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.

• 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.

• 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.

• 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) = r₀.
• 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?

• Explore the 2-strand braid group and explain why it is the same as (is isomorphic to) the integers with addition.

• 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^kba^-k where a = (1 2 ... n) and b = (1 2).
• Show that (1 2) and (1 2 ... n) generate S_n.

• 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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