A corollary of greatest common divisor and division.

Say you have some \(b|a\) such that:

\begin{equation} a = bq + r \end{equation}

Now, \(d|a,b \Leftrightarrow d|b,r\) (because \(d|b,r\) implies there’s some \(x, x’\) such that \(a = (dx)q+dx’\), and so \(a = d(xq + x’)\) and so \(d|a\); the logic goes the other way too).

This finally implies that \(\gcd (a,b)= \gcd (b,r)\) because any divisor that works for one works for both.

A type of cell.

Sample eukareotyic cell gene:

• TATA box promoter
• 5’ non-coding sequence
• Non-coding introns interlaced between exons, unique to eukareotyic cells. Bacteria (prokateotic cells don’t contain introns or have small them)
• 3’ non-coding sequence

The Euler-Bernoulli Theory is a theory in dynamics which describes how much a beam deflect given an applied load.

Assumptions

For Euler-Bernoulli Theory to apply in its basic form, we make assumptions.

• The “beam” you are bending is modeled as a 1d object; it is only long and is not wide
• For this page, \(+x\) is “right”, \(+y\) is “in”, and \(+z\) is “up”
• Probably more, but we only have this so far.
• the general form of the Euler-Bernoulli Theory assumes a freestanding beam

Basic Statement

The most basic for the Euler-Bernoulli Equation looks like this:

\begin{equation} f(x) = e^{ix} = \cos (x) + i\sin (x) \end{equation}

this brings a circle of radius one, because in every point, velocity is orthogonal to where you are (because \(f’(x) = if(x)\), and multiplying by \(i\) accounts for a rotation of 90 degrees.

And so,

\begin{equation} z = re^{i\theta} \end{equation}

gives any point in the imaginary polar plane.