1831
How Many People? Ancient Demography
One of the first questions anyone asks about past societies is, ‘how many people were there?’ It is one of those basic bits of information and generally speaking those of us who teach t…
Topics /
Math & Algorithms
The most popular blog posts about math & algorithms on Hacker News. 2,394 posts.
1832
Is π the Same in Every Universe?
Mathematician: That depends on what you mean by “universe.” Here’s a framing: A circle of radius R centered at a point P is the set of all points in the plane with distance R fro…
1833
Division by zero in type theory: a FAQ
How can a theorem prover use the convention that 1/0=0 and still be consistent?
1834
Floating points between zero and one
floating-points-between-zero-and-one - Chad Nauseam Home
1835
A Trampoline
I didn’t want to buy a trampoline. It wasn’t the money.
1836
Taming floating-point sums
Suppose you have an array of floating-point numbers, and wish to sum them. You might naively think you can simply add them, e.g. in Rust:
1837
1838
Why cryptography is not based on NP-complete problems
Cryptographic schemes are based on the computational difficulty of solving some ‘hard’ problem. For example, RSA (specifically, RSA-2048) is based on the difficulty of a problem like this:
1839
Four Kinds of Optimisation
Premature optimisation might be the root of all evil, but overdue optimisation is the root of all frustration. No matter how fast hardware becomes, we find it easy to write programs which run too slow. Often this is not immediately apparent. Users can go for years without considering a program’s performance to be an…
1840
The inefficiency of RL, and implications for RLVR progress
And implications for RLVR progress
1841
What is the roundest country?
We can define roundness in many ways. For example, as you may know, the circle is the shape that given a fixed perimeter maximizes the area. This definition has many problems. One of the problems is that countries generally have chaotic perimeters (also known as borders), so they tend to be much longer than they seem…
1842
The 17x17 problem solved
THE 17x17 PROBLEM HAS BEEN SOLVED!!!!! On Nov 30, 2009 I posted here the following challenge: If someone emails me a 4-coloring of...
1843
Why Deep Learning Works II: the Renormalization Group
Deep Learning is amazing. But why is Deep Learning so successful? Is Deep Learning just old-school Neural Networks on modern hardware? Is it just that we have so much data now the methods work b…
1844
An interactive review of the Oklab perceptual color space
Blog of Raph Levien.
1845
Magic: The Gathering Is Turing Complete (v5)
I'm assuming you're familiar with the basic idea of what a Turing machine is . There are a few parts to building one in Magic: a tape extending in two directions storing cells of a few different "colours"; a processing head that can read the current cell's colour, write a new colour and change its state if necessary…
1846
Introduction to Mean Shift Algorithm
Its been quite some time since I wrote a Data Mining post . Today, I intend to post on Mean Shift – a really cool but not very well known algorithm. The basic idea is quite simple but the results a…
1847
Random Points on a Sphere
John D. Cook, Greg Egan, Dan Piponi and I had a fun mathematical adventure on Twitter. It started when John Cook wrote a program to compute the probability distribution of distances $latex |xy R…
1848
Inspecting C's qsort Through Animation
The C standard library includes a qsort() function for sorting arbitrary buffers given a comparator function. The name comes from its original Unix implementation, “quicker sort,” a variation of the well-known quicksort algorithm. The C standard doesn’t specify an algorithm, except to say that it may be unstable (C99…
1849
Algebraic graph calculus (2021)
We describe a graph-theoretic analogue of vector calculus. The linear operators of vector calculus (gradient, divergence, laplacian) correspond to the matrices naturally associated to graphs (incidence matrix, adjacency matrix). This analogy is useful for formalizing the discretization of some problems in image and…
1850
Sparse Matrices (2019)
A sparse matrix is a matrix that has a value of 0 for most elements. If the ratio of N umber of N on- Z ero ( NNZ ) elements to the size is less than 0.5, the matrix is sparse. While this is the mathematical definition, I will be using the term sparse for matrices with only NNZ elements and dense for matrices with all…
1851
Fine, I'll Play With Skiplists
Follow me on Bluesky! A Log-Structured Merge tree, or LSM, is a popular data structure for storage engines. It’s what is used by RocksDB, which is sort of...
1852
1853
An Introduction to Lagrange Multipliers
Lagrange multipliers are used in multivariable calculus to find maxima and minima of a function subject to constraints (like "find the highest elevation along the given path" or "minimize the cost of materials for a box enclosing a given volume"). It's a useful technique, but all too often it is poorly taught and…
1854
Summary of all the MIT Introduction to Algorithms lectures
As you all may know, I watched and posted my lecture notes of the whole MIT Introduction to Algorithms course. In this post I want to summarize all the topics that were covered in the lectures and point out some of the most interesting things in them. Actually, before I wrote this article, I had started writing an...
1855
1 kilobyte is precisely 1000 bytes?
We usually learn that 1 kilobyte is 1024 bytes, 1 MB is 1024 kilobytes, etc. But is this true or a kilobyte is precisely 1000 bytes?
1856
1857
EU proposing to regulate the use of Bayesian estimation
The European Commission just released their Proposal for a Regulation on a European approach for Artificial Intelligence . They finally get around to a definition of “AI” on page 60 of the report (link above):
1858
Implementing the Elo Rating System (2020)
In this article we’ll look at how to implement the Elo rating system in code.
1859
Time Travel (2006)
Last time we talked about free will, superintelligent predictors, and Dr. Evil planning to destroy the earth from his moon base. Today I'd like to talk about a more down-to-earth topic: time travel. The first point I have to make is one that Carl Sagan made: we're all time travelers—at the rate of one second per…
1860
