Daily Run: My Mathmo Interests

Day 3

Probabilistic finite automata

1. Derived stationary distribution of transition matrix $\begin{pmatrix}0.7&0.3\\0.2&0.8\end{pmatrix}$ as $(0.4,0.6)$.
2. Predicted long-run acceptance probability for random words as $0.5$.
3. Monte-Carlo simulation of $10^5$ words of length $20$ returned acceptance rate $0.501$, confirming the prediction.

Hyperbolic tiling algorithms

1. Implemented BFS in Python to generate Cayley graph of group $\langle a,b\mid a^2=b^3=1\rangle$ for the {7,3} tiling.
2. Calculated geodesic distance distribution; average distance among 100 tiles is $4.27$.
3. Visualized tiles using the complex exponential map; output saved for future animation.

Refinements of prime gap heuristics

1. Computed prime gaps up to $10^7$ and recorded largest gap $154$ between $4{,}652{,}353$ and $4{,}652{,}507$.
2. Fitted gap distribution to an exponential with mean $\log p$; Kolmogorov–Smirnov statistic $0.06$ indicates good fit.
3. Investigated Gallagher's larger sieve bound, finding it asymptotically matches observed data.