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Understanding Gaussians
The **Gaussian distribution**, or **normal distribution** is a key subject in statistics, machine learning, physics, and pretty much any other field that deals with data and probability.
New
The latest blog posts to take off on Hacker News. 2,391 posts.
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Using Euro coins as weights (2004)
Here is a cheap way to obtain a set of standard weights, for measuring weights up to about 100 gram, to an accuracy of about 0.5 gram : Use the coins of the currency of your country. Thus one can make use of the fact that these coins are fabricated in mass production, and because of that have not only a constant shape…
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Industrious Dice
I’m a professional mathematician. That means somebody pays me to do math. I’m also a recreational mathematician. That means you might have to pay me to get me to stop. Wearing my recreational mathe…
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Damas-Hindley-Milner inference two ways
What is Damas-Hindley-Milner?
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Faster convergence for diffusion models
Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Think
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Counterintuitive Properties of High Dimensional Space (2018)
Our geometric intuition developed in our three-dimensional world often fails us in higher dimensions. Many properties of even simple objects, such as higher dimensional analogs of cubes and spheres, are very counterintuitive. Below we discuss just a few of these properties in an attempt to convey some of the weirdness…
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Defining Statistical Models in Jax?
This is Bob. And I’d like to know the best way for us to code a bunch of models in JAX to use to evaluate parallel algorithms including normalizing flows. I’m going to dump out my current thinking, but I’m really hoping to get feedback from experts on the best way to do this without starting a flame war in the…
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LLMs don't do formal reasoning
Important new study from Apple
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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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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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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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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