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Gradients Are the New Intervals
At the New England Symposium on Graphics , James Tompkin compared graphics researchers to magpies: they're easily distracted by shiny objects and pretty renderings.
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Revisiting the algorithm that changed horse race betting (2023)
Explore Bill Benter’s iconic horse betting strategy, enhanced with modern code and contrasted against three decades of evolving data.
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From Zero to QED: An informal introduction to formality with Lean 4
An informal introduction to formality in Lean 4
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50 years of proof assistants
Crackpots ranging from billionaire Peter Thiel to random YouTube influencers claim that science has been stagnating for the past 50 years. They admit that computing is an exception: they don’t pretend that my personal 32GB laptop is not an advance over the 16MB mainframe that served the whole Caltech community when I…
125
Monte Carlo Crash Course: Quasi-Monte Carlo
We’ve learned how to define and apply Monte Carlo integration—fundamentally, it’s the only tool we need. In the remaining chapters, we’ll explore ways to reduce variance and successfully sample difficult distributions.
126
Packing the world for longest lines of sight
I have a side project to find the longest line of sight on the planet 1 using a novel Total Viewshed algorithm . At a resolution of 3 arc-seconds (~92m x 92m), the planet contains around 4.5 billion elevation samples 2 . Now obviously we don’t need to calculate the visibility between literally every single one of…
127
De Bruijn notation, and why it's useful
At least a familiarity with the lambda calculus, including how it is evaluated. Some base knowledge of programming languages is also assumed.
128
The Math Behind GANs (2020)
Generative Adversarial Networks refer to a family of generative models that seek to discover the underlying distribution behind a certain data generating process. This distribution is discovered through an adversarial competition between a generator and a discriminator. As we saw in an earlier introductory post on…
129
Cache-friendly, low-memory Lanczos algorithm in Rust
How a bit of algorithm engineering and low-level details can alter what seems like a straightforward trade-off on a blackboard.
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Optimizing a Math Expression Parser in Rust
Optimizing a math expression parser for speed and memory.
131
The probability of a hash collision (2022)
Exploring the math behind hash collisions.
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How has mathematics gotten so abstract?
What's the meaning of "numbers" and "arithmetic operations"? We consult Georg Cantor's turtles and look at Giuseppe Peano's code.
133
Decomposing a Factorial into Large Factors
I’ve just uploaded to the arXiv the paper “Decomposing a factorial into large factors”. This paper studies the quantity $latex {t(N)}&fg=000000$, defined as the largest quanti…
134
What is an elliptic curve? (2019)
The informal definition of an elliptic curve is simple. The formal definition is more involved.
135
Algorithms for making interesting organic simulations
Article explaining simulation algorithms that produce complex organic behaviours, starting with the classic physarum algorithm from Jeff Jones.
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The number line freaks me out (2016)
“Welcome to math class, boys and girls! Today we’ll be drawing a picture that gestures beyond the limits of mortal comprehension.”
137
Lace: A New Kind of Cellular Automata Where Links Matter
This article is about a new kind of simple computational rule ("LACE rules" running on LACE, the Link Automata Computing Engine platform) which, when applied locally on a grid of cells, demonstrates fascinating emergent "artificial life" behavior.
138
Random Attractors – Found using Lyapunov Exponents (2001)
Contribution by Philip Ham: attractor.basic and Python implementation by Johan Bichel Lindegaard.
139
Sharing everything I could understand about gradient noise
You've most likely heard about gradient noise through the name Perlin noise , which refers to one particular implementation with various CPU optimizations. Because it's an incredible tool for creative work, it's used virtually everywhere: visual effects, video games, procedural mathematical art, etc. While getting it…
140
God created the real numbers
W. Jherek Swanger writes in his introduction to Camillo 7 Agrippa's "A Treatise on the Science of Arms, with a philosophical Dialogue (1553)" that:
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A triangle whose interior angles sum to zero
In hyperbolic geometry, the sum of a the interior angles of a triangle can be any value less than π. If the sum is zero the triangle has area π.
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The Block Stacking Problem
John D. Norton Department of History and Philosophy of Science University of Pittsburgh https://sites.pitt.edu/~jdnorton/jdnorton.html
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Solving LinkedIn Queens with SMT
For sure easier than solving it in SAT!
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Five Kinds of Nondeterminism
Or four kinds, or six kinds, I'm not picky about how you count them
146
An overview of gradient descent optimization algorithms (2016)
Gradient descent is the preferred way to optimize neural networks and many other machine learning algorithms but is often used as a black box. This post explores how many of the most popular gradient-based optimization algorithms such as Momentum, Adagrad, and Adam actually work.
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The Ruliology of Lambdas
Stephen Wolfram explains the rich ruliology of lambdas, made particularly significant by their connection to practical computing. Covers basic computations to undecidability to multiway graphs and evaluation strategies.
148
Lemma for the Fundamental Theorem of Galois Theory
This post illustrates a key lemma that is used in proving the fundamental theorem of Galois theory (FTGT). Note that FTGT is not covered in this post. The focus of this post is on understanding and proving this lemma only. Here is the lemma from the book Galois Theory , 5th ed. by Stewart (2023):
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A shower thought turned into a Collatz visualization
I recently went on a nice long SCUBA diving trip with my wife and daughters. Lots of diving implies lots of showers, and lots of showers means lots of shower-thoughts! [1] An especially interesting one I had turned into a nice way to visualize some aspects of the Collatz Conjecture.
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A Random Walk in 10 Dimensions (2021)
The geometry of random walks in high dimensions provides the power behind deep learning and may be the secret to intelligence.