Mathematics 4513-002 - Senior Seminar
- Spring 2009
Here I will post miscellaneous items related to homework exercises.
Exercises, Jan 22:
- How many inscribed squares are there in a regular n-gon for n even?
For n odd?
- Revised: How many for n a multiple of 4?
- Prove that an ellipse (that is not a circle) has exactly one inscribed
square.
- Here is a
pdf file with a picture of the curve mentioned in class. Is there an
inscribed square? How many do you think there are?
- Prove that if C is a simple closed curve and C is differentiable at
x, then there is an inscribed equilateral triangle with one corner at x.
Feb. 4:
- Show that for any simple closed curve in the plane, the integral of
the (unsigned) curvature (that is, kappa) is at least 2pi.
- Show that equality holds in the above problem if and only if the
curve is convex (use any convenient definition for convexity).
- Consider a closed curve in 3-space parametrized by arc length, and
its derivative (a curve on the unit sphere). Can you say anything about
the center of mass of the derivative curve, based on the fact that the
original curve is closed? Can you prove your conjecture?
- Among curves on the sphere satisfying the condition of the previous
exercise, what is the minimum length? Which of these curves have exactly
this length? These questions lead to a proof of Fenchel's theorem:
a closed curve in 3-space has total curvature at least 2pi, with equality
if and only if the curve is planar and convex.
Feb. 11:
- Show that if there is a path of length n from 0 to k, then there
exist integers a,b with n = a + b and k = a - b. (Use the picture!)
- In the situation above, show that N_{n}(0 --> k) is "n choose
a".
- Use the previous problem to prove the second equality in the Ballot
Theorem.
Feb. 16:
For these exercises, the random walk operates in such a way that a step
of +1 has probability p, and a step of -1 has probability q (with p + q =
1).
- Show that the ruin probabilities r_{i} satisfy the
difference equation x_{i} = q x_{i-1} + p
x_{i+1}, and the boundary conditions r_{-a}
= 1,
r_{b} = 0.
- Show that two solutions to the difference equation are x_{i}
= 1 and x_{i} = (q/p)^{i}, so that
the general solution is given by x_{i} = C + D
(q/p)^{i}. Then use the boundary conditions to find C and
D, and find r(a,b) = r_{0}.
- Suppose that the game is favorable to you (p > q), but your opponent
has unlimited financial resources (b --> infinity).
(Eg. you wish to open a casino.)
Show that then r(a) = lim_{b --> inifnity} r(a,b) = (q/p)^{a}.
How much capital "a" would you need if you give yourself a hopefully
inconspicuous edge of just p = 0.51 and you want to run a risk of at most
P = 0.01 of going broke?
Mar. 4:
- Explore the 2-strand braid group and explain why it is the same as
(is isomorphic to) the integers with addition.
Mar. 13:
- Recall the function A_{n} --> (S_{n} - A_{n})
taking a permutation s to (1 2)s. Show that this function
is a bijection.
- Express (1 2 3)(4 5)(1 6 7 8 9)(1 5) as a product of disjoint
cycles.
- Compute aba^{-1} where a = (1 3 5)(1 2)
and b = (1 5 7 9).
- Compute a^{k}ba^{-k} where a = (1 2
... n) and b = (1 2).
- Show that (1 2) and (1 2 ... n) generate S_{n}.
Apr. 22:
- Compute (using the skein relations) the bracket polynomial of the
figure eight knot.
- Compute the X-polynomial and the Jones polynomial of the figure
eight knot.
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