Questions of Uniqueness and Existance are important elements in Differential Equations.

Here’s a very general form of a differential equations. First, here’s the:

function behavior tests

continuity

Weakest statement.

A function is continuous if and only if:

\begin{equation} \lim_{x \to y} f(x) =f(y) \end{equation}

Lipschitz Condition

Stronger statement.

The Lipschitz Condition is a stronger test of Continuity such that:

\begin{equation} || F(t,x)-F(t,y)|| \leq L|| x- y|| \end{equation}

for all \(t \in I\), \(x,y \in \omega\), with \(L \in (0,\infty)\), named “Lipschitz Constant”, in the dependent variable \(x\).

A constructor built out of quantum theory which can replicate itself. It is considered a universal computer.

bit string encoding

encode a finite string in \(\Sigma^{*}\) as a bit string: encoding each character in \(\log | \Sigma |\) bits (by basically IDing each bit).

For \(x \in \Sigma^{*}\), define \(b_{\Sigma}(x)\) to be its binary encoding.

For \(x,y \in \Sigma^{*}\), to encode the pair of strings \(x\) and \(y\), we can encode tuples by adding just a separate symbol \(\Sigma’ = \Sigma \cup \qty {,}\), and then we can write \(x,y\). (if we want to know how long \(x\) is ahead of time we can encode it by show you how long it is through zeros early—\(0^{|b_{\Sigma} (x)|} 1 b_{\Sigma}(x) b_{\Sigma}(y)\).