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posted by Fnord666 on Monday February 05 2018, @05:36AM   Printer-friendly
from the common-sense dept.

Mathematician Keith Devlin writes about how the capabilities to work with maths have changed since the late 1960s. He summarizes what he considers to be the essential skills and knowledge that people can focus on as more and more is turned over to software.

The shift began with the introduction of the digital arithmetic calculator in the 1960s, which rendered obsolete the need for humans to master the ancient art of mental arithmetical calculation. Over the succeeding decades, the scope of algorithms developed to perform mathematical procedures steadily expanded, culminating in the creation of desktop and cloud-based mathematical computation systems that can execute pretty well any mathematical procedure, solving—accurately and in a fraction of a second—any mathematical problem formulated with sufficient precision (a bar that allows in all the exam questions I and any other math student faced throughout our entire school and university careers).

So what, then, remains in mathematics that people need to master? The answer is, the set of skills required to make effective use of those powerful new (procedural) mathematical tools we can access from our smartphone. Whereas it used to be the case that humans had to master the computational skills required to carry out various mathematical procedures (adding and multiplying numbers, inverting matrices, solving polynomial equations, differentiating analytic functions, solving differential equations, etc.), what is required today is a sufficiently deep understanding of all those procedures, and the underlying concepts they are built on, in order to know when, and how, to use those digitally-implemented tools effectively, productively, and safely.

Source : What Scientific Term or Concept Ought to be More Widely Known?


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  • (Score: 3, Insightful) by c0lo on Monday February 05 2018, @07:09AM (5 children)

    by c0lo (156) Subscriber Badge on Monday February 05 2018, @07:09AM (#633177) Journal

    A very important tool in numerical modeling is the Runge-Kutta method.

    Doh. Try some stiff equations [wikipedia.org] and come back to tell me how a straight Runge-Kutta (or any other basic ODE numeric ODE solver) performs.

    You'll need to add a predictor-corrector to RK and transform it into an implicit RK [wikipedia.org] for this to work.

    Roughly what it does is minimizing the accumulation of error by alternatingly underestimating than overestimating the result of a physical process.

    Roughly... it doesn't quite do it as you described. There's no "optimisation" step (either minimization or maximization) in there - it only increases the number of evaluation points and consider them in the computations of the next "func-value delta".
    Not dissimilar with retaining more terms in Taylor expansion.

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    https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
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  • (Score: 2) by PiMuNu on Monday February 05 2018, @11:17AM (4 children)

    by PiMuNu (3823) on Monday February 05 2018, @11:17AM (#633235)

    Better use a symplectic integrator. Paper by Yoshida.

    I was having a conversation on a bus a few months ago about how Kerbal Space Programme does the orbit calculation. We were discussing RK vs symplectic. A random stranger stuck their head over the partition and told us it just assumes a conic section. I was disappointed!

    • (Score: 2) by c0lo on Monday February 05 2018, @01:51PM (3 children)

      by c0lo (156) Subscriber Badge on Monday February 05 2018, @01:51PM (#633267) Journal

      Better use a symplectic integrator.

      Unfortunately, not all DE systems admit a Hamiltonian formulation.

      Lotka-Volterra (predator–prey equations) is one of the cases. Interesting outside ecology - e.g. see iodine pit [wikipedia.org] where it usually produces stiff ODE due to concentrations (the variables in ODE systems) in 1015-20 range and constant coefficients in 10-28 range (neutron capture cross-sections in barns [wikipedia.org]). Nasty thing to model numerically when dealing with heaps of radioactive species and decay channels in the reactor soup.

      --
      https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
      • (Score: 2) by PiMuNu on Monday February 05 2018, @02:05PM (2 children)

        by PiMuNu (3823) on Monday February 05 2018, @02:05PM (#633270)

        Understood - original post was referring to elliptic orbits; I think orbit modelling can be described in the general case by Hamiltonians.

        • (Score: 2) by c0lo on Monday February 05 2018, @02:21PM (1 child)

          by c0lo (156) Subscriber Badge on Monday February 05 2018, @02:21PM (#633276) Journal

          I think orbit modelling can be described in the general case by Hamiltonians.

          I'm not sure about the propulsive stage - the entire energy of the system is conserved, of course, but I doubt the potential chemical energy can be captured in the Hamiltonian. But maybe it can, with an artifice like "the inverse of a dissipative/friction force" (well, long time already since I knew the answer.)

          --
          https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
          • (Score: 2) by PiMuNu on Monday February 05 2018, @03:00PM

            by PiMuNu (3823) on Monday February 05 2018, @03:00PM (#633285)

            Good point - you are probably right. Certainly atmospheric effects (e.g. low earth orbit) clearly create a dissipative force.