### Research

- Academia
- Books and Journals
- Business and Economics
- Constants, Identities, and Variations
- Databases and Archives
- Dictionaries
- Encyclopedias
- History
- Learning
- Linguistics
- Mathematics
- Analysis of a Complex Kind
- Ford Circles
- Heurist on Desmos
- Heurist on Geogebra
- Heurist on Overleaf
- Heurist on Wolfram
- Imaginary Numbers Are Real – Welsh Labs
- LaTeX Test
- Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
- MIT A 2020 Vision of Linear Algebra, Spring 2020
- Original Form
- Riemann Surfaces
- Riemann Surfaces 2nd Group
- The Riemann-Hurwitz Formula for Regular Graphs

- Other Niches
- Personalities
- Philosophy
- Reference
- Research
- Science
- Search
- Social Science
- Technology
- Wikidata

### 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

- New Season, New Beginnings at Creative Commons 2021-04-06Dear CC Community, This past year has been full of change and challenges for all of us, and I’ve never been more grateful for (and amazed by!) the people that make up our CC community. I joined Creative Commons in 2016 as a newbie to the open movement, eager to learn from this incredible community […]Alison Pearce
- Her Story: Embracing the Here and Now 2021-04-05For over 40 years, millions across the globe have collectively celebrated the achievements, histories, ideas, and contributions of women on March 8 and increasingly, throughout March for Women’s History Month using #HerStory and #BecauseOfHerStory. This year, we wanted to do something special to celebrate this annual event, so we reached out to several members of […]Victoria Heath
- A New Era of Open? COVID-19 and the Pursuit for Equitable Solutions 2021-04-02In response to the COVID-19 pandemic, Creative Commons published an article titled, “Now Is the Time for Open Access Policies—Here’s Why” in March 2020. We felt it imperative to underscore the importance of open access, specifically open science, in times of crisis. A lot has changed since March of last year and it’s important to […]Victoria Heath
- Her Story: Promoting Inclusivity and Equity 2021-03-22For over 40 years, millions across the globe have collectively celebrated the achievements, histories, ideas, and contributions of women on March 8 and increasingly, throughout March for Women’s History Month using #HerStory and #BecauseOfHerStory. This year, we wanted to do something special to celebrate this annual event, so we reached out to several members of […]Victoria Heath
- Open Minds Podcast: Coraline Ada Ehmke on Ethical Source 2021-03-19We’ve been getting such great feedback about our new podcast, Open Minds…from Creative Commons. Thank you for listening! On today’s episode, Sarah Pearson, CC’s Senior Counsel, talks to Coraline Ada Ehmke about her work at the intersection of open source and social good. Ehmke is a developer, writer, speaker, musician, and activist. She’s the creator […]Eric Steuer
- Our 2020 State of the Commons Report Is Here! 2021-03-182020 was a year none of us will forget—and while there are many reasons to look back on last year with sadness and anger, we’ve chosen a different path: one of optimism and hope. In our 2020 State of the Commons report, we take you through what we accomplished last year, from effectively unlocking hundreds […]Victoria Heath
- Meet Your New Global Network Council Executive Committee! 2021-03-16In December 2020 the Creative Commons Global Network Council (GNC) voted on the new Executive Committee (ExCom). The ExCom took up its work in January 2021 and will be working throughout the next year and beyond by supporting the Network, fostering and strengthening connections, and encouraging activities around the new CC Strategy. Meet the six […]Julia Brungs
- Her Story: Transforming Open Through Feminism 2021-03-15For over 40 years, millions across the globe have collectively celebrated the achievements, histories, ideas, and contributions of women on March 8 and increasingly, throughout March for Women’s History Month using #HerStory and #BecauseOfHerStory. This year, we wanted to do something special to celebrate this annual event, so we reached out to several members of […]Victoria Heath
- Meet CC Nepal, Our Next Feature for CC Network Fridays! 2021-03-12After introducing the CC Italy Chapter to you in July, the CC Netherlands Chapter in August, CC Bangladesh Chapter in September, CC Tanzania Chapter in October, and the CC India Chapter in November, the CC Mexico Chapter in December, and CC Argentina Chapter in January, and CC South Africa Chapter in February, we are now […]Julia Brungs
- Our Response To Canada’s Copyright Term Extension Consultation 2021-03-09On 29 January 2020, the Canadian federal government introduced Bill C-4, “An Act to Implement the Agreement between Canada, the United States of America and the United Mexican States” (CUSMA).1 The bill includes a proposal to extend copyright’s term of protection2 by 20 years, moving it from “life of the author + 50 years” (the […]Brigitte Vézina

Featured Blog Posts – Data Science Central

- How to Improve Content Marketing Results with Big Data 2021-04-09Companies and businesses around the world have been forced during the last year to shift their activity online. This is because people are spending more time online than ever. There, you can promote your business, organize contests, connect with customers, and improve your brand awareness. At the base of all these…Joseph McLean
- Four Alternative Data Trends to Watch in 2021 2021-04-09Given the recent pandemic’s uncertainty and disturbances, it is not surprising that investors seek more information before making a decision. Although alternative data is not new, it has been the recent focus for many investing firms. A…Tom Wilson
- 5 Dominating IoT Trends Positively Impacting Telecom Sector in 2021 2021-04-09The Internet of Things (IoT) has matured to support the telecom industry by becoming an integral part of concepts like self-driving vehicles and industrial IoT. The Telecom industry, in fact, is amongst the biggest players in the IoT.…Josiah Salser
- Vue.js vs AngularJS Development in 2021: Side-by-Side Comparison 2021-04-09What is the first name that comes to your mind when we talk about web development? And why is it JavaScript? Well, it is because JavaScript is the fastest growing language and is ruling the development industry, be it web or…Aria Barnes
- Building Effective Site Taxonomies 2021-04-08Several years ago, the typical company website fit into a predefined template - a home or landing page (usually talking about how innovative the company was), a products page, a business client testimonials page, a blog, and an "about us" page. However, as the number of products or services have multiplied, and as the demands […]Kurt A Cagle
- Simple Machine Learning Approach to Testing for Independence 2021-04-08We describe here a methodology that applies to any statistical test, and illustrated in the context of assessing independence between successive observations in a data set. After reviewing a few standard approaches, we discuss our methodology, its benefits, and drawbacks. The data used here for illustration purposes, has known theoretical…Vincent Granville
- Leveraging SAP’s Enterprise Data Management tools to enable ML/AI success 2021-04-07Background In our previous blog post, “Master Your ML/AI Success with Enterprise Data Management”, we outlined the need for Enterprise Data Management (EDM) and…Mahesh Pethe
- 6 essential steps of healthcare mobile app development 2021-04-07So, you’ve done your research and selected the market niche and the type of your mHealth app. Now it’s time for planning and estimating the project scope, budget, and main features of your product. Healthcare mobile app development can be daunting and time-consuming unless you’re well prepared. Follow these steps to make sure you don’t…Yulia Kondratyuk
- Using Amazon S3 for Object Storage 2021-04-07Image by Mohamed Hassan from Pixabay…Data Geek
- Data Analytics: How it Drives Better Decision-Making 2021-04-07Not one or two, but more than enough studies have shown that for businesses to succeed, insights are vital. Insights about customer behavior, market trends, operations, and so much more — the point is that insight is essential. Now, how do you gain these insights then? Data — tons and tons of it — and […]Ryan Williamson
- DSC Weekly Digest 05 April 2021 2021-04-06 Kurt A Cagle
- A Plethora of Machine Learning Tricks, Recipes, and Statistical Models 2021-04-06Source: See article #5, in section 1 Part 2 of this short series focused on fundamental techniques, see here. In this Part 3, you will find several…Vincent Granville
- DataOps: Building an Efficient Data Ecosystem 2021-04-05Data is more present and more powerful…Olha Zhydik
- Deep Learning for Autonomous Driving: A Breakthrough in Urban Navigation 2021-04-05‘ Autonomous vehicle’ is a buzzword that’s been circulating in recent decades. However, the development of such a vehicle has posed a significant challenge for automotive manufacturers. This article describes how deep learning autonomous driving and…Olha Zhydik
- Reinforcement Learning for Dynamic Pricing 2021-04-05Limitations on physical interactions throughout the world have reshaped our lives and habits. And while the pandemic has been disrupting the majority of industries, …Olha Zhydik
- Forewarn: Business growth with current situation of AI in Construction Market 2021-04-05In these uncertain and unprecedented times due to the COVID-19 outbreak, more and more businesses are witnessing a slow-down in their operations. However, the construction market is continuing to be resilient in spite of the tremendous challenges brought about by COVID-19 pandemic. When it comes to construction sites, drive-thru strategies and…Abhishek Peter
- MES: The path to intelligent manufacturing 2021-04-05The adoption of technology in the Manufacturing Industry has been slow, but steady. Technology adoption has been relatively faster when it has helped improve productivity, boost quality, and reduce costs. Industry CXOs are convinced that IT has a major role to play in manufacturing, but less than 30 percent of manufacturers have adopted…Vinaksh
- Introduction to Probabilistic programming 2021-04-04Last week, I saw a nice presentation on Probabilistic Programming from a student in Iran (link below). I am interested in this subject for my teaching at the #universityofoxford. In this post, I provide a brief introduction to Probabilistic programming. Probabilistic programming is a programming paradigm designed to implement and solve…ajit jaokar
- Distributed Artificial Intelligence with InterSystems IRIS 2021-04-03What is Distributed Artificial Intelligence (DAI)? Attempts to find a “bullet-proof” definition have not produced result: it seems like the term is slightly “ahead of time”. Still, we can analyze semantically the term itself – deriving that distributed artificial intelligence is the same AI (see…Sergey Lukyanchikov
- Seeking Out the Future of Search 2021-04-03The future of search is the rise of intelligent data and documents. Way back in 1991, Tim Berners-Lee, then a young English software developer working at CERN in Geneva, Switzerland, came up with an intriguing way of combining a communication protocol for retrieving content (HTTP) with a descriptive language for embedding such links into documents […]Kurt A Cagle

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

- 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
- Harmonic morphisms from cubic graphs of order 8 to a graph of order 4 2020-06-08There are five simple cubic graphs of order 8 (listed here) and there are 6 connected graphs of order 4 (listed here). But before we get started, I have a conjecture. Let be a simple graph on n1 vertices, a simple graph on n2 vertices, and assume there is a harmonic morphism . Call an […]wdjoyner
- Duursma zeta function of a graph 2020-05-28I’m going to start off with two big caveats: This is not Duursma‘s definition, it’s mine. I’m not convinced (yet?) that it’s a useful idea to examine such a zeta function. So that’s your warning – you may be wasting your time reading this! The Duursma zeta function of a linear block (error-correcting) code is […]wdjoyner
- Harmonic morphisms to D_3 – examples 2020-05-01This post expands on a previous post and gives more examples of harmonic morphisms to the tree . This graph is also called a star graph on 3+1=4 vertices, or the bipartite graph . We indicate a harmonic morphism by a vertex coloring. An example of a harmonic morphism can be described in the plot […]wdjoyner
- NCF Boolean functions 2020-04-20I recently learned about a new class of seemingly complicated, but in fact very simple functions which are called by several names, but perhaps most commonly as NCF Boolean functions (NCF is an abbreviation for “nested canalyzing function,” a term used by mathematical biologists). These functions were independently introduced by theoretical computer scientists in the […]wdjoyner

What's all this, then?

- Voting power (7): Quarreling voters 2021-01-23In 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 2021-01-21As 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 2021-01-13We 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 2021-01-09Recently 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 2021-01-05Introduction 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 2021-01-02As 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 2020-12-30Naive 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: def banzhaf1(q,w): n = len(w) […]
- Voting power 2020-12-29After 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 2020-11-06Every 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 2020-01-17The 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 2020-01-14A 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) 2020-01-05The 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 […]
- The Butera-Pernici algorithm (1) 2020-01-03Introduction We know that there is no general sub-exponential algorithm for computing the permanent of a square matrix. But we may very reasonably ask -- might there be a faster, possibly even polynomial-time algorithm, for some specific classes of matrices? For example, a sparse matrix will have most terms of the permanent zero -- can […]
- The size of the universe 2020-01-01As a first blog post for 2020, I'm dusting off one from my previous blog, which I've edited only slightly. I've been looking up at the sky at night recently, and thinking about the sizes of things. Now it's all very well to say something is for example a million kilometres away; that's just a […]
- Permanents and Ryser's algorithm 2019-12-21As I discussed in my last blog post, the permanent of an \(n\times n\) matrix \(M=m_{ij}\) is defined as \[ \text{per}(M)=\sum_{\sigma\in S_n}\prod_{i=1}^nm_{i,\sigma(i)} \] where the sum is taken over all permutations of the \(n\) numbers \(1,2,\ldots,n\). It differs from the better known determinant in having no sign changes. For example: \[ \text{per} \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} =aei+afh+bfg+bdi+cdi+ceg. \] […]
- Speeds of Julia and Python 2019-12-18Introduction Python is of course one of the world's currently most popular languages, and there are plenty of statistics to show it. Of all languages in current use, Python is one of the oldest (in the very quick time-scale of programming languages) dating from 1990 - only C and its variants are older. However, it […]
- Poles of inaccessibility 2019-12-07Just recently there was a news item about a solo explorer being the first Australian to reach the Antarctic "Pole of Inaccessibility". Such a Pole is usually defined as that place on a continent that is furthest from the sea. The South Pole is about 1300km from the nearest open sea, and can be reached […]
- An interesting sum 2019-12-01I am not an analyst, so I find the sums of infinite series quite mysterious. For example, here are three. The first one is the value of \(\zeta(2)\), very well known, sometimes called the "Basel Problem" and first determined by (of course) Euler: \[ \sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}. \] Second, subtracting one from the denominator: \[ \sum_{n=2}^\infty\frac{1}{n^2-1}=\frac{3}{4} \] […]
- Runge's phenomenon in Geogebra 2019-09-14Runge's phenomenon says roughly that a polynomial through equally spaced points over an interval will wobble a lot near the ends. Runge demonstrated this by fitting polynomials through equally spaced point in the interval \([-1,1]\) on the function \[ \frac{1}{1+25x^2} \] and this function is now known as "Runge's function". It turns out that Geogebra […]
- Fitting the SIR model of disease to data in Python 2019-08-08Introduction and the problem The SIR model for spread of disease was first proposed in 1927 in a collection of three articles in the Proceedings of the Royal Society by Anderson Gray McKendrick and William Ogilvy Kermack; the resulting theory is known as Kermack–McKendrick theory; now considered a subclass of a more general theory known […]