Here are my notes for E, Li, and Vanden-Eijnden’s Applied Stochastic Analysis Chapter 5 - Stochastic Processes 5.1 - Axiomatic Construction of Stochastic Process 5.2 - Filtration and Stopping Time 5.3 - Markov Processes 5.4 - Gaussian Processes Chapter 6 - Wiener Process 6.1 - The Diffusion Limit of Random Walks 6.2 - The Invariance Principle

2.2.1 - Definition and first examples Definition 2.2: A symmetric monoidal structure on a preoirder $(X, \leq)$ consists of (i) a monoidal unit, $I \in X$ (ii) a monoidal product $\otimes: X \times X \rightarrow X$ And the monoidal product $\otimes(x_1,x_2) = x_1 \otimes x_2$ must also satisfy the following properties (assume all elements are in $X$) (a) $x_1 \leq y_1$ and $x_2 \leq y_2 \implies x_1 \otimes x_2 \leq y_1 \otimes y_2$ (b) $I \otimes x = x \otimes I = x$ (c) associativity (d) commutivity/symmetry (a) is called monotnoicity and (b) is unitality...

This is a collection of my notes for Brendan Fong and David Spivak’s An Invitation to Appied Category Theory. The first chapter was done through LaTeX, but the rest should be markdown with mathjax. Chapter 1 - Generative effects: Orders and adjunctions Chapter 2 - Resource theories: Monoidal preorders and enrichment Section 2.2 - Symmetric monoidal preorders

I was recently doing a probability puzzle that I can’t quite remember the context of, but I came across the answer that the probability would be $$\mathbb{P}(X) = n p^n \; \quad \forall \: n\in\mathbb{N}, p \in [0,1].$$ But this is obviously wrong! Plug in $p=.9, n=2$, and you get that $\mathbb{P}(X) = 1.62$. Thaat’s not how probability works! However, for $p=0.5$, $\mathbb{P}(X)$ will remain $\leq 1$ for all $n \in \mathbb{N}$....

I have no idea what I am doing. Anyways, here’s a cool equation: $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0$$