Math

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  • Magic tricks created using artificial intelligence for the first time

    Mathematics News -- ScienceDaily
    16 Nov 2014 | 6:10 pm
    Researchers working on artificial intelligence have taught a computer to create magic tricks. The researchers gave a computer program the outline of how a magic jigsaw puzzle and a mind reading card trick work, as well the results of experiments into how humans understand magic tricks, and the system created completely new variants on those tricks which can be delivered by a magician.
  • Origin of Mysterious Portuguese Mathematical and Geographical Tiles Revealed

    Scientific American - Math
    5 Nov 2014 | 9:15 am
    A few months ago I wrote about some mystifying mathematical and geographic tiles I encountered at the National Tile Museum in Lisbon, Portugal. -- Read more on ScientificAmerican.com
  • When vaccines are imperfect: What math can tell us about their effects on disease propagation

    Mathematics News -- ScienceDaily
    20 Nov 2014 | 3:36 pm
    The control of certain childhood diseases is difficult, despite high vaccination coverage in many countries. One of the possible reasons for this is 'imperfect vaccines,' that is, vaccines that fail either due to 'leakiness,' lack of effectiveness on certain individuals in a population, or shorter duration of potency. In a new article, authors use a mathematical model to determine the consequences of vaccine failure and resulting disease dynamics.
  • How Scientific Are Your Instruments?

    Scientific American - Math
    18 Nov 2014 | 6:36 am
    What do coins, a Wii remote, or card games have to do with science? More than you might think. Scientific instruments are devices specifically designed to measure the subject of your research... -- Read more on ScientificAmerican.com
  • MATLAB R2014b Graphics – Part 2: Using Graphics Objects

    Loren on the Art of MATLAB
    Loren Shure
    14 Oct 2014 | 7:55 am
    Today, David Garrison, our guest blogger, will continue his series on the new graphics system in R2014b.Part 1: Features of the New Graphics SystemPart 2: Using Graphics ObjectsPart 3: Compatibility Considerations in the New Graphics SystemHere is Part 2 of the series.ContentsWhat have we learned so far?The MATLAB Graphics SystemPre-R2014b Numeric HandlesR2014b Graphics ObjectsGetting and Setting Object PropertiesUsing set and getHave you starting using graphics objects in R2014b?Next up -- Part 3: Compatibility Considerations in the New Graphics SystemWhat have we learned so far?In Part 1 of…
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    MATH - Google News

  • Dayton students put math skills to the test - WDTN

    22 Nov 2014 | 7:54 pm
    Dayton students put math skills to the testWDTN“This is important because research shows when kids are engaged in competitive mathematics, they're thinking increases and they're excited about math,” said Kiara Williams with Dayton Public Schools. “So we take this very seriously. Ultimately we want and more »
  • TN chooses new test to replace TCAPs in English and math - WBIR-TV

    22 Nov 2014 | 2:08 pm
    TN chooses new test to replace TCAPs in English and mathWBIR-TVAccording to the state's website, those options include multiple choice questions, writing assessments that require students to cite text evidence at all grade levels, questions that test math abilities without a calculator, and math problems that
  • PBS launches math series for kids in 'Odd Squad' - Fresno Bee

    22 Nov 2014 | 12:06 pm
    Fresno BeePBS launches math series for kids in 'Odd Squad'Fresno BeeConsider this math problem: PBS leaves the train station headed west under a full head of steam to find a new series to teach math to youngsters. Tim McKeon and Adam Peltzman leave a train station at top speed headed east with an idea for a show that ...PBS goes into action with math-based 'Odd Squad'Pittsburgh Post Gazetteall 3 news articles »
  • 'Survivor Math' gets kids jazzed about algebra - Wausau Daily Herald

    22 Nov 2014 | 9:46 am
    'Survivor Math' gets kids jazzed about algebraWausau Daily HeraldThe event was inspired by the "Propellerheads" Immunity Challenge, from the 18th season of the television show "Survivor," in which contestants must memorize a series of math symbols and then solve an equation. "Students watched the episode and ...
  • MORENO VALLEY: League, events promote math as fun - Press-Enterprise

    22 Nov 2014 | 12:06 am
    Press-EnterpriseMORENO VALLEY: League, events promote math as funPress-EnterpriseThe event was part of the Moreno Valley Math League, a competition program for sixth-, seventh- and eighth-graders now in its second year in the Moreno Valley Unified School District. Earlier this year, the California Teachers Association Institute for ...
 
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    Search for "math OR mathematics"

  • Changes coming to Common Core Standards

    22 Nov 2014 | 9:34 pm
    The state Board of Education has posted revisions to the Common Core Georgia Performance Standards for the public to review over a 60-day period. The revisions were made following an executive order from Gov. Nathan Deal for a formal evaluation.
  • Engineering careers: Webelos get hands-on look at STEM fields

    22 Nov 2014 | 9:20 pm
    Two dozen Webelos from Packs 78, 79 and 129 from Corning and Painted Post participated in a variety of activities involving science, technology, engineering and math while working on their engineering activity badge, one of 20 activities the Webelos must complete before moving on to become Boy Scouts. Saturday's event included hands-on activities involving electricity, building a catapult out of orange juice cartons, pencils and plastic spoons, designing their dream house and using engineering principles to build model bridges.
  • In NEPA, boy from Afghanistan finds hope, opportunity

    22 Nov 2014 | 9:18 pm
    In the last 13 months, doctors moved the 9-year-old's bladder from outside his body to its proper place. Instead of making bricks back home, he attends first grade at Clarks Summit Elementary School - and dreams of becoming a doctor and returning to Afghanistan to help children, who like him, had little hope.
  • Mackey: Some postgame thoughts from Nassau

    22 Nov 2014 | 9:17 pm
    And maybe that's what most will do after the Penguins managed one point out of a home-and-home series with a Metropolitan Division rival. Those inside the locker room, though, won't be among the group going nutty after Saturday's 4-1 loss to the New York Islanders at Nassau Coliseum.
  • Crow's-Feet Chronicles: School is a stomach ache

    22 Nov 2014 | 9:15 pm
    The ignorance gap that the new math created between parent and child has not even begun to mend. Before the new math, I had a mysterious aura about me.
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    Mathematics News -- ScienceDaily

  • Theater arts research offers insight for designers, builders of social robots

    21 Nov 2014 | 11:12 am
    Researchers have provided insight into human behavior for scientists, engineers who design and build social robots.
  • When vaccines are imperfect: What math can tell us about their effects on disease propagation

    20 Nov 2014 | 3:36 pm
    The control of certain childhood diseases is difficult, despite high vaccination coverage in many countries. One of the possible reasons for this is 'imperfect vaccines,' that is, vaccines that fail either due to 'leakiness,' lack of effectiveness on certain individuals in a population, or shorter duration of potency. In a new article, authors use a mathematical model to determine the consequences of vaccine failure and resulting disease dynamics.
  • Gifted men and women define success differently, 40-year study finds

    19 Nov 2014 | 7:17 am
    Researchers spent four decades studying a group of mathematically talented adolescents, finding that by mid-life they were extraordinarily accomplished and enjoyed a high level of life satisfaction. Gender, however, played a significant role in how they pursued—and defined—career, family and success. Intellectually gifted women tracked for 40 years were found to earn less money, be less present in STEM fields, and work fewer hours than their male counterparts. Despite that, they expressed a high level of personal satisfaction and sense of achievement, defining success more broadly than…
  • Magic tricks created using artificial intelligence for the first time

    16 Nov 2014 | 6:10 pm
    Researchers working on artificial intelligence have taught a computer to create magic tricks. The researchers gave a computer program the outline of how a magic jigsaw puzzle and a mind reading card trick work, as well the results of experiments into how humans understand magic tricks, and the system created completely new variants on those tricks which can be delivered by a magician.
  • Stock market models help researchers predict animal behavior

    12 Nov 2014 | 9:01 am
    Modeling used to forecast fluctuations in the stock market has been discovered to predict aspects of animal behavior. The movement of zebrafish when mapped is very similar to the stochastic jump process, a mathematical model used by financial engineers. The model could improve the effectiveness of experiments, minimize the number of fish used, and allow researchers to make better use of their data following experiments.
 
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    Scientific American - Math

  • How Scientific Are Your Instruments?

    18 Nov 2014 | 6:36 am
    What do coins, a Wii remote, or card games have to do with science? More than you might think. Scientific instruments are devices specifically designed to measure the subject of your research... -- Read more on ScientificAmerican.com
  • A Proof of the Math Fact of Rolle in Short Words

    17 Nov 2014 | 8:40 pm
    This proof of the math fact of Rolle, I wrote it down; here was my goal: Use just words with one part. (So it won’t sound too smart.) Please tell me if you find a hole. -- Read more on ScientificAmerican.com
  • It is Never too Early to Think – and Communicate – Like a Scientist

    14 Nov 2014 | 8:21 am
    When you think about the job of a scientist, what images come into your mind? A chemist wearing a lab coat surrounded by beakers? A geologist out studying rocks in the desert? -- Read more on ScientificAmerican.com
  • Physics Week in Review: November 8, 2014

    8 Nov 2014 | 2:09 am
    It was a big week for physics in the movies, with the premiere of Interstellar, and the Stephen Hawking biopic The Theory of Everything. That translates into lots of pixels commenting on the science... -- Read more on ScientificAmerican.com
  • Origin of Mysterious Portuguese Mathematical and Geographical Tiles Revealed

    5 Nov 2014 | 9:15 am
    A few months ago I wrote about some mystifying mathematical and geographic tiles I encountered at the National Tile Museum in Lisbon, Portugal. -- Read more on ScientificAmerican.com
 
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    Ars Mathematica

  • Learning about Stochastic Processes the Almost Sure Way

    Walt
    9 Nov 2014 | 1:21 pm
    George 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 in a form sufficient for applications. Most of the formal presentations of the subject seem very far away from the informal ones, to an unusual extent. For example, for the important technical notion of semimartingale the Wikipedia definition is the usual one, which has a very different flavor from the naive picture useful in…
  • Arguesian Lattices

    Walt
    23 Sep 2014 | 12:46 pm
    As 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 is a lattice-theoretic analogue of Desargues’ theorem in projective geometry. I read the statement of the theorem several times and I have no intuition about what it means. There is a kind of converse to this result: a complemented lattice can be embedded into the lattice of submodules of a module if and only if it is arguesian. (I found the result in…
  • K2, not the mountain

    Walt
    20 Mar 2014 | 2:18 pm
    Chandan 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 trailer to tell you if you want to invest the time in the movie.
  • Cayley Bacharach Theorem through History

    Walt
    10 Feb 2014 | 3:04 pm
    I 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 techniques is fascinating.
  • Nonassociative Algebras

    Walt
    30 Dec 2013 | 2:00 pm
    I 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 — the octonions, the five exceptional Lie algebras, the exceptional Jordan algebra — that I would like to understand better. John Baez has an article about the direct connection that I post about before, but what I don’t understand about the general theory is how…
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    Loren on the Art of MATLAB

  • MATLAB R2014b Graphics – Part 3: Compatibility Considerations in the New Graphics System

    Loren Shure
    5 Nov 2014 | 11:48 am
    Today, David Garrison, our guest blogger, will continue his series on the new graphics system in R2014b.Part 1: Features of the New Graphics SystemPart 2: Using Graphics ObjectsPart 3: Compatibility Considerations in the New Graphics SystemHere is Part 3 of the series.ContentsWhat have we learned so far?Section 1: Compatibility Considerations for R2014b graphicsWhy did we make these changes?How do the changes affect me?How do I find out more about the changes in R2014b graphics?Section 2: Visual DifferencesThe New Default ColormapLine Colors in PlotsPlot Titles and LabelsSection 3: Changes…
  • Using MATLAB to Detect Cookies

    Loren Shure
    23 Oct 2014 | 2:41 am
    Today, I'd like to introduce a guest blogger, Tom Lowell, who is a program manager here at MathWorks. He works in our hardware connectivity group, and plays with Arduinos and Raspberry Pis in his spare time. He's relatively new to both MathWorks and MATLAB and decided to write a cookie detector as his first MATLAB project.ContentsMotivationSetupCodeTuningThe OutputNotificationDo You Have a Project that Uses an IP camera?MotivationWhen I joined MathWorks, one of the first traditions I learned was that a large plate of home made cookies is delivered every Friday to every floor of every building…
  • Taking the Pulse of MOOCs

    Loren Shure
    16 Oct 2014 | 5:45 am
    Coursera is a technology platform that kickstarted the current MOOCs boom. Even though there are more MOOCs players now, it still remains one of the leading companies in this space. But how are they doing these days for delivering higher education to the masses online?Today's guest blogger, Toshi Takeuchi, would like to share an analysis using Courera's data.I am a big fan of MOOCs and I benefited a lot from free online courses on Coursera, such as Stanford's Machine Learning course. Like many websites these days, Coursera offers its data through REST APIs. Coursera offers a number of APIs,…
  • MATLAB R2014b Graphics – Part 2: Using Graphics Objects

    Loren Shure
    14 Oct 2014 | 7:55 am
    Today, David Garrison, our guest blogger, will continue his series on the new graphics system in R2014b.Part 1: Features of the New Graphics SystemPart 2: Using Graphics ObjectsPart 3: Compatibility Considerations in the New Graphics SystemHere is Part 2 of the series.ContentsWhat have we learned so far?The MATLAB Graphics SystemPre-R2014b Numeric HandlesR2014b Graphics ObjectsGetting and Setting Object PropertiesUsing set and getHave you starting using graphics objects in R2014b?Next up -- Part 3: Compatibility Considerations in the New Graphics SystemWhat have we learned so far?In Part 1 of…
  • Reversal of Sorts – New in Release R2014b

    Loren Shure
    9 Oct 2014 | 5:36 am
    I wanted to show you a glimpse of some of the new math functionality available in R2014b.ContentsThe QuestionMy Original AnswerSolution with R2014bWhat New Math Have You Enjoyed in R2014b?The QuestionRecently on the MATLAB newsgroup, Christoph asked this question:I have a vector A shown below, which has 6 elements. the elements are already sorted in descending order. now i want to create vector C by deleting elements from A, starting with element a1, until the sum of the vector equals or is smaller the value BA= 26 23 20 19 15 14 B=70So, the output should beC= 20 19 15 14Any idea how to do…
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    Homeschool Math Blog

  • Man who had half of his brain removed

    21 Nov 2014 | 4:49 am
    I received this story in an email and thought it was very encouraging and heart-warming! True story. I asked Scott for his permission to post it online.In the beginning he is referencing this article of mine (I sent it to my list last year):http://www.mathmammoth.com/lessons/value_of_mistakes.php Enjoy... and SHARE. :)Hello Maria, I am just NOW ... today reading this email from about one year ago. I know terrible, but late than never, right. Anyway, I wanted to write you and tell you that like Cameron, I had more than half of my brain removed in 1978. I had brain surgery for an AVM, arterial…
  • Will there be Math Mammoth Algebra 1?

    18 Nov 2014 | 6:52 am
    I've had several people ask me if I'm planning to write algebra 1, algebra 2 and other high school math courses.The answer is no. I'm planning to stop at pre-algebra. The main reason is, I've seen some fantastic high school math books other people have written -- people with lots of teaching experience etc. -- and I feel I could not surpass what those people have done. Especially when I don't have lots of experience teaching high school math to students.You see, with elementary math, there exist numerous homeschool math curricula. I've seen most of them and I can tell many of them  are…
  • Math Mammoth Thanksgiving sale

    15 Nov 2014 | 12:31 pm
    My traditional Thanksgiving sale has started: you can get 25% off of all Math Mammoth & Make It Real Learning downloads & CDs at Kagi.Use the coupon code THANKSGIVING. The sale starts TODAY and runs till Monday, December 1, 2014. You can go to www.MathMammoth.com first, then find the links to Kagi's order pages there.Or, if you know what you want to get, you can use these direct links: Light Blue series (complete curriculum)Blue seriesGolden and Green SeriesMake It Real Learning activity workbooksBundles (CDs or downloads)Review workbooksLight Blue Series, South African versionThanks!
  • New pre-algebra topic books for Math Mammoth Blue Series

    15 Nov 2014 | 6:00 am
    Three new books in the Blue Series! These books deal with some pre-algebra level topics.Check out their free samples! Math Mammoth Expressions & EquationsThis is a worktext covering the order of operations, equations, expressions, and simplifying expressions in several different ways in 6th-7th grade level.The main principles are explained and practiced both with visual models and in abstract form, and the lessons contain varying practice problems that approach the concepts from various angles. We also touch on inequalities and graphing on a very introductory level. In order to make the…
  • Group buy for Math Mammoth bundles - only TODAY!

    3 Nov 2014 | 6:05 am
     Homeschool Buyers Co-op is running a GROUP BUY for large Math Mammoth bundles at 40% discount – but just for today, November 3, 2014.So hurry! It's only for ONE day!
 
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    Let's Play Math!

  • Roadmap to Mathematics: 2nd Grade

    Denise Gaskins
    21 Nov 2014 | 11:30 am
    [Feature photo (above) by Loren Kerns. (CC BY 2.0 via Flickr)] A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?” The Council of the Great City Schools has published a handy 6-page pdf summary of second grade math concepts, with suggestions for how parents can support their children’s learning: Parent Roadmap- Supporting Your Child in Grade Two Mathematics Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help your…
  • Roadmap to Mathematics: 1st Grade

    Denise Gaskins
    18 Nov 2014 | 6:32 am
    [Feature photo (above) by woodleywonderworks. (CC BY 2.0 via Flickr)] A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?” The Council of the Great City Schools has published a handy 6-page pdf summary of first grade math concepts, with suggestions for how parents can support their children’s learning: Parent Roadmap- Supporting Your Child in Grade One Mathematics Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help…
  • Education Bloggers: Share Your Post!

    Denise Gaskins
    17 Nov 2014 | 6:48 am
    photo by Omar Omar via flickr If you are a homeschooler or classroom teacher, student or independent learner, or anyone else who writes about math, now is the time to send in your favorite blog post for next week’s Math Teachers at Play (MTaP) math education blog carnival. Click here to submit your blog post. Browse all the past editions of the Math Teachers at Play blog carnival Don’t procrastinate: The deadline for entries is this Friday. The carnival will be posted next week at Triumphant Learning. Click to tweet about the carnival: (No spam, I promise! You will have a chance…
  • Roadmap to Mathematics: Kindergarten

    Denise Gaskins
    13 Nov 2014 | 6:57 am
    [Feature photo (above) by MIKI Yoshihito. (CC BY 2.0 via Flickr)] A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?” The Council of the Great City Schools has published a handy 6-page pdf summary of kindergarten math concepts, with suggestions for how parents can support their children’s learning: Parent Roadmap- Supporting Your Child in Kindergarten Mathematics Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help…
  • Math Teachers at Play #79

    Denise Gaskins
    22 Oct 2014 | 7:00 am
    [Feature photo above by Jimmie, and “79″ image (right) by Steve Bowbrick via flickr (CC BY 2.0).] Do you enjoy math? I hope so! If not, browsing this post just may change your mind. Welcome to the 79th edition of the Math Teachers At Play (MTaP) math education blog carnival — a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college. Let the mathematical fun begin! By tradition, we start the carnival with a puzzle, game, or trivia tidbits. If you would like to jump…
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    Computational Complexity

  • A November to Remember

    20 Nov 2014 | 5:25 am
    The Imitation Game starring Benedict Cumberbatch as Alan Turing opens in the US on November 28. If you read this blog you should see that movie. If one challenged British scientist biograph is not enough for you, The Theory of Everything with Eddie Redmayne as Stephen Hawking opened earlier this month. Also this month has the physics-laden Interstellar and the nerd-hero robot adventure Big Hero 6. Science rules the box office, though likely to be clobbered by a Mockingjay. Speaking of Turing, The ACM Turing Award will now come with a $1 million dollar prize, up from $250K, now…
  • A Fly on the wall for a Harvard Faculty meeting: Not interesting for Gossip but interesting for a more important reason

    17 Nov 2014 | 1:51 pm
    I was recently in Boston for Mikefest (which Lance talked about  here)  and found time to talk to  my adviser Harry Lewis at Harvard (adviser? Gee, I already finished my PhD. Former-Advisor? that doesn't quite sound right. I'll stick with Adviser, kind of like when they refer to Romney as Governor Romney, or Palin as half-governor Palin). He also invited me to goto a Harvard Faculty meeting. NO, I didn't see anything worth gossiping about. NO I am not going to quote Kissinger ``Academic battles are so fierce because the stakes were so low'' NO I am not going to say that under…
  • From Homework Solution to Research Paper

    13 Nov 2014 | 8:22 am
    Inspired by the Dantzig Story  I occasionally put an open problem on a class assignment. Never worked, though I did have a student get a research paper from solving a homework question the hard way. Teaching in the early 90's, I showed Valiant's proof that computing the permanent of a 0-1 matrix was #P-complete, including showing that the 0-1 permanent was in #P, the class of functions representable as the number of accepting paths of a nondeterministic polynomial-time Turing machine. I gave a homework assignment to show that the permanent of a matrix with non-negative integer…
  • Non controversial thoughts on rankings

    11 Nov 2014 | 7:49 am
    US News has a ranking of CS depts and various subcategories. Recently MohammadTaghi Hajiaghay and Luca Trevisan have suggested alternative rankings here (Moh) and here (Luca). These rankings inspire me to record some thoughts about rankings. When making a ranking one must ask: What is it for? For Academic depts it may be to help guide potential ugrads or grads in their choice of where to go. Rankings of the most influential people of all time (Michael Harts book here), or in a given year (Time magazine does this) are made to (I think) organize our thoughts and start debates. Michael Hart…
  • George Dantzig >= 100

    8 Nov 2014 | 5:34 am
    We celebrate the 100th anniversary of the birth of George Dantzig today. In his obituary post we talked about his work on optimization, particularly the simplex method for solving linear programs. For this centenary let's recall the urban legend of the famous mathematician (I heard it as John von Neumann) who as a student wasn't paying close attention in class. The professor wrote down four of the major open problems in the field. von Neumann wrote them down thinking they were homework problems. The next day he went back to the professor, ashamed that he could only solve two of them.
 
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    Mathematics and Computation

  • A HoTT PhD position in Ljubljana

    Andrej Bauer
    22 Nov 2014 | 4:16 am
    I am looking for a PhD student in mathematics. Full tuition & stipend will be provided for a period of three years, which is also the official length of the programme. The topic of research is somewhat flexible and varies from constructive models of homotopy type theory to development of a programming language for a proof assistant based on dependent type theory, see the short summary of the Effmath project for a more detailed description. The candidate should have as many of the following desiderata as possible, and at the very least a master’s degree (or an equivalent one): a…
  • TEDx “Zeroes”

    Andrej Bauer
    16 Oct 2014 | 12:01 am
    I spoke at TEDx University of Ljubljana. The topic was how programming influences various aspects of life. I showed the audence how a bit of simple programming can reveal the beauty of mathematics. Taking John Baez’s The Bauty of Roots as an inspiration, I drew a very large image (20000 by 17500 pixels) of all roots of all polynomials of degree at most 26 whose coefficients are $-1$ or $1$. That’s 268.435.452 polynomials and 6.979.321.752 roots. It is two degrees more than Sam Derbyshire’s image,  so consider the race to be on! Who can give me 30 degrees? The code…
  • Reductions in computability theory from a constructive point of view

    Andrej Bauer
    19 Jul 2014 | 5:50 am
    Here are the slides from my Logic Coloquium 2014 talk in Vienna. This is joint work with Kazuto Yoshimura from Japan Advanced Institute for Science and Technology. Abstract: In constructive mathematics we often consider implications between non-constructive reasoning principles. For instance, it is well known that the Limited principle of omniscience implies that equality of real numbers is decidable. Most such reductions proceed by reducing an instance of the consequent to an instance of the antecedent. We may therefore define a notion of instance reducibility, which turns out to have a…
  • Seemingly impossible constructive proofs

    Martin Escardo
    8 May 2014 | 7:15 am
    In the post Seemingly impossible functional programs, I wrote increasingly efficient Haskell programs to realize the mathematical statement $\forall p : X \to 2. (\exists x:X.p(x)=0) \vee (\forall x:X.p(x)=1)$ for $X=2^\mathbb{N}$, the Cantor set of infinite binary sequences, where $2$ is the set of binary digits. Then in the post A Haskell monad for infinite search in finite time I looked at ways of systematically constructing such sets $X$ with corresponding Haskell realizers of the above omniscience principle. In this post I give examples of infinite sets $X$ and corresponding constructive…
  • Brazilian type checking

    Andrej Bauer
    6 May 2014 | 2:59 am
    I just gave a talk at “Semantics of proofs and certified mathematics”. I spoke about a new proof checker Chris Stone and I are working on. The interesting feature is that it has both kinds of equality, the “paths” and the “strict” ones. It is based on a homotopy type system proposed by Vladimir Voevodsky. The slides contain talk notes and explain why it is “Brazilian”. Download slides: brazilian-type-checking.pdf GitHub repository: https://github.com/andrejbauer/tt Abstract: Proof assistants verify that inputs are correct up to judgmental…
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    Natural Blogarithms

  • Get “Things” for Free (An Excellent To-Do list app)

    SplineGuy
    21 Nov 2014 | 7:42 am
    Before I moved all my action items to Trello, the “Things” app was my favorite To-Do list for getting things done (‪#‎GTD‬). It’s free this week if you want to try it out. I think they have separate apps for iPhone and iPad, so if you have both you might go ahead and get it now to try out later. Remember, if you get a free app, it’s yours for good, even if they raise the price again later. So you could get it this week, then delete, but re-install it anytime in the future for free. Give thanks to Apple for offering Things as its free App of the Week on the App…
  • Blackboard Grader App for iOS

    SplineGuy
    7 Nov 2014 | 2:28 pm
    [Cross-posted on the E-Learning Pioneer – Blog of the Virtual Campus] There is finally an app for Blackboard that has been designed with the online instructors in mind. The Blackboard Grader App is now available and gives instructors an option for reviewing, providing feedback, and grading student submissions to Blackboard Learn Assignments. With the new Bb Grader app, instructors can view a detailed list of all submissions to any assignment and can sort based on category of information.  Instructors can add comments directly to student previews, annotate  paper submissions, and…
  • Connect the Dots Like a Numerical Analyst

    SplineGuy
    4 Nov 2014 | 3:06 pm
    I have to say, teaching Numerical Analysis is one of the highlights of my job. Granted, my primary responsibility at Wayland is the Virtual Campus Director, and I will never teach Numerical Analysis online. Nevertheless, I LOVE it. In fact, the course banner that I use in Blackboard reinforces that fact to my students every time they log in: Just as a for instance, I was able to get them to “solve” the age-old Connect-the-Dot problem. What is that, you ask? Well, simple: We all know, from the time we are toddlers, how to complete a Connect the Dot worksheet: BUT, what is the…
  • The Importance of Your Worldview

    SplineGuy
    2 Sep 2014 | 8:06 pm
    This week, I have the privilege and honor to lead the discussion in the Faith and Science course at Wayland.  The topic of discussion will be the importance of your worldview.  We start with a discussion on the 19th century masterpiece, “Flatland: A Romance of Many Dimensions” by Edwin A. Abbot. Then we’ll discuss a couple of readings: Are Scientists Biased by Their Worldview The Importance of Worldview Slides for guided discussion: This is an embedded Microsoft Office presentation, powered by Office Online.
  • 9 Essential Settings for the Teacher’s iPad

    SplineGuy
    19 May 2014 | 8:20 pm
    When using your iPad to teach, particularly in the one-iPad classroom, you can run into a few frustrations with the technology. In spite of all the exciting new features you bring to the classroom with the iPad, there are also some headaches that come along with it. Here are some of the settings that our teachers have discovered and implemented to help to alleviate many of those frustrations. 1. Use Side Switch to Lock Rotation Tap the Settings icon on your home screen and go to the General tab. You can configure the side switch to either “Lock Rotation” or “Mute.” It…
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    IMA Journal of Applied Mathematics - current issue

  • Orbital stability of periodic waves

    Natali, F., Neves, A.
    21 Nov 2014 | 6:19 am
    In this paper, we deal with the existence and orbital stability of periodic waves for a large class of dispersive equations. We use the classical theory of orbital stability to propose a method of obtaining orbital stability that can be applied to classes of dispersive equations which include Korteweg–de Vries (KdV)-type equations and non-linear Schrödinger equations. An important feature of the method is that the sufficient conditions for stability can be checked numerically, so that method can be used in problems where the waves are not known explicitly. As applications we study…
  • Real-time active wave control with preservation of wanted field

    Utyuzhnikov, S. V.
    21 Nov 2014 | 6:19 am
    In active wave control, a quite arbitrary bounded domain is protected from the field (noise), generated outside, via implementation of additional sources. The secondary sources can be situated at the boundary of the domain to be shielded. Apart from the noise, the existence of the required field, generated inside the protected domain, is also allowed. The solution to the problem is obtained in a rather general formulation. The input information to the secondary sources is based on the total field at the perimeter of the shielded domain only. In the previous publications, the input data for…
  • Blowing-up solutions to systems of fractional differential and integral equations with exponential non-linearities

    Kadem, A., Kirane, M., Kirk, C. M., Olmstead, W. E.
    21 Nov 2014 | 6:19 am
    We consider two separate systems of fractional differential equations with exponential non-linearities. We also consider the corresponding systems of non-linear Volterra integral equations. We present results on the non-existence of global solutions for each system. Bounds on the blow-up time for each system are provided along with the asymptotic growth near blow-up. Each system can be regarded as a model for the interaction of two weakly diffusive media subjected to Arrhenius-type reactions. Our results indicate that a thermal blow-up cannot be avoided under such conditions.
  • Dead-time effects on the voltage spectrum of a PWM inverter

    Moore, D. C., Odavic, M., Cox, S. M.
    21 Nov 2014 | 6:19 am
    An inverter converts a direct-current power supply to an alternating-current power supply. This conversion is achieved by switching the output between the inputs at high frequency. The resulting output voltage may be described by a high-frequency train of variable-width pulses. Pulse widths are slowly modulated so that this output waveform contains a prescribed low-frequency component, which may then be isolated by an appropriate filtering regime. Techniques for determining the full harmonic spectrum of input and output voltages and currents are well established, at least for an idealized…
  • Stability of a time-periodic flow incorporating a simple model for noise

    Roberts, M. E., Bassom, A. P.
    21 Nov 2014 | 6:19 am
    A horizontal layer of Boussinesq fluid is known to be susceptible to vortex instability when the lower surface of the fluid is subjected to a time-periodic heating of sufficient strength. In this paper a quasi-steady technique is used to trace the stability of the layer right from a standard linear theory to the problem when the vortices are so strong so as to change the thermal profile at leading order. Furthermore, this system is used to assess the possible influence of the noise that inevitably arises in many physical and laboratory situations. Even though in many problems a fluid flow is…
 
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    WordPress Tag: Mathematics

  • The Only Word Game You Can NEVER LOSE! Brian explores a form of...

    scientiflix
    19 Nov 2014 | 3:06 pm
    View video here.
  • Basic Facts Assessments

    debeerkate
    19 Nov 2014 | 2:33 pm
    I haven’t learnt much for basic facts this term, but I have only been revising as I had forgotten some skills. My results have improved a lot this term compared to the rest of the year, especially on the year 7 test. In the year 7 test I got 50/5 and last term I got 50/3. In the year 8 test I got 98/100 and last term I got 96/100.
  • Algebra and Number Assessment

    debeerkate
    19 Nov 2014 | 2:29 pm
    This term I think I did well in my algebra and number assessment and I learnt a lot. I learnt how to solve two step equations and how to plot coordinates. I also learnt about integers, multiplying, adding, subtracting and dividing fractions as well as working out written problems. For this assessment I got an E.
  • VIDEO: Mathematics, Philosophy and Giving Answers to Our Questions

    CuratedInterestingness
    19 Nov 2014 | 1:37 pm
    A reminder of the depth, beauty and philosophy that can be derived from numbers.
  • You’ll Never Amount to Anything

    an Siorrachd
    19 Nov 2014 | 12:39 pm
    Ancient wisdom has it that we must never speak ill of the dead, and so the depths of bitterness we harbour for certain people now numbered among the departed must be muted by a resented pauciloquence. As those who damned the name of Bin Laden demanded and enforced silence for Thatcher’s requiem, I will permit myself some few cathartic words on a deceased terrorist from my past. Now that he has been immortalised in a school memorial prize, and for the sake of his bereaved loved ones I shall name the assassin only as ‘McDonald’ – a mathematics teacher and a murderer. Casting our eyes…
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    MathNotations

  • Implementing The Core: B lives twice as far from A as from C. Draw that!

    Dave Marain
    8 Nov 2014 | 8:26 am
    From twitter.com/dmarain 11-8-14... A,B,C live on a straight road. B lives 5 times as far from A as from C. If AC=12 draw,determine all possible distances! COREFLECTIONS 1. 140 characters make the writing and interpretation of the problem challenging. But within each group of students there will usually be a few who will make more sense of it and they should be allowed to convince others in their group. When the inevitable hands go up and they ask "Do you mean...?" it's tempting to clarify but don't! Unless everyone is lost of course. The confusion will resolve itself in the…
  • Implementing The Core: Draining A Tank - A Real-World (?) Quadratic Model Problem

    Dave Marain
    3 Nov 2014 | 10:38 am
    From twitter.com/dmarain today (of course the wording of the problem will exceed 140 characters!)...Water is flowing out of a tank. The number of gallons after t min is given by the function V(t) = k-2t-t^2. [Assume t≥0 and other suitable restrictions] If 153 gallons remain after 3 min, in how many additional min will the tank empty?I'll even provide an answer: 9 minCOREFLECTIONSProblems like these which *artificially* model the real world are common these days on standardized tests but let's go beyond assessment issues.Before throwing this problem out to the class I usually began with some…
  • Implement The Core: No *Mean* Tricks!

    Dave Marain
    31 Oct 2014 | 6:43 am
    Halloween Twitter Problem (@dmarain)    Treats__Kids           1 ___4           2____5           3____4           4____4 M🎃an treats/child? M🎃dian? COREFLECTIONS (1) This question fits where in Common Core? Grade levels? (2) What questions could you ask before calculation to develop number sense/conceptual thinking? Some ideas... Why is this sometimes referred to as a frequency table? OR Which is easier to determine -- mean or median? OR If  frequency = 4 kids for all # of treats, mean = ? Mental Math!! OR Explain to your partner why mean >…
  • A Dose of Reality -- My Latest Common Core Rant

    Dave Marain
    22 Oct 2014 | 9:55 am
    I'm reproducing my comment to the post, "Who Needs Algebra?"on Mr. Honner's outstanding blog... http://mrhonner.com/archives/14291#comment-10579. I strongly recommend you  read all of his excellent pieces. The current one is compelling for all math educators not to mention the public... MY COMMENTS... First of all requiring an in-depth conceptual understanding of algebra for all students shows complete insensitivity to special needs students and their longsuffering teachers and parents. Sure just modify the curriculum for them. Go ahead. Show me exactly what that looks like…
  • Just How Common is our Core?

    Dave Marain
    22 Oct 2014 | 7:54 am
    Borrowing a problem from the comments in the excellent blog CorkboardConnections. Hope that's ok... http://corkboardconnections.blogspot.com/2014/08/common-core-math.html?showComment=1413889358852#c7893872002833512194 THE PROBLEM Mdm Shanti bought 1/3 as many chocolates as sweets. She gave each of her neighbours' children 4 chocolates and 3 sweets, after which she had 6 chocolates and 180 sweets left. (a) How many children received the chocolates and sweets? (b) how many sweets did she buy? ans: 18 children; 234 sweets. FROM THE COMMENTER ON THE BLOG ABOVE This is the questions our 12…
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    Neoformix

  • Stars and Stones

    5 Nov 2014 | 3:00 am
    I just launched my first mobile app. It's a game, called Stars and Stones, and you can download it now on the Apple App Store for free. I enjoy games that have a simple natural user interaction, are easy to learn, but have a rich depth of play - an elegant complexity. That's what I attempted to create and I think I came close in many respects. Stars and Stones is a series of dynamic puzzles that get progressively more challenging. There are over 100 levels and the first 50 are free. In each puzzle you drag a token around to try and capture five stars while avoiding stones. The stones move as…
  • A Short Reflection on Storytelling in Data Visualization

    28 Apr 2014 | 6:15 am
    The role of storytelling in Data Visualization has become much discussed over the last year or so. One reason I find this aspect of Data Visualization so interesting is that my own natural tendencies are to focus on exploratory visualization. Much of my own past data visualization work is weak in the storytelling side of things. Coming from a scientific background and personally enjoying the act of discovering patterns in data means my default approach is to build exploratory tools. For me, personally, this whole storytelling aspect seems a rich area to mine in order to improve my work. I…
  • Markham Winter of 2014

    1 Apr 2014 | 3:30 am
    Winter has finally ended in Markham where I live and it has seemed a very long and cold season this year. I decided to take a look at the weather data from Environment Canada and see whether my impression is supported by the data. The result is the graphic below. Click on it to see a larger version. Yes, 2014 was the coldest winter in Markham since 1994. We had an average temperature during the winter of -8.2 C this year and in 1994 it was -9.2 C. Both last year and especially 2012 were warmer than usual so it likely felt that much worse in comparison. We also had the 4th most snow in the…
  • Toronto Visible Minorities

    27 Sep 2013 | 4:30 am
    Toronto is the most multicultural city in the world. According to the 2011 National Household Survey, 46% of the population were foreign-born immigrants and 47% are members of a visible minority. (ref) These immigrants come from a wide variety of places across the globe and their diversity makes the city a truly remarkable place. I have created a Dot Map that shows a single point for every person in the Toronto area, coloured by visible minority status. There are 5,700,628 in all and they are positioned at their place of residence and coloured based on the information from the 2011 census and…
  • Toronto 311 Visualization

    6 Sep 2013 | 4:20 am
    The calls people make into the 311 service line in Toronto give an interesting glimpse into the pulse of the city. The City of Toronto makes this data available through their Open Data initiative. I did some analysis and design work with it to produce a visualization for illuminating time-based patterns during 2012. The visualization is a set of small multiple calendar heatmaps, one for each data series. The one shown above is for reports about 'long grass and weeds'. I was inspired to use this visual form by this example: Vehicles involved in fatal crashes by Nathan Yau. I experimented with…
 
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    Math Bootcamps

  • How to Determine if a Vector is a Linear Combination of Other Vectors

    Jerimi
    12 Nov 2014 | 5:47 pm
    The idea of a linear combination of vectors is very important to the study of linear algebra. We can use linear combinations to understand spanning sets, the column space of a matrix, and a large number of other topics. One of the most useful skills when working with linear combinations is determining when one vector is a linear combination of a given set of vectors. Suppose that we have a vector and we want to know the answer to the question “is a linear combination of the vectors , , and ?”. Using the definition of a linear combination of vectors, this question can be restated…
  • Using the NFL to Understand Hypothesis Testing

    Jerimi
    10 Nov 2014 | 2:07 pm
    The logic of hypothesis testing can be really confusing. Why do we “reject ” or “fail to reject “, and what does that really mean? Did we prove the null hypothesis when we didn’t reject it? These are common questions for any student studying these ideas. It turns out that we have a very nice example of the logic behind hypothesis testing right in your everyday NFL or college football game. The basic idea comes from the “challenge” that a coach can use to dispute a call on the field by a referee. For example, a referee may say that the other team has…
  • The Tools Needed for Math Screencasting

    Jerimi
    9 Nov 2014 | 3:55 pm
    While many of you are students, I know that a few of you are math teachers. This update is just for you! Over on my personal homepage (jerimiannwalker.com), I have started a series that will cover the tools and tricks I have learned about creating high quality math screencasts – the same screencasts I produce for this website and for my online courses. Part 1 of the series specifically talks about the software and hardware tools you would need if you were just getting started. You can read that article here: Math Screencasting – Part 1: Tools The post The Tools Needed for Math…
  • Understanding Exponents

    Jerimi
    9 Nov 2013 | 7:44 am
    When working with exponents, there are two new words that come up: exponent and base. As an example, in , 2 is the exponent while is the base and in , 12 is the exponent while is the base. As long as the exponent is a positive whole number, you can think of it as telling you how many times you should multiply the base by itself.       As you can see,   means “4 multiplied by itself 12 times” and that is a really big number. Many values that you calculate from exponents will be quite large. There are a couple of things to note: Anything with an exponent of zero is defined to…
  • Confidence Intervals for the Mean Using the TI83 or 84 Graphing Calculator

    Jerimi
    3 Nov 2013 | 5:26 pm
    In order to estimate the value of an unknown population mean, we can use a confidence interval based on the standard normal distribution (z-interval) or the t-distribution (t-interval). The choice is usually determined by the following rule: Use a z-interval if: the population standard deviation is known and either the population is normally distributed or the sample size is larger than 30. Use a t-interval if: the population standard deviation is unknown and either the population is normally distributed or the sample size is larger than 30. In both cases, you can either use the formula to…
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    MathFour

  • Indian Corn Craft with Fractions, Decimals and Percent Practice!

    Bon
    14 Nov 2014 | 3:13 am
    I found this awesome activity on fractions, decimals and percents a few months ago that was perfect for understanding, well… fractions, decimals and percents! Last night, browsing pinterest, I got the idea of putting it into a math craft for Indian Corn. #lightbulb So just for you, kind reader, here is the free downloadable handout with all the great goodies. (That’s on Teachers Pay Teachers, btw, but it’s free.) Supplies The free downloadable template Crayons or colored pencils Scissors Glue or glue stick Optional: one piece of paper or construction paper…
  • Geometric Turkeys – Live AND Baked with FREE Downloads!

    Bon
    1 Nov 2014 | 10:06 pm
    I had this great idea to create a geometric turkey (the baked kind). Then I found a ton of live foul on Pinterest – so I made one of those too! Halloween Math is over, but Thanksgiving Math is just beginning. (And look for more Christmas Math and even Hanukkah Math soon!) You’ve got four choices of free downloadable templates: Live Turkey for young kids Live Turkey for older kids Baked (or fried) Turkey for young kids Baked Turkey for older kids Supplies One of the free downloadable templates above Construction paper in your prefered colors Scissors Glue or glue stick Crayons,…
  • Bee-Bot Floor Robot – Teaching Basic Programming

    Bon
    25 Oct 2014 | 7:11 am
    A friend gave me a Bee-Bot floor robot on “permanent loan.” She wanted me to figure out some good ways to teach the basics of programming as well as some algebra. I was happy to take on the challenge. What is the bee-bot? I handed the bee-bot to the kids with the prompt, “What do you think?” When they asked about it, I answered: “Figure it out.” They mashed some buttons. Some students figured it out. But some needed my prompts: Press clear. Now press two arrow buttons. Put it down and press “Go.” With this they got enough of what it is, a…
  • Using the Hundreds Chart to Teach Beginning Multiplication

    Bon
    21 Oct 2014 | 5:50 am
    We typically think of beginning multiplication as memorizing your math facts for 2’s, 3’s, 4’s and 5’s. But in my new job, things aren’t so typical. So I’m trying something new. Sieve of Eratosthenes It started with a brilliant idea to teaching the Sieve of Eratosthenes. I began by asking the students to color in (or cross out) each of the multiples of 2, 3, 5 and 7. Alas, the plan went awry quickly. Students got confused when they needed to cross out a number that was previously crossed out (like 6). So they moved to the next number and crossed it out:…
  • Really Big Numbers: The 100 Dots Project

    Bon
    13 Oct 2014 | 10:26 am
    Note: this post has affiliate links. If you use these, you support my work in giving you free content. The AMS sent me a copy (for free, yay me!) of their first publication ever – Really Big Numbers by Evan Schwartz. I loved reading the first few pages with K8, and it gave me an idea. What if I used this in class to introduce numbers? I started teaching at a private, special education school recently. I quickly figured out that everything I know about math education is wrong. I thought this book (and the activity that spawned from it) might work well. And for once, with these kids, I…
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    CSE Blog - quant, math, computer science puzzles

  • Pebble Placement Puzzle 2

    Pratik Poddar
    9 Nov 2014 | 5:30 am
    Source: AUSTMS Gazette 35 Related Problem: Pebble Placement Puzzle 1 Problem: Peggy aims to place pebbles on an n × n chessboard in the following way. She must place each pebble at the center of a square and no two pebbles can be in the same square. To keep it interesting, Peggy makes sure that no four pebbles form a non-degenerate parallelogram. What is the maximum number of pebbles Peggy can place on the chessboard?
  • Pebble Placement Puzzle 1

    Pratik Poddar
    6 Nov 2014 | 5:27 am
    Source: AUSTMS Gazette 35 Problem: There are several pebbles placed on an n × n chessboard, such that each pebble is inside a square and no two pebbles share the same square. Perry decides to play the following game. At each turn, he moves one of the pebbles to an empty neighboring square. After a while, Perry notices that every pebble has passed through every square of the chessboard exactly once and has come back to its original position. Prove that there was a moment when no pebble was on its original position.
  • Diminishing Differences Puzzle

    Pratik Poddar
    18 Oct 2014 | 3:05 am
    Source: Australian Mathematical Society Gazette Puzzle Corner 34 Problem: Begin with n integers x1, . . . , xn around a circle. At each turn, simultaneously replace all of them by the absolute differences Repeat this process until every number is 0, then stop. Prove that this process always terminates if and only if n is a power of 2. Shameless plug: Follow CSE Blog on CSE Blog - Twitter and CSE Blog on Quora. :-)
  • Balancing Act Puzzle

    Pratik Poddar
    19 Sep 2014 | 1:21 am
    Source: Australian Mathematical Society Gazette Puzzle Corner 35 Problem:There are some weights on the two sides of a balance scale. The mass of each weight is an integer number of grams, but no two weights on the same side of the scale share the same mass. At the moment, the scale is perfectly balanced, with each side weighing a totalof W grams. Suppose W is less than the number of weights on the left multiplied by the number of weights on the right.Is it always true that we can remove some, but not all, of the weights from each side and still keep the two sides balanced?
  • "Flawless Harmony" Puzzle

    Pratik Poddar
    26 Aug 2014 | 2:40 am
    Source: AUSTMS Puzzle Corner 35 Problem: Call a nine-digit number flawless if it has all the digits from 1 to 9 in some order. An unordered pair of flawless numbers is called harmonious if they sum to 987654321. Note that (a, b) and (b, a) are considered to be the same unordered pair. Without resorting to an exhaustive search, prove that the number of harmonious pairs is odd. Update (23 Oct 2014): Solution: Posted by me (Pratik Poddar) in comments!
 
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    Marauders of the Lost Sciences

  • Poynting wants to know how energy is transfered

    Editors
    22 Nov 2014 | 2:10 am
    The Poynting vector represents energy in the electromagnetic fields and is given by     Students often read it as the “pointing vector”, but Poynting was a person, not a direction. Here are his own words. A SPACE containing electric … Continue reading →
  • Merton was on to Itó’s formula before it was cool

    Editors
    21 Nov 2014 | 2:10 am
    Judging from his published papers Merton was already expert at continuous time finance before the 1973 option pricing papers came out. A Digression on Itó Processes To apply the dynamic programming technique in a continuous-time model, the state variable dynamics … Continue reading →
  • Metropolis inadvertently creates a major tool for statistics in a physics paper.

    Editors
    20 Nov 2014 | 2:10 am
    While trying to do Statistical Mechanics simulations a group of physicists at Los Alamos faced the problem of simulating from the canonical distribution. The method they created was generalized by Hastings to the become the Metropolis-Hastings workhorse of statistics. Notice … Continue reading →
  • Heisenberg on turbulence

    Editors
    19 Nov 2014 | 2:10 am
    Heisenberg writing shortly after World War II had some deep insights into turbulence. A few remarks may be added with regard to the physical picture of turbulence presented in the recent papers. In the earlier years one thought that turbulence … Continue reading →
  • Pigou on the stationary state

    Editors
    18 Nov 2014 | 2:10 am
    It’s left to the reader to decide how relevant the following passage is to the current state of the economy. In his Fiscal Policy and Business Cycles Professor Hanson writes: “The classicals were quite right when they argued that without … Continue reading →
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