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Conway's Gradient of Life
What happens if we start from this configuration and take a single step of the game? Here is a snapshot of the next generation:
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The latest blog posts to take off on Hacker News. 266 posts.
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An Intuitive Explanation of Black–Scholes
The Black–Scholes formula is the crown jewel of quantitative finance. The formula gives the fair price of a European-style option, and its success can ultimately be measured by its impact on option markets. Before the formula’s publication in 1973 (Black & Scholes, 1973; Merton, 1973) , option markets were relatively…
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52 Factorial
They seem harmless enough, 52 thin slices of laminated cardboard with colorful designs printed on their sides. Yet, as another illustration of the mantra that complexity begins from the most simple systems, the number of variations that these 52 cards can produce is virtually endless. The richness of most playing card…
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The bunkbed conjecture is false
What follows is an unusual story of perseverance. We start with a conjecture and after some plot twists end up discussing the meaning of truth. While the title is a spoiler, you might not be able t…
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What P vs. NP is about
We recently made a Polylog video about a nonstandard way of understanding the P vs NP problem. As usual, our goal was to present an underrated idea in a broadly understandable way, while being slig…
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Math from Three to Seven
Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers, Alexander Zvonkin (Moscow Center for Continuing Mathematical Education, 2007).
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Solving climate change by abusing thermodynamic scaling laws
A wintertime climate change solution
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Too much efficiency makes everything worse (2022)
This blog is intended to be a place to share ideas and results that are too weird, incomplete, or off-topic to turn into an academic paper, but that I think may be important. Let me know what you think! Contact links to the left.
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A pilot project in universal algebra to explore new ways to collaborate
Traditionally, mathematics research projects are conducted by a small number (typically one to five) of expert mathematicians, each of which are familiar enough with all aspects of the project that…
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The centrality of stupidity in mathematics
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Randomness extractors: making fair coins out of biased coins
I discuss various models of biased bit sequences, and how to extract uniform random (or close to it) output bit sequences from them, illustrated with Python code.
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Fractran: Computer architecture based on the multiplication of fractions
By Devine Lu Linvega
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Better-performing “25519” elliptic-curve cryptography
Automated reasoning and optimizations specific to CPU microarchitectures improve both performance and assurance of correct implementation.
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Learning to Reason with LLMs
We are introducing OpenAI o1, a new large language model trained with reinforcement learning to perform complex reasoning. o1 thinks before it answers—it can produce a long internal chain of thought before responding to the user.
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Possibly all the ways to get loop-finding in graphs wrong
In my puzzle collection , there are many games in which you have to connect points together by edges, making a graph, and the puzzle rules say you must avoid making any loop in the graph. Examples are Net , Slant , and some configurations of Bridges (although not the default one). Loopy and Pearl also care about…
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Breaking Bell's Inequality with Monte Carlo Simulations in Python
The article explains the Bell inequality using Monte Carlo simulations in Python, and shows how non-local action-at-a-distance can be used to break it with entangled qubits.
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A new rare high-rank elliptic curve, and an orchard of Diophantine equations
I got an email the other day from mathematician Bogdan Grechuk, whose book Polynomial Diophantine Equations: A Systematic Approach ( was recently released. This is to my mind a rather remarkable bo…
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The expected value of the game is positive regardless of Ballmer’s strategy
A few days ago John Graham-Cumming posted about “Steve Ballmer’s incorrect binary search interview question” which drew a lot of attention on Hacker News. The Ballmer’s favorite brain teaser goes like this:
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How does cosine similarity work?
Cosine similarity measures the angle between two vectors — the dot product divided by the product of their magnitudes. Here's what each part of the formula does.
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Diffusion Is Spectral Autoregression
A deep dive into spectral analysis of diffusion models of images, revealing how they implicitly perform a form of autoregression in the frequency domain.
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Understanding the Y Combinator
This article is about the Y combinator, a fixed-point combinator that provides a way of using recursion in scenarios and programming languages where symbols are not available. The concept of the Y combinator itself is closely related to lambda calculus (λ-calculus) and functional programming languages. More…
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Strange Attractors (2018)
Interactive Mathematical Applets and Animations
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Bresenham's Circle Drawing Algorithm (2021)
Once upon a time I was given the following problem for a technical programming interview:
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Floating points between zero and one
floating-points-between-zero-and-one - Chad Nauseam Home
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Using Fibonacci numbers to convert from miles to kilometers and vice versa
I recently learned an interesting fact about Fibonacci numbers while watching a random number theory video on YouTube. Fibonacci numbers can be used to approximately convert from miles to kilometers and back. Here is how. Take two consecutive Fibonacci numbers, for example 5 and 8. And you're done converting. No...
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Poor Foundations in Geometric Algebra
EDIT: Be warned that this post is angry. If you’re easily offended when someone goes on a rant, then leave now because I’m very unforgiving in several places below. The tone of my writing is unusually harsh, but the people whose work I’m talking about deserve it, and here’s why. Even though I had been working with…
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Generating Mazes
Visualizations and techniques for different maze generation algorithms.