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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?