A direct sum is a sum of subspaces (not just subsets!!) where there’s only one way to represent each element.

constituents

subspaces of \(V\) named \(U_1, \dots, U_{m}\)

requirements

The sum of subsets of \(U_1+\dots+U_{m}\) is called a direct sum IFF:

each element in \(U_1+\dots +U_{m}\) can only be written in one way as a sum \(u_1 +\dots +u_{m}\) (as in, they are linearly independent?)

We use \(\oplus\) to represent direct sum.

additional information

why is it called a direct sum?

Something is not a direct sum if any of its components can be described using the others. Its kind of line linear independence but! on entire spaces.

Directed Evolution is a process of recreating Darwinian processes in a lab setting

1. mutation: make select mutation
2. selection: selection specific changes
3. replication: make more of it

Examples: PACE

Softmax Method

Pull arm \(a\) with probability \(\propto \exp (\lambda \rho_{a})\), where \(\lambda \geq 0\) is the “precision parameter”.

When \(\lambda \to 0\), this system uses the same rate for each of the actions, so you are essentially randomly sampling; when \(\lambda \to \infty\), the system will use only the greedy action because only the element with the biggest \(\rho_{a}\) gets selected.

For a multi-state case:

\begin{equation} \propto \exp (\lambda Q(s,a)) \end{equation}

regular expressions are the same thing, but can be treated in two senses

• the tool: regular expression (string processing)
• complexity theory: regular expression (complexity)

discourse features are marks of fluency/etc. which mark one’s speech.