-free rate, \(\sigma_{p}\) is the portfolio returns, and \(\sigma_{t}\) is standard-deviation of the market returns.

Sharpe Ratio

The Sharpe Ratio is a measure of the risk-adjusted performance of an asset—given the rate of return of some risk-free asset.