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  • Joseph Nebus 3:00 pm on Tuesday, 5 April, 2016 Permalink | Reply
    Tags: , , solutions, square root day, ,   

    JH van ‘t Hoff and the Gaseous Theory of Solutions; also, Pricing Games 

    Do you ever think about why stuff dissolves? Like, why a spoon of sugar in a glass of water should seem to disappear instead of turning into a slight change in the water’s clarity? Well, sure, in those moods when you look at the world as a child does, not accepting that life is just like that and instead can imagine it being otherwise. Take that sort of question and put it to adult inquiry and you get great science.

    Peter Mander of the Carnot Cycle blog this month writes a tale about Jacobus Henricus van ‘t Hoff, the first winner of a Nobel Prize for Chemistry. In 1883, on hearing of an interesting experiment with semipermeable membranes, van ‘t Hoff had a brilliant insight about why things go into solution, and how. The insight had only one little problem. It makes for fine reading about the history of chemistry and of its mathematical study.

    In other, television-related news, the United States edition of The Price Is Right included a mention of “square root day” yesterday, 4/4/16. It was in the game “Cover-Up”, in which the contestant tries making successively better guesses at the price of a car. This they do by covering up wrong digits with new guesses. For the start of the game, before the contestant’s made any guesses, they need something irrelevant to the game to be on the board. So, they put up mock calendar pages for 1/1/2001, 2/2/2004, 3/3/2009, 4/4/2016, and finally a card reading \sqrt{DAY} . The game show also had a round devoted to Pi Day a few weeks back. So I suppose they’re trying to reach out to people into pop mathematics. It’s cute.

    • Marta Frant 5:27 am on Thursday, 7 April, 2016 Permalink | Reply

      Questions, questions, questions… The constant ‘why’ is what makes the world go around.


      • Joseph Nebus 2:07 am on Saturday, 9 April, 2016 Permalink | Reply

        ‘Why’ is indeed one of the big questions. ‘What’ and ‘The Heck?’ are also pretty important.

        Liked by 1 person

  • Joseph Nebus 2:35 pm on Wednesday, 15 July, 2015 Permalink | Reply
    Tags: , , inverse problems, problems, solutions, , variations   

    A Summer 2015 Mathematics A To Z: well-posed problem 

    Well-Posed Problem.

    This is another mathematical term almost explained by what the words mean in English. Probably you’d guess a well-posed problem to be a question whose answer you can successfully find. This also implies that there is an answer, and that it can be found by some method other than guessing luckily.

    Mathematicians demand three things of a problem to call it “well-posed”. The first is that a solution exists. The second is that a solution has to be unique. It’s imaginable there might be several answers that answer a problem. In that case we weren’t specific enough about what we’re looking for. Or we should have been looking for a set of answers instead of a single answer.

    The third requirement takes some time to understand. It’s that the solution has to vary continuously with the initial conditions. That is, suppose we started with a slightly different problem. If the answer would look about the same, then the problem was well-posed to begin with. Suppose we’re looking at the problem of how a block of ice gets melted by a heater set in its center. The way that melts won’t change much if the heater is a little bit hotter, or if it’s moved a little bit off center. This heating problem is well-posed.

    There are problems that don’t have this continuous variation, though. Typically these are “inverse problems”. That is, they’re problems in which you look at the outcome of something and try to say what caused it. That would be looking at the puddle of melted water and the heater and trying to say what the original block of ice looked like. There are a lot of blocks of ice that all look about the same once melted, and there’s no way of telling which was the one you started with.

    You might think of these conditions as “there’s an answer, there’s only one answer, and you can find it”. That’s good enough as a memory aid, but it isn’t quite so. A problem’s solution might have this continuous variation, but still be “numerically unstable”. This is a difficulty you can run across when you try doing calculations on a computer.

    You know the thing where on a calculator you type in 1 / 3 and get back 0.333333? And you multiply that by three and get 0.999999 instead of exactly 1? That’s the thing that underlies numerical instability. We want to work with numbers, but the calculator or computer will let us work with only an approximation to them. 0.333333 is close to 1/3, but isn’t exactly that.

    For many calculations the difference doesn’t matter. 0.999999 is really quite close to 1. If you lost 0.000001 parts of every dollar you earned there’s a fine chance you’d never even notice. But in some calculations, numerically unstable ones, that difference matters. It gets magnified until the error created by the difference between the number you want and the number you can calculate with is too big to ignore. In that case we call the calculation we’re doing “ill-conditioned”.

    And it’s possible for a problem to be well-posed but ill-conditioned. This is annoying and is why numerical mathematicians earn the big money, or will tell you they should. Trying to calculate the answer will be so likely to give something meaningless that we can’t trust the work that’s done. But often it’s possible to rework a calculation into something equivalent but well-conditioned. And a well-posed, well-conditioned problem is great. Not only can we find its solution, but we can usually have a computer do the calculations, and that’s a great breakthrough.

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