Based on the wise words of a crab, I will start writing down some Proof Design Patterns I saw over Axler everything.

• inheriting properties (splitting, doing, merging) “complex numbers inherit commutativity via real numbers”

• construct then generalize for uniqueness and existence

• try to remember to go backwards

• to prove IFF

• zero is cool, and here too!, also \(1-1=0\)

  • \(0v = 0\)
  • \(1-1 = 0\)
  • \(v-v=0\) a.k.a. \(v+(-v)=0\)
  • \(v+0 = v\)

• distributivity is epic: it is essentially the only tool to connect scalar multiplication and addition in a vector space

Suppose \(\qty(G,s,t) \in \neg \text{STCONN}\), meaning no path \(s \to t\) exists in \(G\).

setup

For \(l \in \qty {0,1, \dots, n}\), define \(R_{l} \subseteq V\) is the set of all vertices readable from \(s\) in \(\leq l\) steps. \(R_0 \subseteq R_1 \subset … R_{n}\). Meaning \(R_0 = \qty {s}\). Goal: convince \(V\) that \(t \not \in R_{n}\).

Define: \(r_{l} = |R_{l}|\). What’s the size of \(r_{l}\)? It’s at most \(O\qty(\log \qty(n))\) size. Notice that storing \(R_{l}\) takes \(O\qty(n \log n)\) or at least \(O(n)\) space by storing either IDs or at least bitstring of everything.

propaganda is a form of advertising which:

• propaganda persuades people into believe in a cause
• often defies reason to reach into ??

See examples:

• US WWII Propaganda

techniques for propaganda

• Name calling
• Generalities
• Transferring of authority
• Public testimonial
• Attachment to plane folks
• Bandwagoning (FOMO)
• Fear
• Bad logic
• Unwanted extrapolation

We define the optimal proposal distribution as the one that minimizes the variance of the estimator of the Probability of Failure.

Sadly, the best proposal distributions is…

\begin{equation} q^{*}\qty(\tau) = \frac{p\qty(\tau) 1\qty {\tau \not \in \psi}}{p_{\text{fail}}} = \frac{p\qty(\tau) 1\qty {\tau \not \in \psi}}{\int 1 \qty {\tau \not\in \psi} p\qty(\tau) \dd{\tau }} \end{equation}

but wait this is just the Failure Distribution! But our entire point is trying to estimate \(p_{\text{fail}}\).

notice that this is exactly the DEFINITION OF THE FAILURE DISTRIBUTION. et, we were trying to estimate \(p_{\text{fail}}\) in the first place? Recall; we are able to sample from the Failure Distribution, fit a model and nice.