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A Unified Calculus?
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."
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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Yeah...mod me offtopic but yo...leave it to the germans to come up with published works titled "Ein Masskettenkalkul mit Andwendung auf Zentrumsmannigfaltigkeiten"...good god...Zentrumsmannigfaltigkeiten???
You'll have that sometimes...
...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
I remember from math competitions way back one of my favorite tricks was when an iterative problem looked like it lended itself to a difference equation, to solve the related continuous calculus problem, and then use that solution as a starting point for the difference-equation solution. Always worked much faster than anything else I could think of... Of course I was no expert in the calculus of difference equations, but this sounds really neat. And given how much application both calculus and difference equations have had in other areas of science, this could have big implications once somebody figures out what they are :-)
Energy: time to change the picture.