2071
You need much less memory than time
Just as I was complaining that we haven't seen many surprising breakthroughs in complexity recently, we get an earthquake of a result to st...
Top posts
Every blog post that took off on Hacker News, ranked by the votes of its best submission. 2,851 posts.
2072
Reverse-engineering the Intel 8087's tangent algorithm: more than CORDIC
I hope you're not tired of the 8087, because I have another article about Intel's floating-point chip. 1 In 1980, Intel introduced the 8087...
2073
An alternative construction of Shannon entropy
The underpinning of information theory is Shannon’s formula for the entropy H of a discrete probability distribution P. Derived in 1948 in the paper “A mathematical theory of communication”, this is given by \[ H[P] = -\sum_{i=1}^N p_i \log p_i \] where \( p_i = p(x_i) \) is the probability of sampling \( x_i \) from…
2074
SIMD Vectorization in Julia
A portfolio of HPC tools help data scientists, researchers, and developers efficiently build cross-architecture applications.
2075
Succinct Data Structures: Cramming 80,000 words into a Javascript file
<img src="http://zwibbler.com/shared/1024.png"/> jQuery creator John Resig needs a little help storing lists of words in his side project. Let's go overkill and explore a little known branch of computer science called Succinct Data Structures.
2076
Geometrically understanding calculus of inverse functions (2023)
The Legendre transformation tells you how the inverse map acts on integrals
2077
Elsevier journals: has anything changed?
Greg Martin, a number theorist at UBC (the University of British Columbia in Vancouver) doesn’t think so, so he has decided to resign from the editorial board of Elsevier’s Journal of N…
2078
Progress on longstanding open problems
For those who haven’t seen it: Aubrey de Grey, better known to the world as a radical life extension researcher, on Sunday posted a preprint on the arXiv claiming to prove that the chromatic …
2079
The field of “useful reals” between rational and real numbers (2019)
Most schools teach number systems the same way. First they teach natural numbers (or “counting numbers”), then integers, rationals, reals, and complex numbers, in that order. Rationals are countable and reals are not, but that’s okay. Once you start taking roots, you need a number system more powerful than a countable…
2080
Gerrymandering with geographically compact districts
The following tweet has been making the rounds on social media: Notice that since the Democrats won overall, it would be impossible for Republicans to win all seven districts, no matter how the map…
2081
How Chordcat works – a chord naming algorithm
chordcat is a chord-naming application and a pet project of mine. It is written in C++ and SFML.
2082
The Scientific Case for P≠NP
Out there in the wider world—OK, OK, among Luboš Motl, and a few others who comment on this blog—there appears to be a widespread opinion that P≠NP is just “a fashionable dogma of…
2083
Ergodicity, What's It Mean
Ensemble averages and time averages
2084
The Future of Mathematics
[This is a guest post by Jeremy Avigad. This blog post was initially written in a different file format and converted using AI. — T.] “Mathematics underwent, in the nineteenth century, …
2085
Rediscovering Quaternions
How do we represent a rotation in 3 dimensions? This is a surprisingly deep question, and one we can learn a lot from. Our goal will be to “rediscover” quaternions from first principles, building them up by exploring different representations and fixing the failures along the way.
2086
Ten Signs a Claimed Mathematical Breakthrough is Wrong
Yesterday several people asked my opinion of a preprint claiming to solve the Graph Isomorphism problem in deterministic polynomial time. I responded: If I read all such papers, then I wouldn’t hav…
2087
Ancient Babylonian Algorithms: The Earliest Programs
An interactive, illustrated timeline of historic moments in humankind's quest for information. With annotations by Jeremy Norman.
2088
Solving First Order Differential Equations with Julia
In this tutorial, we will learn about solving Differential Equations with Julia. Differential Equations are everywhere in Science and Engineering problems. And, being able to solve Differential Equations using computers is very convenient. We will use the DifferentialEquations.jl package for solving Differential…
2089
Let's Play Ancient Greek Geometry
Today I’m releasing Let’s Play: Ancient Greek Geometry . It’s a Compass and Straightedge tool/puzzle game written in JavaScript. I’ve always thought Geometric Construction felt like a puzzle, so to me this pairing was quite natural. Compass and Straight edge is a technique for constructing shapes out of circles…
2090
No, RSA Is Not Broken
I have been seeing this paper by cryptographer Peter Schnorr making the rounds: “Fast Factoring Integers by SVP Algorithms.” It describes a new factoring method, and its abstract ends with the provocative sentence: “This destroys the RSA cryptosystem.” It does not. At best, it’s an improvement in factoring—and I’m not…
2091
Homology
Xoilac TV trực tiếp bóng đá HD hàng đầu Việt Nam, đường truyền ổn định, đa dạng link trực tuyến, đa dạng giải đấu, BLV chuyên môn cao và hoàn toàn miễn phí.
2092
2093
Operational Transformation (algorithm behind Wave, Google Docs)
Almost exactly a year ago, Google made one of the most remarkable press releases in the Web 2.0 era. Of course, by "press release", I actually mean keynote at their own conference, and by "remarkable" I mean potentially-transformative and groundbreaking. I am referring of course to the announcement of Google Wave , a…
2094
Set theory with types
It is known that mathematics is heavily reliant on set theory, but no one can agree on what set theory is. Many people today understand that we have a choice between set theory and type theory , but they don’t know what type theory is either. Many think that type theory refers to some sort of dependent type theory, as…
2095
2D Signed Distance Functions
Tutorials and articles of Inigo Quilez on computer graphics, fractals, math, art, demoscene, shaders and more.
2096
2097
Adventure Games and Eigenvalues (2017)
Do adventure games have dead ends? The author of The Teeny Tiny Mansion proves the absence of dead ends in that game via formal verification (in particular, with the help of the cbmc model-checking program ). Here I’ll demonstrate how to prove the same thing via Markov theory. The Markov approach requires more…
2098
The Mathematics of Crochet
I remember thinking at school when will I ever in my “real life” use maths. Well much to my utter surprise I’ve come to the stark realisation that there is a link between mathematics and crochet. …
2099
Machine Learning Trick of the Day: Instrumental Thinking
· Read in · 1499 words · All posts in series · [dropcap]This[/dropcap] trick is unlike the others we’ve conjured. It will not reveal a clever manipulation of a probability or an integral or a…
2100
The Asynchronous Computability Theorem
In this post I describe the Asynchronous Computability Theorem, which uses tools from Algebraic Topology to show whether a task is solvable…