Daily Run: My Mathmo Interests

Day 2

Probabilistic finite automata

1. Implemented a 2-state PFA accepting strings over {a,b} with transition matrices $P_a$ and $P_b$.
2. Computed acceptance probability for the word "abba" as $0.433$ using matrix multiplication.
3. Proved that any 2-state PFA language is regular by constructing an equivalent deterministic automaton.

Hyperbolic tiling algorithms

1. Encoded group generators for a $(5,4)$ tiling and used breadth-first search to list tiles within radius 3.
2. Confirmed exponential growth rate by counting 61 tiles at radius 3 versus 12 at radius 2.
3. Outlined algorithm to embed tiling coordinates using complex Möbius transformations.

Refinements of prime gap heuristics

1. Applied the Riemann $R$ function to estimate $\pi(10^6)$ and found relative error below $0.1\%$.
2. Modeled gaps with the Cramér random model, predicting expected gap $(\log 10^6)^2 \approx 191$.
3. Observed actual gap 114 near $10^6$, supporting the $O((\log p)^2)$ behaviour.