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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

- SDG Academy & Creative Commons 2021-06-22+ By Cable Green: Director of Open Education, Creative Commons, and Chandrika Bahadur: Director, SDG Academy Open access to knowledge has never been more important than it is today. The promise of connectivity and the democratization of knowledge has made it possible for anyone, anywhere, to learn. In an increasingly connected and complex world, society […]Cable Green
- CC Copyright Platform Members Share the Stories of Their Projects 2021-06-14Last year, six projects were carried out thanks to funding made available to the Creative Commons Copyright Platform members to drive policy issues affecting the open movement. In this blog post, we’re glad to share the engaging, inspiring articles the project lead wrote on CC’s Medium publication We Like to Share. Preparing Bulgarian GLAMs for […]Brigitte Vézina
- We’re Launching the CC Open GLAM Program 2021-06-10Following our recent announcement of a major grant from Arcadia to advance open access at galleries, libraries, archives, and museums (GLAMs), we’re excited to officially launch Creative Commons’ Open GLAM program. In this post, we share an overview of the program’s rationale and briefly introduce our key program areas. ——— GLAMs’ public interest mission, rooted […]Brigitte Vézina
- Creative Commons Receives $5M Grant from Arcadia to Advance Open Access at Galleries, Libraries, Archives, and Museums 2021-06-03Today Creative Commons (CC) announced that it has received a five-year $5 million grant from Arcadia, a charitable fund of Lisbet Rausing and Peter Baldwin, to advance open access in the Galleries, Libraries, Archives, and Museums (GLAM) sector. The generous grant will enable CC to develop a robust Open GLAM Program that builds upon the […]Eric Steuer
- The 2021 CC Global Summit Call for Proposals Is Now Open 2021-06-02The 2021 CC Global Summit will be going virtual again this year – join us 20-24 September, 2021 for an extra special event – Creative Commons is turning 20! We’ll be building off of the successes from our 2020 Summit to bring you another week filled with discussion and debate, workshops and planning, talks and […]Alison Pearce
- Four Creative Commons Working Groups Will Explore Policy Issues. Meet Their Leads! 2021-05-25Throughout 2021, four working groups of the Creative Commons Copyright Platform are undertaking an exploration of policy issues affecting the open ecosystem, in line with the Creative Commons 2021-2026 strategy. In this blog post, we present the four working groups and introduce you to their leads. Generally, the Platform is a space for copyright and […]Brigitte Vézina
- We’re Turning 20! What’s Happened Since 2001? 2021-05-24Creative Commons is turning 20! We are celebrating with a special Better Sharing campaign, honoring 20 years of commitment to open access and better sharing. We invite you to join us. We have an ambitious goal to raise over $15 million in support. When Creative Commons was founded in 2001, the internet was a budding universe with high potential, […]Eric Steuer
- Announcing Our 20th Anniversary “Better Sharing” Campaign 2021-05-24Creative Commons is turning 20! We are delighted to be celebrating this milestone with our global community, honoring our commitment as a nonprofit to creating a world where everyone has access to knowledge and creativity. As we considered CC’s goals for the next 20 years, we kept returning to a simple idea: Better Sharing for […]Catherine Stihler
- The Future of Museums Is Open 2021-05-18It’s International Museum Day and at Creative Commons (CC) we are thrilled to once again celebrate the institutions that acquire, conserve, research, communicate, and exhibit the world’s heritage for education, study, and enjoyment. This year’s theme is The Future of Museums: Recover and Reimagine and in this blog post, we imagine a future where museums […]Brigitte Vézina
- Creative Commons Calls on the EU to Show Clear Support for Waiving COVID Vaccine Patents 2021-05-06The news yesterday from US trade Ambassador Katherine Tait that the Biden-Harris administration supports waiving IP protections for COVID vaccines is not just welcome, it is laying a stake in the ground for others to follow. The hesitancy of both the EU and UK to support the US places them on the wrong side of […]Catherine Stihler

Featured Blog Posts – Data Science Central

- The Most Common Problems and Solutions with Your CRM Data 2021-06-25Source: istockphoto Your Customer Relationship Management (CRM) software is the mother source you leverage for effective client communication. It is also the key to achieving personalized sales targeting and…Rochelle Williams
- Face Detection Explained: State-of-the-Art Methods and Best Tools 2021-06-24So many of us have used different Facebook applications to see us aging, turned into rock stars, or applied festive make-up. Such waves of facial transformations are usually accompanied by warnings not to share images of your faces – otherwise, they will be processed and misused.…Oleksandr Tyron
- How to Use Data Science for Search Engine Optimization 2021-06-23Data science is one of the hottest topics in the market nowadays. It is one of those industries that has revolutionized the world. It associates two chief technologies, big data and artificial intelligence, and utilizes them to examine and process datasets. It also uses machine learning, which helps to strengthen artificial intelligence. Data science has […]Saajan Sharma
- Ways Artificial Intelligence Impacts the Banking Sector 2021-06-23"75 percent of respondents at banks with over $100 billion in assets say they're currently implementing AI strategies, compared with 46 percent at banks with less than $100 billion in assets," UBS Evidence Lab reports. Artificial Intelligence (AI) has become an integral part of the most demanding and fast-paced…Aileen Scott
- DSC Weekly Digest 21 June 2021 2021-06-22 Kurt A Cagle
- Top Eleven Skills to Look Out For While Hiring DevOPS Engineer 2021-06-22Companies that incorporate DevOps practices get more done. It is as simple as that. The technical benefits include continuous delivery, easier management, easier to manage, and faster problem-solving. In addition to this, there are cultural benefits like more productive teams, better employee engagement, and better development…Amit Dogra
- Good source of coding puzzles for programming interviews 2021-06-20Here is a paper which gives a set of coding puzzles which could be useful for technical interviews in data science. The paper introduces a new type of programming challenge called programming puzzles, as an objective and comprehensive evaluation of program synthesis, and release an open-source dataset of Python Programming Puzzles…ajit jaokar
- The Lost Art of Decile Analysis 2021-06-20Image Source: Author “Logistic Regression is not Regression but a Classification Algorithm”.…Ridhima Kumar
- More Fun Math Problems for Machine Learning Practitioners 2021-06-19This is part of a series featuring the following aspects of machine learning: Mathematics, simulations, benchmarking algorithms based on synthetic data (in short,…Vincent Granville
- In a Cloud-Native World, It’s Time to Rethink Data Storage 2021-06-18Digital transformation has created new product and service capabilities and untold additional yottabytes of data. It has become increasingly clear that data is a key creator of value. Take, for example, the realm of digital entertainment. For proof, just scan your monthly credit…Paul Speciale
- Applying Regression-based Machine Learning to Web Scraping 2021-06-18Whenever we begin dealing with machine learning, we often turn to the simpler classification models. In fact, people outside of this sphere have mostly seen those models at work. After all, image recognition has become the poster child of machine learning.…Aleksandras Sulzenko
- The Role of Big Data in Banking 2021-06-18How do modern banks use Big Data? Recently, we have been hearing about Big Data more and more often. In today's digital world, this technology is being actively used in the financial industry as…Artsiom Balabanau
- Clickless Analytics is the Future of Business User Analytics 2021-06-18If your business is trying to incorporate data analytics into the fabric of day-to-day work, you will need to get your users to adopt analytical tools. The way forward is not all that complicated. The solution you choose must take an augmented analytics approach, one that includes simple search analytics, ala Google search. Natural Language […]Kartik Patel
- Angular vs React: Which is Best for your Business? 2021-06-18When it comes to choosing the right JavaScript framework for developing an exceptional web application, developers have many options. These include Angular, React, Vue, etc. However, it is quite difficult for developers to decide as each of these frameworks has its pros and…OLIVIA CUTHBERT
- Why was Power BI considered the best BI tool? 2021-06-17Power BI was chosen by Gartner as the best BI tool in the world. This has been happening for the twelfth consecutive year, which reinforces the platform's power. When it comes to Business Intelligence and…Cleverson Alexandrini Falciano
- Artificial Intelligence to Take User Experience to the Next Level 2021-06-17Just imagine that you walked into a restaurant and are welcomed by the hotel staff. Later, one of the waiters directs you to a convenient seat, tells you their special dishes, and helps you order food by understanding your preference. He makes sure that you are attended well, serves you good food, and asks your […]Kedar Supekar
- How a good data visualization could save lives 2021-06-17Ignaz Philipp Semmelweis (1818-1865) was a Hungarian physician and scientist, now known as an early pioneer of antiseptic procedures. Described as the "saviour of mothers", at 1846 Semmelweis discovered that the incidence of puerperal fever could be drastically cut by the use of hand disinfection in obstetrical clinics (this fever was…Lana S
- How COVID-19 Is Accelerating Smart Home Automation 2021-06-17Modern technologies have tapped varied sectors and the residential sector is the most prominent one. The growing influence of technologies like the Internet-of-Things (IoT) and others across a large number of applications at home have increased the demand and popularity of home automation to a considerable extent. The rising popularity of these systems will…chaitali gawande
- Why Open Data is Not Enough 2021-06-16Periods of crisis create a greater need for transparency. In the age of Open Data, this observation is more true now that everyone can access massive amounts of data to help make better decisions. We all would like to know more about…Charles Miglietti
- DSC Weekly Digest 14 June 2021 2021-06-16 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

- 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
- 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

What's all this, then?

- High precision quadrature with Clenshaw-Curtis 2021-04-20An 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 2021-04-09Note: 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 2021-01-23In 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: \[ \sum_{n=2}^\infty\frac{1}{n^2-1}=\frac{3}{4} \] […]