Recent content by alanlu

  1. A

    Disjoint dyadic squares in unit disk

    I asked the author for some clarification and he has chosen to grace me with a response! Here's the quote: In short, I'm not quite done yet.
  2. A

    Disjoint dyadic squares in unit disk

    Alright, thank you haruspex. I'm starting to think the stricter interpretation has an interesting upper bound in its own right, but I don't think this would not be the forum for that discussion. :)
  3. A

    Disjoint dyadic squares in unit disk

    I believe the squares include their boundaries, and by definition of the disjoint condition two adjacent squares would have a line/vertex (nonempty) intersection and so would not be disjoint. If I interpreted the question wrong and things are fine, then ... haha yeah, there isn't much difficulty...
  4. A

    Disjoint dyadic squares in unit disk

    Homework Statement Given ε > 0, show that there is a collection of disjoint dyadic squares in the unit disk that has a total area which exceeds π - ε. Homework Equations Define a dyadic interval as an interval of the form [a, b] such that a = p/2k and b = (p + 1)/2k, p and k are integers. A...
  5. A

    Intercepts in quadric surfaces?

    What if it was an elliptic paraboloid which was rotated so that its axis of revolution was not parallel to an axis? Couldn't it then achieve 6 intercepts?
  6. A

    Bounded continous implies uniformly continuous

    Ah thanks! Actually, I did arrive at inf { d(x) }, but I wasn't sure how to turn that into something that is guaranteed to be > 0.
  7. A

    Bounded continous implies uniformly continuous

    Was not aware of this characterization until now. My text does not mention this, so there must be another way.
  8. A

    Bounded continous implies uniformly continuous

    I'm trying to show that continuous f : [a, b] -> R implies f uniformly continuous. f continuous if for all e > 0, x in [a, b], there exists d > 0 such that for all y in [a, b], ¦x - y¦ < d implies ¦f(x) - f(y)¦ < e. f uniformly continuous if for all e > 0, there exists d > 0 such that for...
  9. A

    What is the distance a particle has moved if I know the work done

    I agree with RGV. It may be easier to think about using a different variable name for the integral than the x you're trying to solve for. Try plugging in the limits of integration as if you are evaluating the integral.
  10. A

    What is the distance a particle has moved if I know the work done

    What is the integral of x2 + 2x?
  11. A

    Countably infinite set: odd integers

    How do you know that m is even when n is even? Likewise, how do you know m is odd when n is odd?
  12. A

    Countably infinite set: odd integers

    Yep you showed that f is onto correctly. Can you show that f is 1-1, or that f(m)=f(n) implies m=n?
  13. A

    Countably infinite set: odd integers

    I'm not convinced. It doesn't say why m must equal f(n) for some n, but just that it could. Consider using n for odd n and -n + 1 for even n as your f. It will lead to a simpler proof. A strategy for examining piecewise functions can be to look at the pieces of the domain where the...
  14. A

    Express 3x3 matrix as projection + shearing

    http://en.wikipedia.org/wiki/Shear_mapping
  15. A

    Countably infinite set: odd integers

    You're fine. Actually, you can define the same function in two pieces: n - 2 for odd n and -n - 1 for even n. Furthermore, n for odd n and -n + 1 for even n works as well. Same way you should show 1-1 and onto for any other function. Show f(n) = f(m) implies n = m. Then show for every odd...