### Syndications of Interest

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Christopher Olah's Blog

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### 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

### 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 […]

### What's all this, then?

- Image dithering: a very simple error diffusion matrix 2023-07-10We have seen a number of different error diffusion matrices (and there are others we haven't discussed); the simplest of our matrices has been "Sierra Lite" \[\frac{1}{4}\begin{bmatrix} 0& *& 2\\ 1& 1& 0 \end{bmatrix}\] However, there is an even simpler one which seems to give very good results. The simplest dither matrix usually given is […]
- Image dithering: a very simple error diffusion matrix 2023-07-10We have seen a number of different error diffusion matrices (and there are others we haven't discussed); the simplest of our matrices has been "Sierra Lite" \[\frac{1}{4}\begin{bmatrix} 0& *& 2\\ 1& 1& 0 \end{bmatrix}\] However, there is an even simpler one which seems to give very good results. The simplest dither matrix usually given is […]
- Image dithering (2): error diffusion 2023-07-08A totally different approach to dithering is error diffusion. Here, the image is scanned pixel by pixel. Each pixel is thresholded t0 1 or 0 depending on whether the pixel value is greater than 0.5 or not, and the error - the difference between the pixel value and its threshold - is diffuse across neighbouring […]
- Image dithering (1): half toning 2023-07-07Image dithering, also known as half-toning, is a method for reducing the number of colours in an image, while at the same time trying to retain as much of its "look and feel" as possible. Originally this was required for newspaper printing, where no shades of grey were possible, and only black and white could […]
- The Pegasus and related methods for solving equations 2023-07-06In the previous post, we saw that a small change to the method of false position provided much faster convergence, while retaining its bracketing. This was the Illinois method which is only one of a whole host of similar methods, some of which converge even faster. And as a reminder, here's its definition, with a […]
- The Illinois method for solving equations 2023-07-05Such a long time since a last post! Well, that's academic life for you ... If you look at pretty much any modern textbook on numerical methods, of which there are many, you'll find that the following methods will be given for the solution of a single non-linear equation \(f(x)=0\): direct iteration, also known as […]
- Carroll's "improved" Doublets: allowing permutations 2022-11-07Carroll originally invented his Doublets in 1877, they were published in "Vanity Fair" (the magazine, not the Thackeray novel) in 1879. Some years later, in an 1892 letter, Carroll added another rule: that permutations were allowed. This allows very neat chains such as: roses, noses, notes, steno, stent, scent Because the words stay the same […]
- Super Doublets: more word ladders with Julia 2022-11-05Apparently there's a version of Doublets (see previous post) which allows you to add or delete a letter each turn. Thus we can go from WHEAT to BREAD as WHEAT, HEAT, HEAD, READ, BREAD which is shorter than the ladder given in that previous post. However, we can easily adjust the material from that post […]
- Word ladders with Julia 2022-11-03Lewis Carroll's game of Doublets Such a long time since my last post! Well, that's the working life for you. Anyway, recently I was reading about Lewis Carroll - always one of my favourite people - and was reminded of his word game "Doublets" in which one word is turned into another by changing one […]
- Every academic their own text-matcher 2022-06-19Plagiarism, text matching, and academic integrity Every modern academic teacher is in thrall to giant text-matching systems such as Ouriginal or Turnitin. These systems are sold as "plagiarism detectors", which they are not - they are text matching systems, and they generally work by providing a report showing how much of a student's submitted work […]
- More mapping "not quite how-to" - Voronoi regions 2022-06-18What this post is about In the previous post we showed how to set up a simple interactive map using Python and its folium package. As the example, we used a Federal electorate situated within the city of Melbourne, Australia, and the various voting places, or polling places (also known as polling "booths") associated with […]
- A mapping "not quite how-to" 2022-06-11Message about the underlying software NOTE: much of the material and discussion here uses the Python package "folium", which is a front end to the Javascript package "leaflet.js". The lead developer of leaflet.js is Volodymyr Agafonkin, a Ukrainian up until recently living and working in Kyiv. Leaflet version 1.80 was released on April 18, "in […]
- Further mapping: a win and a near miss 2022-06-09In this post we look at two Divisions from the recent Federal election: the inner city seat of Melbourne, and the bayside seat of Macnamara. Up until the recent election, Melbourne was the only Division to have a Greens representative. Macnamara, previously known as "Melbourne Ports" has been a Labor stronghold for all of its […]
- Post-election mapping 2022-06-05This continues on from the previous post, trying to make some sense of the voting in my electorate of Wills and the neighbouring electorate of Cooper. Both these electorates (or more formally "Divisions"), as I mentioned in the previous post, are very similar in their geography, demography, and history. Last post I simply showed a […]
- Post-election swings 2022-05-22So the Australian federal election of 2022 is over as far as the public is concerned; all votes have been cast and now it's a matter of waiting while the Australian Electoral Commission tallies the numbers, sorts all the preferences, and arrives at a result. Because of the complications of the voting system, and of […]
- 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 […]