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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

- You can now rewatch the 2021 CC Global Summit Panel Presentations! 2021-10-22Last week, we released the five keynotes from CC Summit 2021. This week, we’re back with seven panel presentations from the event. Our Summit panels this year featured thought leaders, pioneers and community members who shared their insights on emerging trends and innovations in Open. Whether you missed them live or want to replay them […]Ony Anukem
- A time of new beginnings and heartfelt farewells 2021-10-18I wanted to welcome and introduce some new team members to our broader community, as well as publicly thank those for their service who have recently moved on to new and exciting opportunities. We are delighted to announce our new Director of Communications, Dee Harris, has joined the CC team today. Dee has a wealth […]Catherine Stihler
- Join us at ‘The Future of Open’ Webinar on 9 November! 2021-10-15CC Policy Working Groups In May this year, we announced that four working groups of the Creative Commons Copyright Platform would examine policy issues affecting the open ecosystem. For the past five months, they have worked vigorously to provide clearer articulation of the issues and consider possible solutions in four areas: Working Group 1 — […]Brigitte Vézina
- The 2021 CC Global Summit Keynotes Are Here! 2021-10-14We have exciting news…we published the keynotes from the 2021 CC Global Summit! Alongside the 170+ sessions that took place at this year’s virtual event, we hosted five keynotes from global leaders in the open movement, who shared their work in open data, science and health, software and law. We’re excited to share these recordings […]Ony Anukem
- Reflections from the 2021 CC Global Summit 2021-10-141600+ attendees | 300+ presenters | 170+ sessions | 90+ countries The 2021 CC Global Summit (20-24 September) was our biggest and boldest event yet! The CC Global Summit is an annual highlight for the Creative Commons vast global network, this year being CC’s 20th anniversary and a milestone year for the Open movement more […]Ony Anukem
- License Stewardship consultation 2021-10-12The CC license suite was foundational to the creation of Creative Commons. To this day, the stewardship of CC licenses and public domain tools remains one of the core functions of the organization. At this milestone of the 20th anniversary of Creative Commons, we are embarking on a public consultation around CC’s stewardship. Specifically, we […]Sarah Hinchliff Pearson
- Introducing InclusiveAccess.org 2021-10-06The price of college textbooks in the United States (and many other countries) has skyrocketed over the past 30 years. Two-thirds of college students say that they’ve delayed buying a textbook because it was too expensive—even though 90% of those students worry that not having the textbook will affect their course grade. In addition, more […]Cable Green
- Creative Commons Announces New Board Members: Angela Oduor Lungati, Glenn Otis Brown & Jeni Tennison 2021-10-06On behalf of the entire Creative Commons Board of Directors, I am delighted to announce the appointment of three new members of the Board: Glenn Otis Brown, Angela Oduor Lungati, and Jeni Tennison. Glenn is one of CC’s original architects: he was our Executive Director from 2002-2005, and then served on the Board from 2009-2012. […]Molly Van Houweling
- Creative Commons’ Response to Wikimedia Foundation’s WIPO Observer Application Being Blocked 2021-10-06Yesterday, we were disappointed to learn that the Wikimedia Foundation’s application for observer status at the World Intellectual Property Organization (WIPO) was blocked due to opposition by China for the second time after its initial application in 2020. Creative Commons has been an accredited observer at WIPO since 2005. WIPO is the UN agency that […]Ony Anukem
- CC Global Summit to Promote Open Access and Better Sharing 2021-09-20More than a thousand educators, lawyers and tech experts from over 65 countries will come together this week for a global summit to promote open access and better sharing of research and resources. Creative Commons, the US-based global non-profit organization behind free copyright licenses, is hosting the event to develop solutions for the post-Covid world. […]Ony Anukem

Featured Blog Posts – Data Science Central

- A New Machine Learning Optimization Technique - Part I 2021-10-24In this series, we discuss a technique to find either the minima or the roots of a chaotic, unsmooth, or discrete function. A root-finding technique that works well for continuous, differentiable functions is successfully adapted and applied to piece-wise constant functions with an infinite number of discontinuities. It even works if the function has no […]Vincent Granville
- Understanding the Role of Project Managers 2021-10-22Most people think that a Project Manager actually can’t do anything and knows nothing. All this person does is manage other people and the project. In fact, a PM has a lot of specific knowledge in various fields in order to be able to manage the project as a whole. In this article,…Yuri Filatov
- DSC Weekly Digest 19 October 2021 2021-10-22 Kurt A Cagle
- How Big Data and Geolocation Solutions Can Help Your Business 2021-10-22The modern age market is decidedly lucrative for businesses and customers alike. However, things are a…Ryan Williamson
- RetailZoom Uses Sisense to Help Beer Lovers Find the Perfect Brew 2021-10-21Beer drinkers today are more spoiled for choice than ever before. From suburban supermarkets to big city bodegas, cooler cases are filled to bursting: IPAs, pilsners, stouts, lagers, goses, and countless other varieties and variations. …Matt Madden
- Big Data Analytics: How it Enables Digital Transformation 2021-10-19The digital way of life is no longer a distant dream but our current reality and it has served everyone well too. For businesses, especially, it has far-reaching consequences, enabling them to reach newer heights every day. But how does one achieve such digital transformation, you ask? With help from big data; you see, the […]Ryan Williamson
- Data-Driven Manufacturing: A Quick Guide 2021-10-19Manufacturing is a highly intricate process, especially in today’s day and age, where countless different factors must be considered before an efficient manufacturing process can be executed. Thankfully, this complexity can be easily addressed with help from technologies and technology-driven solutions, including data-driven manufacturing. It is…Ryan Williamson
- 5G IIOT to unleash Digital Transformation In Manufaturing 2021-10-195G IIOT Market: Introduction 5G is the fifth-generation wireless technology that brings three things to the table: Lower latency (Responsiveness), Wider channel (Speed), and more bandwidth (connect to IIOT devices). After the incredible achievement of 2G and 3G mobile networks and the rapid growth of 4G, innovative mobile networks…chaitali gawande
- MLflow Installation 2021-10-19In this article, we cover How to install MLflow. Before we dive into the process, let's begin with introducing MLOps MLOps By definition, MLOps is a cross-functional, collaborative, and continuous process that focuses on operationalizing data science use cases by managing statistical, machine learning models as reusable, highly…Prajwal LM
- Choosing Between MEAN Or MERN Stack for Web Development 2021-10-19Looking for a solid web development stack but aren't sure between MEAN and MERN? Don't worry; this article will help you decide. MEAN and MERN both have advantages and disadvantages that make them…Varun Bhagat
- The role of SmartNICs in 5G’s drive for new network architecture 2021-10-185G, cloud and edge computing, and IoT – any one of these monumental technology developments could be considered one of the biggest networking evolutions in human history. And yet we're experiencing all of them simultaneously. As a combined force, they are radically changing the way people…Jarrod J.S. Siket
- Training Deep Neural Networks using a low-code app in MATLAB 2021-10-18In this blog post I will show how to use a low-code app in MATLAB, the Deep Network Designer, for two different tasks and design paradigms: creating a network from scratch vs. using transfer learning. The process of building deep learning…OGE MARQUES
- The Future of Pandemic Modeling 2021-10-18A hundred years of modeling didn't help forecast the extent of the current pandemic. Hundreds of thousands of deaths could have been prevented with better models. The quantitative economic epidemiological model may be the future for pandemic modeling. Many countries, including the UK and US, mounted inefficacious…Stephanie Glen
- SAP BW Data Mining Analytics: Clustering Reporting (Part 4, final) 2021-10-18Summary Clustering analysis is another standard method available with SAP BW Data Mining. The clustering models based on this method may apply various combinations of parameters (e.g., maximum number of clusters, minimum fraction of inter-cluster hops per iteration, etc.) in order to implement various…Sergey Lukyanchikov
- Shopify Development Trends To Prioritize In 2021 2021-10-18Shopify has a 23% market share of the e-commerce landscape in 2021. The contrast that Shopify enjoys when compared to other platforms such as Magento or OpenCart is that it provides a website builder module. This module simplifies the entire development process to a great extent. The adoption is high and the benefits are evident. […]Amit Dogra
- Do algorithms introduce bias or do they promote transparency? 2021-10-17Background Every industry will be affected by algorithms and algorithmic bias. But we could rethink this as – Do algorithms introduce bias or do they introduce transparency? An interesting paper illustrates this point. Assessing Algorithmic Biases for Musical Version…ajit jaokar
- DSC Weekly Digest 12 October 2021 2021-10-15Announcements Build statistical and analytical expertise as well as the management and…Kurt A Cagle
- Equivalence class clustering and bottom-up lattice traversal (ECLAT) 2021-10-15This article has been excerpted from my book, Models and Algorithms for Unlabelled Data. Next time you visit a nearby grocery…Vaibhav Verdhan
- Healthcare Administrative Automation & Digitization for Better Management 2021-10-14Introduction Lack of technology or Healthcare Automation solutions has been creating havoc in the healthcare industry. According to Health Affairs, 44,000 deaths are caused in the U.S. each year due to clinical errors. While every 6 out of 10 deaths are caused due to the lack of technological…Swarali garud
- Object Detection Technology - How It Works And Where Is It Used? 2021-10-14 John Levis

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?

- 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: \[ \sum_{n=2}^\infty\frac{1}{n^2-1}=\frac{3}{4} \] […]