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I used to not like analog clocks because they unecessarily made it harder to tell time in a world where digital clocks are a reality. Now, I appreciate them a lot more for all the mathematical fun that they present. So, here’s a very simple puzzle I thought of while looking at one. The Puzzle It is 3:00 right now on an analog clock. How much longer do I have to wait to see the minute and the hour hands cross each other?...

This project was done as our final project for William Gilpin’s Graduate Computational Physics Course. Our complete GitHub repository, with instructions on how to replicate our results, can be found here. Introduction The goal of this project is to simulate the behavior of a bike-sharing system in a network of stations and destinations, and then optimize the positions of the stations. We approach the simulation of the bike-sharing system with Agent Based Modeling (ABM)....

Notes on the paper Planting Undetectable Backdoors in Machine Learning Models by Shafi Goldwasser, Michael P. Kim, Vinod Vaikuntanathan, and Or Zamir. This paper was recommended to me by Scott Aaronson if I wanted to better understand some earlier, more cryptographic/theoretical work in backdooring neural networks. I am also reading through Anthropic’s Sleeper Agents paper, which is more recent and practical in its approach to backdooring current LLMs, those notes will be posted soon as well....

About two years ago, I attended a seminar given by Dr. Sid Redner of the Santa Fe Institute titled, “Is Basketball Scoring a Random Walk?” I was certainly skeptical that such an exciting game shared similarities with coin flipping, but, nevertheless, Dr. Redner went on to convince me–and surely many other audience members–that basketball does indeed exhibit behavior akin to a random walk. At the very end of his lecture, Dr....

Let $\{{\xi }_m{\}}_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables such that $\mathbb{E}[{\xi }_n]=0$ and $\mathbb{E}[{\xi }_n^2]=1$. Then, define $$S_0=0,S_N=\sum _{i=1}^N{\xi }_i$$ and by the Central Limit Theorem, rescaling $S_N$ by $\sqrt{N}$, we get that $$\frac{S_N}{\sqrt{N}}\stackrel{d}{\to }\mathcal{N}(0,1)$$ (the $\stackrel{d}{\to }$ means convergence in distribution) as $N\to \mathrm{\infty }$. Using this, we can define a continuous random function $W_t^N$ on $t\in [0,1]$ such that $W_0^N=0$ and $$W_t^N=\frac{1}{\sqrt{N}}(\theta S_k+(1-\theta )S_{k+1}),Nt\in (k,k+1],k=0,1,\dots ,N-1$$ where $\theta =\lceil Nt\rceil -Nt$....

The Puzzle Inspired by true events Alice is assigned to be the 56th passenger to board a full plane with 60 seats. However, a panic causes all the passengers–including Alice–to arrange themselves radomly in line to board. As Alice was originally 56th, she decides that she would be happy as long as passengers with the assigned spots 57, 58, 59, and 60 are not in front of her. What is the probability that Alice will be happy?...

Random Walk Let $\{{\xi }_i\}$ be i.i.d. random variables such that ${\xi }_i=\pm 1$ with probability $1/2$. Then, define $$X_n=\sum _{k=1}^n{\xi }_k,X_0=0.$$ $\{X_n\}$ is the familiar symmetric random walk on $\mathbb{Z}$. Let $W(m,n)=\mathbb{P}(X_N=m)$. It is easy to see that $$W(m,n)=(\genfrac{}{}{0ex}{}{N}{(N+m)/2}){\left(\frac{1}{2}\right)}^N$$ and that the mean and std are $$\mathbb{E}[X_N]=0,{\sigma }_{X_N}^2=N$$ Diffusion Coefficient Definition 6....

Definition 5.9: A stochasitc process $\{X_t{\}}_{t\ge 0}$ is a Gaussian Process if its finite dimensional distributions are consistent Gaussian measures for any $0\le t_1<t_2<\dots <t_k$. Recall that a Gaussian random vector $\mathbf{X}=(X_1,X_2,\dots ,X_n{)}^T$ is completely characterized by its first and second moments $$\mathbf{m}=\mathbb{E}[\mathbf{X}],\mathbf{K}=\mathbb{E}[(\mathbf{X}-\mathbf{m})(\mathbf{X}-\mathbf{m}{)}^T]$$ Meaning that the characteristic function is expressed only in terms of $\mathbf{m}$ and $\mathbf{K}$ $$\mathbb{E}\left[e^{i\xi \cdot \mathbf{X}}\right]=e^{i\xi \cdot \mathbf{m}-\frac{1}{2}{\xi }^T\mathbf{K}\xi }$$ This means that for any $0\le t_1<t_2<\dots <t_k$, the measure ${\mu }_{t_1,t_2,\dots ,t_k}$ is uniquely determined by an $\mathbf{m}=(m(t_1),\dots ,m(t_k))$ and a covariance matrix ${\mathbf{K}}_{ij}=K(t_i,t_j)$....

Markov processes in continuous time and space Given a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ and the filtration $\mathbb{F}=({\mathcal{F}}_t{)}_{t\ge 0}$, a stochastic process $X_t$ is called a Markov process wrt ${\mathcal{F}}_t$ if $X_t$ is ${\mathcal{F}}_t$-adapted For any $t\ge s$ and $B\in \mathcal{R}$, we have $$\mathbb{P}(X_t\in B|{\mathcal{F}}_s)=\mathbb{P}(X_t\in B|X_s)$$ Essentially, this is saying that history doesn’t matter, only the current state matters....

Filtration Definition 5.3: (Filtration). Given a probability space, the filtration is a nondecreaseing family of $\sigma$-algebras $\{{\mathcal{F}}_t{\}}_{t\le 0}$ such that ${\mathcal{F}}_s\subset {\mathcal{F}}_t\subset \mathcal{F}$ for all $0\le s<t$. Intuitively, the filtration is a sigma algebra of events that can be determined before time $t$ (we can’t lose information by foing forward in time). A stochastic process is called ${\mathcal{F}}_t$-adapted if it is measurable with respect to ${\mathcal{F}}_t$; that is, for all $B\in \mathcal{R}$, $X_t^{-1}(B)\in {\mathcal{F}}_t$....