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A Unified Calculus?

Posted by chrisd on from the limit-as-it-approaches-my-grades-in-school dept.

DeAshcroft writes "Science Daily is reporting that one Martin Bohner's work, "Asymptotic Behavior of Dynamic Equations on Time Scales," has made significant waves (ahem) in the mathematical community. The work is "part of a fairly new and exciting effort to unify continuous and discrete calculus" I guess it's time to re-learn long division."

4 of 31 comments (clear)

  1. It turns out this topic is not entirely new: by jerkface · · Score: 5, Informative

    The relationship between the discrete time scales approach and the unification of calculus has been widely known since S. Hilger published "Ein Masskettenkalkul mit Andwendung auf Zentrumsmannigfaltigkeiten" in 1988as his Ph.D. thesis. The problem remained for other to, um, elaborate on the connection. Martin Bohner, as one of the few individuals taking a great interest in this somewhat narrow area of the field, turns out to be the prime mover in the progress in the field. The really important development is that more people are going to take interest now and they will publish new and interesting results. Bohner's key accomplishment so far is proving to the community that this topic is worthy of more interest.

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  2. In case you're wondering... by Bazzargh · · Score: 4, Informative

    ...what this is all about, after a little digging on Martin's site I found this paper "Basic Calculus on Time Scales and some of its applications"

    Its readable enough if you can remember your calculus from first year at Uni.

    The gist: normally we do calculus with the set of real numbers, and difference equations with integers. The 'time scales' notion is that instead of having even gaps between numbers like the integers, you can have independently varying gaps, down to infinitesimal ones. Thus, timescales are really just arbitrary subsets of the reals. An example of a time scale might be:

    1_2 3_4 5_6

    (the underscore indicates a chunk of real numbers, the space a gap of numbers we don't use, and so on)

    It's hopefully obvious that the set of integers and the set of reals are special cases of timescales. So, if you derive the fundamental theorems in calculus using timescales, you find the equivalent theorems for reals and integers are special cases.

    Cheers,
    Baz

  3. Re:Jesus Christ! by Anonymous Coward · · Score: 1, Informative

    Center manifolds

  4. Re:cool by trixillion · · Score: 2, Informative

    Linear difference equations can be solved methodologically using the Z-Transform. This is dual to the use of the Laplacian Transform with linar differential equations. Find an advanced book on signal processing for more details. Similarly there are methods for handling coupled difference equations in a manner dual to coupled differential equations.