A blog about machine learning, math, math history, and probably a lot of other stuff.
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Pell’s Equation, Greek mathematics, and the square root of 2
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How did we think about math before algebra existed?
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Can LLaMa Reason? Investigations with Large Language Models and PrOntoQA
“…That being said, LLaMa’s outputs are close to reasoning. The space of possible outputs is effectively constrained such that it still contains most reasonable responses in higher-probability regions…”
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Describing Alien Empires with the LLaMa Language Model
“…Am I arguing at a dataset of 30 examples is optimal? No, definitely not. But part of what I wanted to investigate with this project was how small a training dataset can be. Can I hack together 50 examples by hand in one evening and then train a model to automate the task? Big if true…”
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A Short Proof About Circular Motion
“…All this is to say that I found myself straining to say anything interesting about rotational motion. Hence, the following discovery…”
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Bayesian Multinomial Models: A Tragedy (In Runescape)
“…Originally I set out in this project to design an interesting Bayesian model which could predict opponent behavior in player-vs-player Old School Runescape combat. I will spoil the outcome for you now. After a long and arduous journey, I discovered that my derived model absolutely blows…”
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Weakly Supervised Video Segmentation In Runescape
“…This paper presents a weakly-supervised machine learning algorithm for video segmentation. By making certain assumptions, this algorithm reduces the problem of video segmentation to a problem of many low-resolution image classifications. The associated training data can be collected easily for certain video games by using an automated procedure. This is demonstrated for the video game ‘Old School Runescape’…”
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Teaching ChatGPT to Play Runescape
“…Some games have elaborate mechanics relying heavily on common sense, which reinforcement learning does not have… NLP might be more successful under these circumstances due to its general knowledge…”
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Newton’s Law of Cooling
“…So Newton’s logic was essentially “if the temperature change is $T_w - T_b$, then it stands to reason that the amount of heat exchanged is probably proportional.”
Totally unrigorous, and yet, he pretty much nailed it…”
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Bernoulli Diffusion Derivations
“…The first time I saw this, I had no idea how to even interpret it, and the authors never even mention it beyond “oh, yeah, that’s the posterior.”…”
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The Mystery of the Frog Riddle
“…So here we are. I’ve tipped my hand. You know everything I know. One question remains.
- Suppose that tomorrow I’m out wandering in a clearing. Behind me, I hear a croak. I turn around and see two frogs. What is the probability that at least one of the frogs is female?…”
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What is an object? Object Detection and Philosophy… In Runescape
“…Suppose you have an image containing some object. Suppose the object might be partially or totally obscured by other objects in the image, but you aren’t sure. In fact, suppose you don’t even know what the object is supposed to be. However, suppose there is one thing you know: You know where the object would be in the image if it were visible in the image. Can you guess whether the object is visible by looking at the image?…”
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Egyptians and the Rhind Mathematical Papyrus: Egyptian Arithmetic (2)
“…To divide 47 by 33, the Egyptians would, in their own words, “treat 33 so as to obtain 47.” For example, they might simply observe that
\[47 = 1*33 + \frac{1}{3}*33 + \frac{1}{11}*33 \implies\] \[47 = (1+\frac{1}{3}+\frac{1}{11})33 \implies\] \[\frac{47}{33} = 1+\frac{1}{3}+\frac{1}{11}"\]
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Egyptians and the Rhind Mathematical Papyrus: Historical Context (1)
“…Recently I had the idea that I might appreciate mathematics more if I understood how it originated. “I’m so clever,” I thought, “I’ll just Google ‘Ancient Egyptian Mathematics’ and then everything will make sense.” Surprise! Nobody has a complete understanding of Egyptian mathematics. There are many mysteries about how Egyptians performed mathematical operations and how Egyptians understood mathematics…”
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Where Am I? Estimating Location In Runescape With The Mini-Map
“Runescape is a boring video game for losers. In the game, your location is represented to you in both a world map and a mini-map. My goal was to estimate the player’s location on the world map using only the mini-map. Real-time location estimation is the first step in breaking the game so that it can play itself and we can all go home.”
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Elliptic Functions Revisited: What Were They Thinking
“Mathematicians quickly ran into difficulties with integrals involving square roots of polynomials larger than degree two. These were natural objects of interest, being marginally more complex than the integrals previously studied, while still appearing in natural physical problems, such as the arc length of the lemniscate of Bernoulli:
\[\int_0^x \frac{1}{\sqrt{1-t^4}}dt\]Thus begins the study of elliptic integrals…”
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Lagrangians, Diff Geo, and the Isochronous Curve of Leibniz
“I’ve been working on building intuitions for advanced classical mechanics. I find that there is an unfortunate trade-off at the development of the Lagrangian. On one hand, setting up and solving problems becomes (hypothetically) easier and more streamlined. On the other, the intuition can become obscured.”
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History of Modular Forms 1: The First Era, 1800s
“The rabbit-hole of modular forms began with elliptic functions at the start of the 1800s, $\pm 30$ years. It was the twilight years of the Enlightenment Era, about a century after Newton’s Principia and 20 years after Euler’s death. The telescope had been used by Galileo 200 years ago, but innovations by Newtons and others during the 1700s greatly improved the quality of observations.”
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The Inscribed Rectangle Proof, Visually
“…Draw a smooth loop on a piece of paper. Now, pick a rectangle of some length:height proportion. A simple example might be 1:1, which is a square. 2:1 would be a little more rectangle-y. The claim is that you can find 4 points on your loop which make a rectangle of that proportion…”
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What You Need To Know To Understand Quantum Modeling
“…We show that modeling consists of building a Hamiltonian with a potential function, and then we talk about how quantum allows us to describe what we should expect from each possible kind of measurement, like momentum or position…”
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What’s This All About?
“…Beep Boop…”