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Is Probability Real?
Today, I want to address an issue with statements involving chance. To demonstrate, let’s first consider a statement that doesn’t involve chance:
Top posts
Every blog post that took off on Hacker News, ranked by the votes of its best submission. 2,851 posts.
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The Most Common Errors in Undergraduate Mathematics (2009)
This page describes the errors that I have seen most frequently in undergraduate mathematics, the likely causes of those errors, and their remedies.
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How Tail Call Optimization Works
Most undergraduate computer sciences courses teach students about tail call optimization (TCO) , and even if you don't have a formal computer science background the concept is talked about enough that you might be familiar with it anyway, especially if you've ever done any functional programming. However, I think the…
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Feynman on Fermat's Last Theorem
I write about Startups, coding and Physics
845
Tritone Substitutions
A ‘tritone substitution’ is a popular trick for making your music sound more sophisticated. I’ll show you a couple of videos with lots of examples. But since I’m mathematici…
846
Upper stage impacting the moon on 2026 August 5
On 2026 August 5, within a few minutes of 06:35 UTC, an upper stage (section) of a rocket used for a lunar mission will hit the moon. This may be of some (probably minor) scientific interest, and we may learn some things from it. It doesn't present any danger to anyone, though it does highlight a certain carelessness…
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Hashed sorting is typically faster than hash tables
Benchmarks and theoretical explanation of why and when hashed radix sort beats hash tables.
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A High-Level Technical Overview of Homomorphic Encryption
About two years ago, I switched teams at Google to focus on fully homomorphic encryption (abbreviated FHE, or sometimes HE). Since then I’ve got to work on a lot of interesting projects, learning along the way about post-quantum cryptography, compiler design, and the ins and outs of fully homomorphic encryption. If…
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The faster-than-fast Fourier transform
For a large range of practically useful cases, MIT researchers find a way to increase the speed of one of the most important algorithms in the information sciences.
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Operation Costs in CPU Clock Cycles
Quote: Back in 80s, it was possible to calculate the speed of the program just by looking at assembly. Another Quote: keep in mind that these days compilers ...
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The Riemann Hypothesis Says 5040 Is the Last
Note: These pages make extensive use of the latest XHTML and CSS Standards . They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in…
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Anxiety as an algorithm
Founder, Investor, Author
855
Secrets of smooth Béziers
I haven’t posted here much lately, and I admit, it’s because I’ve gotten sidetracked thinking about curves again. I did my PhD thesis on curves, so spent years thinking about them, then put that on hold for a while, aside from some work in crunching font file sizes down.
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Programmer's guide to polynomials and splines
This is a brief introduction into polynomials. From how to make a polynomial run through your set of points to how to make it into a spline.
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How to walk through walls using the 4th dimension [video]
An explanation of how walking through walls would actually look like if you could move in 4D.
859
Category Theory Illustrated – Sets
Ready, set , begin… (you don’t know how hard I tried to resist to making that pun). We begin our inquiry with the theory of sets. Set theory and category theory share many similarities. We can view category theory as a generalization of set theory. That is, it’s meant to describe the same thing as set theory…
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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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Seven basic rules for causal inference
Personal website of Dr. Peder M. Isager
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Symmetry in Chaos
The following is based upon systems of equations found in "Symmetry in Chaos, A Search for Pattern in Mathematics, Art, and Nature" (the second edition) by Michael Field and Martin Golubitsky. They ask the question whether there are aperiodic chaotic attractors that generate symmetric patterns? Can there be order…
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Attacking Merkle Trees with a second preimage attack
Hacking, hardware, software, and general curiosities of computing machinery! Docendo discumus.
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Reinforcement Learning – Bandit Problems
Table of Contents
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An Introduction to Cache-Oblivious Data Structures
If you read my recent post about Postgres you may have noted that Postgres operates primarily with fixed-size blocks of memory called “pages.” By default, Postgres pages are 8KB. This number is tuned to match operating system page sizes which are tuned to match hardware cache sizes. If you were to run Postgres on…
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Mapping the internet with Hilbert curves
The internet is big. Really big . You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think the /22 you got as a LIR was big, but that’s just peanuts to the internet.
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Cache Oblivious Algorithms
Cache-Oblivious Algorithms Cache-oblivious algorithms seemed incredible to me when I first heard about it in 6.854 (Advanced Algorithms). It’s relatively straightforward to imagine algorithms that utilizes information about page size $B$ and cache size $M$ to create efficient data structures. However, cache-oblivious…
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NP Complete 3rd Grade Math Problems (2018)
Leo’s third grade class got to try a Noyce Foundation math worksheet [1] the other day. They didn’t fare so well, but I gotta tell you, some of the problems are REALLY hard! In fact, on…