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    Re: A simple three-body fix puzzle
    From: Tom Sult
    Date: 2010 Dec 11, 13:42 -0600
    With my 2 years of calc now many many years rusty I thank you for that. I Am always perplexed be what seems to be well reasoned mathematical arguments that defy common sense. But as you have shown us we simple need the right math tools to get to the answer. 

    Thomas A. Sult, MD
    Sent from iPhone

    On Dec 10, 2010, at 23:48, P H <pmh099@yahoo.com> wrote:

    Glad to hear it but I must say I am still confused by the assertion that the probability of being at the center of a perfect crossing 3 body fix is "0"
    Thomas A. Sult, MD

    ------------------------------------------------------------------------------------------------------

    I believe that part of the problem lies in the adopted terminology.  If I am right, then this is also a good example of when digging deeper into the mathematical underpinnings of a problem is in fact necessary for its understanding.  Intuition and "good enough" may work OK in practice 99% of the time, until it doesn't.  This maybe an example from that remaining 1%.

    We need to distinguish here between "probability" and "probability density."  The former is given by a number between 0 and 1 (i.e. 0% and 100%).  The probability DENSITY, however, is probability distributed over some region, which in this case is the surface of the Earth.  This latter quantity is not expressed in % but instead carries units of % per unit area (e.g. % per square mile).  These two quantities are related by:

    Probability of being in a given area (in %)    =    Average probability DENSITY throughout that area (in % per sq. mile)      x     that area (in sq. miles)

    or, to shorten the notation,

    P = PD x A

    P and PD resemble each other in shape when plotted (because often all A's have the same size).  But they are different things expressed in different units. Not distinguishing between the two is, I believe, at the root of these difficulties.

    Choosing a smaller A (e.g. square meter instead of a sq. mile) makes for better resolution.  Getting the maximum resolution by taking A to the limiting case of zero is in fact the basis of calculus.  The total probability P of being somewhere on the Earth is Ptot = 100% (we should hope so!) which must be the result of adding up all the PD x A products arising from the entire distribution PD and all the little squares A covering the entire Earth surface.

    A mathematical point has A = 0, hence P will be zero for any point, including the one that looks like a perfect LOP intersection.  And yet, along with Tom we intuitively understand that with a perfect LOP crossing at a single point we should have the 100% probability assigned to that single special point, and 0% to all the others.  On the other hand, points are only eligible to be paired up with values from PD, not P; therefore asking: "What is the P at this point?" is really an ill-posed question.  Instead, a valid question is: "What is the P of being in this area A surrounding this point?"

    So what gives?

    In order to resolve this apparent paradox we must "take it to the limit one more time" (with a nod and perhaps apologies to "Eagles") and make a step beyond ordinary calculus into the portion of mathematics called "theory of distributions."  As we shrink A -> 0 to get at our special single point of intersecting LOPs, then in order for all the probability P = Ptot = 100% (which is an immutable constraint) to remain concentrated in that ever shrinking area A, we must have PD -> Infinity.  At that limit, PD ceases to be an ordinary function and becomes a "distribution" commonly called the Dirac delta-function.

    Thus in that mathematically abstract case Tom's intuition is correct; we CAN have all 100% concentrated at a single point, and there indeed exists mathematical machinery designed to handle situation like these.  In real life, however, this 100% would be concentrated in a "point" (which is really a very small area), for example, the size of the vessel's bridge.  :-)

    The delta-function can be modeled as the limit of a sequence of ever-narrower and ever-taller Gaussians with the area under the curve kept constant.
    http://en.wikipedia.org/wiki/Dirac_delta_function

    To conclude on a lighter node, when I saw an Eagles concert in 2001, Glenn Frey stated that his wife calls "Take It To The Limit" the "credit card song."


    Peter Hakel


       
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