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Monday, December 18, 2006

Math Matters, December 2006

Unfortunately, that is the only kind of math that I have given any thought to lately.

But here are some things to think about over break, (while I am finishing a referee job)

  • Geometric Topology: there is a way to form a wild knot as an intersection of nested solid tori, and I think that I can prove that if two knots from this class are equivalent, then the defining sequences must also be equivalent up to some sort of cofinality condition. If I can do this, then perhaps I can come up with some invariants that distinguish very ill behaved will knots (that is, knots that are so wild that they fail to pierce a disk anywhere)
  • College level mathematics: suppose one has the differential equation y”+y = f(t) and f(t) is both bounded and periodic of period 2pi as well as piecewise smooth. Then any solution to this differential equation is bounded on [0, infinity) if and only if the k = 1 term of the Fourier series expansion vanishes. Sufficiency is easy to show; necessity is where the work lies.
    Of course, this is just a mild extension of the concept of resonance; for example the solution to y” + y = sin(t), y(0)=0, y’(0)= -1/2 is y(t) = -(1/2)tcos(t), which has unbounded oscillations as t goes to infinity, even though the driving function is bounded. For those who like to see examples, I suggest surfing to the interactive differential equations website. Click enter, and then go to the menu; there is tons of stuff there. You can use sliders to vary parameters and you can watch the phase plane as the systems evolve with time.
  • Ok, now I’ve said it. Now what am I going to do about it?


    Blogger Jim said...

    Dear Ollie,

    I have recently started an optical thin films blog site. So yesterday when I was working on a math problem and got stuck I thought I might search for a math related blog and see who is out here!

    I apologize in advance for posting this in the comments although it may be appropriate. Perhaps one of your students may help me as well and it could be one more problem added to the couple in your post. (By the way, its been a few years since diff-eq and I am not sure what it means to be bounded on [0, infinity) or I would enjoy taking your challenge!

    My problem, if I may, is to solve the following for theta where R, r, k, and d, are all constants. I suppose this cannot be solved except numerically but I'm hoping there might be a way using an expansion or such that you might suggest?

    By the way, this is a real world problem related to what is called planetary geometry (dual axis of rotation) used commonly in the deposition of optical thin films. I am interested to find, for a point travelling on a dual axis rotation system, what fraction of the path is contained within an anular ring of known size and centered about the primary axis of rotation. I have worked the problem down to a difficult problem to separate cosines and sines of related but different angles.

    So here is the trouble:

    cos(theta)cos(k*theta) + sin(theta)sin(k*theta) > b

    I hope this will be of interest to you. Please feel free to contact me via email if that is preferred ( And if a student solves this problem please post it so Professor Nanyes will know!

    Jim Walker

    8:36 AM  
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