## Dynamics

The mathematical approach to change over time. Most dynamical systems are nonlinear and generally unsolvable, and though deterministic are often unpredictable.

## Boundaries

Trajectories of any dynamical equation may stay bounded or else diverge towards infinity. The borders between bounded and unbounded trajectories can take on spectacular fractal geometries.

## Foundations

### Primes are unpredictable

$\lnot \exists n, m : (g_n, g_{n+1}, g_{n+2}, ... , g_{n + m - 1}) \\ = (g_{n+m}, g_{n+m+1}, g_{n+m+2}, ..., g_{n + 2m - 1}) \\ = (g_{n+2m}, g_{n+2m+1}, g_{n+2m+2}, ..., g_{n + 3m - 1}) \\ \; \; \vdots$

### Aperiodicity implies sensitivity to initial conditions

$f(x) : f^n(x(0)) \neq f^{n+k}(x(0)) \forall k \implies \\ \forall x_1, x_2 : \lvert x_1 - x_2 \rvert < \epsilon, \; \\ \exists n \; : \lvert f^n(x_1) - f^n(x_2) \rvert > \epsilon$

### Aperiodic maps, irrational numbers, and solvable problems

$\Bbb R - \Bbb Q \sim \{f(x) : f^n(x(0)) \neq f^k(x(0))\} \\ \text{given} \; n, k \in \Bbb N \; \text{and} \; k \neq n \\$

### Irrational numbers on the real line

$\Bbb R \neq \{ ... x \in \Bbb Q, \; y \in \Bbb I, \; z \in \Bbb Q ... \}$

### Discontinuous aperiodic maps

$\{ f_{continuous} \} \sim \Bbb R \\ \{ f \} \sim 2^{\Bbb R}$

### Poincaré-Bendixson and dimension

$D=2 \implies \\ \forall f\in \{f_c\} \; \exists n, k: f^n(x) = f^k(x) \; if \; n \neq k$

### Computability and Periodicity I: the Church-Turing thesis

$\\ \{i_0 \to O_0, i_1 \to O_1, i_2 \to O_2 ...\}$

### Computability and Periodicity II

$x_{n+1} = 4x_n(1-x_n) \implies \\ x_n = \sin^2(\pi 2^n \theta)$

### Reversibility and periodicity

$x_{n+1} = rx_n(1-x_n) \\ \; \\ x_{n} = \frac{r \pm \sqrt{r^2-4rx_{n+1}}}{2r}$

## Physics

As for any natural science, an attempt to explain observations and predict future ones using hypothetical statements called theories. Unlike the case for axiomatic mathematics, such theories are never proven because some future observation may be more accurately accounted for by a different theory. As many different theories can accurately describe or predict any given set of observations, it is customary to favor the simplest as a result of Occam’s razor.

### Quantum Mechanics

$P_{12} \neq P_1 + P_2 \\ P_{12} = P_1 + P_2 + 2\sqrt{P_1P_2}cos \delta$

## Biology

The study of life, observations of which display many of the features of nonlinear mathematical systems: an attractive state resistant to perturbation, lack of exact repeats, and simple instructions giving rise to intricate shapes and movements.

## Deep Learning

Machine learning with layered representations. Originally inspired by efforts to model the animalian nervous system, much work today is of dubious biological relevance but is extraordinarily potent for a wide range of problems. For some of these pages and more as academic papers, see here.

### Input Representation III: Language Models

$\mathtt{This \; is \; a \; prompt \; sentence} \\ \mathtt{channelAvailability \; is \; a \; prompt \; sentence} \\ \mathtt{channelAvailability \; millenn \; a \; prompt \; sentence} \\ \dots \\ \mathtt{redessenal \; millenn-+-+DragonMagazine}$

## Small Projects

### Programs to compute things

$\; \\ \begin{vmatrix} a_{00} & a_{01} & a_{02} & \cdots & a_{0n} \\ a_{10} & a_{11} & a_{12} & \cdots & a_{1n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n0} & a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{vmatrix} \; \\$ $\; \\ 5!_{10} = 12\mathbf{0} \to 1 \\ 20!_{10} = 243290200817664\mathbf{0000} \to 4 \\ n!_k \to ? \; \\$

## High voltage

High voltage engineering projects: follow the links for more on arcs and plasma.