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### Syndications of Interest

Ars Mathematica

- Nine Chapters on the Semigroup Art 2015-02-28While Googling something or other, I came across Nine Chapters on the Semigroup Art, which is a leisurely introduction to the theory of semigroups. (While the document is labelled “lecture notes”, the typography is quite beautiful.)Walt
- What Did Grothendieck Do? 2015-01-01Happy New Year! The publicity in the wake of Grothendieck’s death has left a certain number of non-mathematicians with the question of what it was exactly that he did. I wrote an answer elsewhere that people seemed to find informative, … Continue reading →Walt
- Learning about Stochastic Processes the Almost Sure Way 2014-11-09George Lowther at Almost Sure has written a terrific series of posts explaining stochastic processes and the stochastic calculus. Stochastic calculus is widely used in physics and finance, so there are many informal introductions that get across the main ideas … Continue reading →Walt
- Arguesian Lattices 2014-09-23As is well-known, the lattice of submodules of a module is modular. What I did not know is that the converse is not true, and that lattices of submodules must satisfy a stronger property, the arguesian law. The Arguesian law … Continue reading →Walt
- K2, not the mountain 2014-03-20Chandan Singh Dalawat has a nice survey article about K2. It just gives the highlights of the theory, without proofs, so it’s closer to a teaser trailer than it is to full-length movie. But sometimes you just want a teaser … Continue reading →Walt
- Cayley Bacharach Theorem through History 2014-02-10I came across this terrific article that describes a sequence of results beginning with Pappas’ theorem through the Cayley-Bacharach theorem to modern formulations in terms of the Gorenstein (!) condition. The connection between classical topics in algebraic geometry and modern … Continue reading →Walt
- Nonassociative Algebras 2013-12-30I periodically feel like I should learn more about nonassociative algebra. (I’ve studied Lie algebras, and technically Lie algebras are non-associative, but they’re pretty atypical of nonassociative algebras.) There’s a mysterious circle of “exceptional” examples that are all related — … Continue reading →Walt
- Determinacy 2013-11-30One of my ambitions in life is to understand projective determinacy. Fortunately, Tim Gowers has written a series of posts to explain Martin’s proof that Borel sets are determined. The main source of interest in determinacy is that results suggest … Continue reading →Walt
- A Generalized Fermat Equation 2013-08-31I came across a number theory paper Twists of X(7) and Primitive Solutions of x2 + y3 = z7 that I find completely fascinating. I find it fascinating because a) the question is so easy, b) the answer is so … Continue reading →Walt
- Linear Bestiary of Francois Pottier 2013-07-09Ugh, I suck at this blogging thing. I periodically get ambitious, and make big plans. That doesn’t actually lead to any completed posts, just many long half-finished posts, and hundreds of open tabs in Firefox. I think I’ll start with … Continue reading →Walt

Christopher Olah's Blog

- Deep Learning, NLP, and Representations 2014-07-08In the last few years, deep neural networks have dominated pattern recognition. They blew the previous state of the art out of the water for many computer vision tasks. Voice recognition is also moving that way. But despite the results, we have to wonder… why do they work so well? This post reviews some extremely […]colah
- Fanfiction, Graphs, and PageRank 2014-07-07On a website called fanfiction.net, users write millions of stories about their favorite stories. They have diverse opinions about them. They love some stories, and hate others. The opinions are noisy, and it’s hard to see the big picture. With tools from mathematics and some helpful software, however, we can visualize the underlying structure. […]colah
- Neural Networks, Manifolds, and Topology 2014-04-09Recently, there’s been a great deal of excitement and interest in deep neural networks because they’ve achieved breakthrough results in areas such as computer vision. However, there remain a number of concerns about them. One is that it can be quite challenging to understand what a neural network is really doing. If one trains it well, it […]colah
- Visualizing Functions On Groups 2014-01-16Functions of the form or , where is a group, arise in lots of contexts. One very natural way this can happen is to have a probability distribution on a group, . The probability density of group elements is a function . Another way this can happen is if you have some function and has […]colah
- The Death of a Squirrel 2013-08-25(Trigger warning: descriptions of severe animal injury.) Today a squirrel was hit by a car a few feet away from me while I was walking down the side walk. Three of its legs kept twitching. I thought it had a broken leg. I came out of my stupor and went to grab it and pull […]colah
- Order Statistics 2013-08-16What is the distribution of the maximum of random variables? What started out a utilitarian question in my exploration of some generalized versions of the secretary problem turns out to be quite a deep topic. (Note that I have little background in probability and statistics. Please forgive (and inform me of, so I can fix!) […]colah
- Topology Notes 2013-06-14I’ve been talking about writing a topology textbook introductory notes on topology for years. Basically since I wrote my Rethinking Topology (or a Personal Topologodicy) post 2 years ago — it’s hard to believe it’s been that long! In any case, I finally started writing it. I’ve done a mild review of existing introductions to general topology (ie. I […]colah
- How My Neural Net Sees Blackboards (Part 2) 2013-06-09Previously, I discussed training a neural net to clean up images. I’m pleased to say that, using more sophisticated techniques, I’ve since achieved much better results. My latest approach is a four layer convolutional network. Sadly, the convolution throws away the sides of the images, so we get a black margin. In any case, compare […]colah
- I’m Sick and Tired of 3D Printed Guns 2013-05-29For the last few months, every time someone hears that I work with 3D printers they bring up 3D printed guns. I can’t say how many times it has happened in this month alone. And I’m getting really really tired of it. “They’re the killer app of 3D printers.” What a great pun. You don’t know […]colah
- How My Neural Net Sees Blackboards 2013-05-11For the last few weeks, I’ve been taking part in a small weekly neural net study group run by Michael Nielsen. It’s been really awesome! Neural nets are very very cool! They’re so cool, I had to use them somehow. Having been interested in mathematical handwriting recognition for a long time, I decided to train […]colah

Creative Commons

- CC at SCCR 42 — A Look Back at the WIPO Copyright Meeting 2022-05-16From 9 to 13 May 2022, Creative Commons (CC) participated in the 42nd session of the World Intellectual Property Organization (WIPO) Standing Committee on Copyright and Related Rights (SCCR) in Geneva, Switzerland. In this blog post, we look back on the highlights of the SCCR/42 week. Super happy to represent @creativecommons at @WIPO #SCCR42 #copyright […]Brigitte Vézina
- Episode 30: Open Culture VOICES – Julia Pagel 2022-05-12Welcome to episode 30 of Open Culture VOICES! VOICES is a vlog series of short interviews with open GLAM (galleries, libraries, archives, and museums) experts from around the world. The Open Culture Program at Creative Commons aims to promote better sharing of cultural heritage in GLAMs collections. With Open Culture VOICES, we’re thrilled to […]Camille Françoise
- Episode 29: Open Culture VOICES – Philippe Rivière 2022-05-12Welcome to episode 29 of Open Culture VOICES! VOICES is a vlog series of short interviews with open GLAM (galleries, libraries, archives, and museums) experts from around the world. The Open Culture Program at Creative Commons aims to promote better sharing of cultural heritage in GLAMs collections. With Open Culture VOICES, we’re thrilled to […]Brigitte Vézina
- Creative Commons condemns rejection of Wikimedia chapters as observers at WIPO SCCR 2022-05-10Yesterday, China blocked the ad-hoc accreditation of the Wikimedia chapters of France, Germany, Italy, Mexico, Sweden, and Switzerland as official observers to the Standing Committee on Copyright and Related Rights (SCCR) of the World Intellectual Property Organization (WIPO). Previously, China rejected the Wikimedia Foundation’s application for observer status to this UN agency. The WIPO SCCR […]Ony Anukem
- Episode 28: Open Culture VOICES – Mariana Ziku 2022-05-06Welcome to episode 28 of Open Culture VOICES! VOICES is a vlog series of short interviews with open GLAM (galleries, libraries, archives, and museums) experts from around the world. The Open Culture Program at Creative Commons aims to promote better sharing of cultural heritage in GLAMs collections. With Open Culture VOICES, we’re thrilled to […]Camille Françoise
- Episode 27: Open Culture VOICES – Simon Tanner 2022-05-06Welcome to episode 27 of Open Culture VOICES! VOICES is a vlog series of short interviews with open GLAM (galleries, libraries, archives, and museums) experts from around the world. The Open Culture Program at Creative Commons aims to promote better sharing of cultural heritage in GLAMs collections. With Open Culture VOICES, we’re thrilled to […]Brigitte Vézina
- Eight case studies show opportunities, challenges, and needs of low-capacity and non-Western cultural heritage institutions 2022-05-04In October 2021, Creative Commons launched a call for case studies on open access in cultural institutions such as galleries, libraries, archives and museums (GLAMs), from low-capacity, non-Western institutions, or representing marginalized,underrepresented communities from various parts of the world. The aim of the open call was to help generate a more global, inclusive, and equitable […]Camille Françoise
- Open Minds Podcast: Damien Riehl & Noah Rubin of All The Music 2022-05-04Hello Creative Commoners! We are back with a brand new episode of CC’s Open Minds … from Creative Commons podcast. In this episode, we sat down with programmer, musician, and copyright attorney, Damien Riehl, and fellow musician and programmer, Noah Rubin—the creators of the All The Music project. Frustrated by accidental copyright infringement lawsuits stifling […]Ony Anukem
- CC welcomes Nate Angell 2022-05-03Here at Creative Commons, we know that the stories we tell and the people we engage are deeply connected, together forming the backbone of who we are and what we do. Our community generates our most powerful stories and also amplifies all the work we do, spreading the benefits of open knowledge and better sharing […]Catherine Stihler
- Episode 26: Open Culture VOICES – Susanna Ånäs 2022-04-28Welcome to episode 25 of Open Culture VOICES! VOICES is a vlog series of short interviews with open GLAM (galleries, libraries, archives, and museums) experts from around the world. The Open Culture Program at Creative Commons aims to promote better sharing of cultural heritage in GLAMs collections. With Open Culture VOICES, we’re thrilled to […]Brigitte Vézina

Planet Sage

- Sébastien Labbé: Tiling a polyomino with polyominoes in SageMath 2020-12-03Suppose that you 3D print many copies of the following 3D hexo-mino at home: sage: from sage.combinat.tiling import Polyomino, TilingSolver sage: p = Polyomino([(0,0,0), (0,1,0), (1,0,0), (2,0,0), (2,1,0), (2,1,1)], color='blue') sage: p.show3d() Launched html viewer for Graphics3d Object You would like to know if you can tile a larger polyomino or in particular a rectangular […]
- William Stein: DataDog: Don't make the same mistake I did -- a followup and thoughts about very unhappy customers 2020-04-13This is a followup to my previous blog post about DataDog billing. TL;DR:- I don't recommend DataDog,- dealing with unhappy customers is hard,- monitoring for data science nerds?Hacker News CommentsDataDog at Google Cloud SummitI was recently at the Seattle Google Cloud Summit and DataDog was well represented, with the biggest booth and top vendor billing […]
- Sébastien Labbé: Computer experiments for the Lyapunov exponent for MCF algorithms when dimension is larger than 3 2020-03-27In November 2015, I wanted to share intuitions I developped on the behavior of various distinct Multidimensional Continued Fractions algorithms obtained from various kind of experiments performed with them often involving combinatorics and digitial geometry but also including the computation of their first two Lyapunov exponents. As continued fractions are deeply related to the combinatorics […]

Yet Another Mathblog

- Let’s do the Landau shuffle 2021-10-23Here’s a shuffle I’ve not seen before: Take an ordinary deck of 52 cards and place them, face up, in the following pattern:Going from the top of the deck to the bottom, placing cards down left-to-right, put 13 cards in the top row:11 cards in the next row:then 9 cards in the next row:then 7 […]wdjoyner
- Coding Theory and Cryptography 2021-10-11This was once posted on my USNA webpage. Since I’ve retired, I’m going to repost it here. Coding Theory and Cryptography:From Enigma and Geheimschreiber to Quantum Theory(David Joyner, ed.) Springer-Verlag, 2000. ISBN 3-540-66336-3 Summary: These are the proceedings of the “Cryptoday” Conference on Coding Theory, Cryptography, and Number Theory held at the U. S. Naval […]wdjoyner
- Shanks’ SQUFOF according to McMath 2021-07-30In 2003, a math major named Steven McMath approached Fred Crabbe and I about directing his Trident thesis. (A Trident is like an honors thesis, but the student gets essentially the whole year to focus on writing the project.) After he graduated, I put a lot of his work online at the USNA website. Of […]wdjoyner
- The truncated tetrahedron covers the tetrahedron 2021-04-29At first, you might think this is obvious – just “clip” off each corner of the tetrahedron to create the truncated tetrahedron (by essentially creating a triangle from each of these clipped corners – see below for the associated graph). Then just map each such triangle to the corresponding vertex of the tetrahedron. No, it’s […]wdjoyner
- A mathematical card trick 2021-03-28If you search hard enough on the internet you’ll discover a pamphlet from the 1898 by Si Stebbins entitled “Card tricks and the way they are performed” (which I’ll denote by [S98] for simplicity). In it you’ll find the “Si Stebbins system” which he claims is entirely his own invention. I’m no magician, by from […]wdjoyner
- Quartic graphs with 12 vertices 2020-10-13This is a continuation of the post A table of small quartic graphs. As with that post, it’s modeled on the handy wikipedia page Table of simple cubic graphs. According to SageMath computations, there are 1544 connected, 4-regular graphs. Exactly 2 of these are symmetric (ie, arc transitive), also vertex-transitive and edge-transitive. Exactly 8 of these are […]wdjoyner
- A footnote to Robert H. Mountjoy 2020-08-27In an earlier post titled Mathematical romantic? I mentioned some papers I inherited of one of my mathematical hero’s Andre Weil with his signature. In fact, I was fortunate enough to go to dinner with him once in Princeton in the mid-to-late 1980s – a very gentle, charming person with a deep love of mathematics. […]wdjoyner
- The Riemann-Hurwitz formula for regular graphs 2020-08-21A little over 10 years ago, M. Baker and S. Norine (I’ve also seen this name spelled Norin) wrote a terrific paper on harmonic morphisms between simple, connected graphs (see “Harmonic morphisms and hyperelliptic graphs” – you can find a downloadable pdf on the internet of you google for it). Roughly speaking, a harmonic function […]wdjoyner
- The number-theoretic side of J. Barkley Rosser 2020-08-13By chance, I ran across a reference to a paper of J Barkey Rosser and it brought back fond memories of days long ago when I would browse the stacks in the math dept library at the University of Washington in Seattle. I remember finding papers describing number-theoretic computations of Rosser and Schoenfeld. I knew […]wdjoyner
- A table of small quartic graphs 2020-07-02This page is modeled after the handy wikipedia page Table of simple cubic graphs of “small” connected 3-regular graphs, where by small I mean at most 11 vertices. These graphs are obtained using the SageMath command graphs(n, [4]*n), where n = 5,6,7,… . 5 vertices: Let denote the vertex set. There is (up to isomorphism) […]wdjoyner

What's all this, then?

- Ramanujan's cubes 2022-04-03This post illustrates the working of Ramanujan's generating functions for solving Euler's diophantine equation \(a^3+b^3=c^3+d^3\) as described by Andrews and Berndt in "Ramanujan's Lost Notebook, Part IV", pp 199 - 205 (Section 8.5). The text is available from Springer. Ramanujan's result is that if \[ f_1(x) = \frac{1+53x+9x^2}{1-82x-82x^2+x^3} = a_0+a_1x+a_2x^2+a_3x^3+\cdots = \alpha_0+\frac{\alpha_1}{x}+\frac{\alpha_2}{x^2}+\frac{\alpha_3}{x^3}+\cdots\] \[ f_2(x) = […]
- Wordle 2022-01-24Wordle is a pleasant game, basically Mastermind with words. You choose an English word (although it can also be played in other languages), and then you're told if your letters are incorrect, correct but in the wrong place, or correct and in the right place. These are shown by the colours grey, yellow, and green. […]
- Five letter words in English 2022-01-23I was going to make a little post about Wordle, but I go sidetracked exploring five letter words. At the same time, I had a bit of fun with regular expressions and some simple scripting with ZSH. The start was to obtain lists of 5-letter words. One is available at the Stanford Graphbase site; the […]
- A new year (2022) 2022-01-01What does one do on the first day of a new year but write a blog post, and in it clearly delineate all plans for the coming year? Well, I'm doing the first part, but not the second, as I know that any plans will not be fulfilled - something always gets in the way. […]
- Exploring Tanh-Sinh quadrature 2021-04-30As is well known, tanh-sinh quadrature takes an integral \[ \int_{-1}^1f(x)dx \] and uses the substitution \[ x = g(t) = \tanh\left(\frac{\pi}{2}\sinh t\right) \] to transform the integral into \[ \int_{-\infty}^{\infty}f(g(t))g'(t)dt. \] The reason this works so well is that the derivative \(g'(t)\) dies away at a double exponentional rate; that is, at the rate […]
- A note on Steffensen's method for solving equations 2021-04-30Steffensen's method is based on Newton's iteration for solving a non-linear equation \(f(x)=0\): \[ x\leftarrow x-\frac{f(x)}{f'(x)} \] Newton's method can fail to work in a number of ways, but when it does work it displays qudratic convergence; the number of correct signifcant figures roughly doubling at each step. However, it also has the disadvntage of […]
- High precision quadrature with Clenshaw-Curtis 2021-04-21An article by Bailey, Jeybalan and LI, "A comparison of three high-precision quadrature schemes", and available online here, compares Gauss-Legendre quadrature, tanh-sinh quadrature, and a rule where the nodes and weights are given by the error function and its integrand respectively. However, Nick Trefethen of Oxford has shown experimentally that Clenshaw-Curtis quadrature is generally no […]
- The circumference of an ellipse 2021-04-10Note: This blog post is mainly computational, with a hint of proof-oriented mathematics here and there. For a more in-depth analysis, read the excellent article "Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, pi, and the Ladies Diary" by Gert Akmkvist and Bruce Berndt, in The American Mathematical Monthly, vol 95 no. 7 (August-September 1988), pages […]
- Voting power (7): Quarreling voters 2021-01-24In all the previous discussions of voting power, we have assumed that all winning coalitions are equally likely. But in practice that is not necessarily the case. Two or more voters may be opposed on so many issues that they would never vote the same way on any issues: such a pair of voters may […]
- Voting power (6): Polynomial rings 2021-01-22As we have seen previously, it's possible to compute power indices by means of polynomial generating functions. We shall extend previous examples to include the Deegan-Packel index, in a way somewhat different to that of Alonso-Meijide et al (see previous post for reference). Again, suppose we consider the voting game \[ [30;28,16,5,4,3,3] \] What we'll […]
- Voting power (5): The Deegan-Packel and Holler power indices 2021-01-14We have explored the Banzhaf and Shapley-Shubik power indices, which both consider the ways in which any voter can be pivotal, or critical, or necessary, to a winning coalition. A more recent power index, which takes a different approach, was defined by Deegan and Packel in 1976, and considers only minimal winning coalitions. A winning […]
- Three-dimensional impossible CAD 2021-01-10Recently I friend and I wrote a semi-serious paper called "The geometry of impossible objects" to be delivered at a mathematics technology conference. The reviewer was not hugely complimentary, saying that there was nothing new in the paper. Well, maybe not, but we had fun pulling together some information about impossible shapes and how to […]
- Voting power (4): Speeding up the computation 2021-01-06Introduction and recapitulation Recall from previous posts that we have considered two power indices for computing the power of a voter in a weighted system; that is, the ability of a voter to influence the outcome of a vote. Such systems occur when the voting body is made up of a number of "blocs": these […]
- Voting power (3): The American swing states 2021-01-03As we all know, American Presidential elections are done with a two-stage process: first the public votes, and then the Electoral College votes. It is the Electoral College that actually votes for the President; but they vote (in their respective states) in accordance with the plurality determined by the public vote. This unusual system was […]
- Voting power (2): computation 2020-12-31Naive implementation of Banzhaf power indices As we saw in the previous post, computation of the power indices can become unwieldy as the number of voters increases. However, we can very simply write a program to compute the Banzhaf power indices simply by looping over all subsets of the weights: 1def banzhaf1(q,w): 2 n = […]
- Voting power 2020-12-30After the 2020 American Presidential election, with the usual post-election analyses and (in this case) vast numbers of lawsuits, I started looking at the Electoral College, and trying to work out how it worked in terms of power. Although power is often conflated simply with the number of votes, that's not necessarily the case. We […]
- Electing a president 2020-11-07Every four years (barring death or some other catastrophe), the USA goes through the periodic madness of a presidential election. Wild behaviour, inaccuracies, mud-slinging from both sides have been central since George Washington's second term. And the entire business of voting is muddied by the Electoral College, the 538 members of which do the actual […]
- Enumerating the rationals 2020-01-18The rational numbers are well known to be countable, and one standard method of counting them is to put the positive rationals into an infinite matrix \(M=m_{ij}\), where \(m_{ij}=i/j\) so that you end up with something that looks like this: \[ \left[\begin{array}{ccccc} \frac{1}{1}&\frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\dots\\[1ex] \frac{2}{1}&\frac{2}{2}&\frac{2}{3}&\frac{2}{4}&\dots\\[1ex] \frac{3}{1}&\frac{3}{2}&\frac{3}{3}&\frac{3}{4}&\dots\\[1ex] \frac{4}{1}&\frac{4}{2}&\frac{4}{3}&\frac{4}{4}&\dots\\[1ex] \vdots&\vdots&\vdots&\vdots&\ddots \end{array}\right] \] It is clear that not only […]
- Fitting the SIR model of disease to data in Julia 2020-01-15A few posts ago I showed how to do this in Python. Now it's Julia's turn. The data is the same: spread of influenza in a British boarding school with a population of 762. This was reported in the British Medical Journal on March 4, 1978, and you can read the original short article here. […]
- The Butera-Pernici algorithm (2) 2020-01-06The purpose of this post will be to see if we can implement the algorithm in Julia, and thus leverage Julia's very fast execution time. We are working with polynomials defined on nilpotent variables, which means that the degree of any generator in a polynomial term will be 0 or 1. Assume that our generators […]