403/503- Homework 3

March 7, 2011

This homework set is due Monday, March 21 at the beginning of lecture. Problems 1-5 are required from everybody, and graduate students should also work on problem 6. (Of course, everybody is more than welcome to work on everything, including the remarks on f mentioned at the end of problem 6.)

  1. Solve problems 5.3, 5.6, 5.8, 5.10, 5.11, 5.14, 5.20, 5.21, 5.23 from the book.
  2. Solve problems 6.6, 6.7, 6.8 from the book.
  3. The taxicab norm on {\mathbb R}^2 is defined by setting \|v\|=|v_1|+|v_2| where v=(v_1,v_2). Show that this is indeed a norm, and that there is no inner product \langle\cdot,\cdot\rangle on {\mathbb R}^2 for which \|v\|=\sqrt{\langle v,v\rangle}. Find two non-congruent non-degenerate triangles with sides of length 1, 1, 2. (Of course, lengths are computed with respect to this norm, not the usual one).
  4. Prove Lagrange’s identity: If P(x_1,\dots,x_n,y_1,\dots,y_n)= \displaystyle \left(\sum_{i=1}^n x_i^2\right)\left(\sum_{i=1}^n y_i^2\right)-\left(\sum_{i=1}^n x_iy_i\right)^2, then P(x_1,\dots,x_n,y_1,\dots,y_n)= \displaystyle \sum_{1\le i<j\le n}(x_iy_j-x_jy_i)^2. Note that this implies the Cauchy-Schwarz inequality for the usual inner product on {\mathbb R}^n.
  5. Prove the Cauchy-Schwarz inequality for the usual product in {\mathbb R}^n as follows: Given u,v\in{\mathbb R}^n, consider the function f(\lambda)=\|\lambda u+v\|^2 as a quadratic in \lambda, and deduce the inequality by examining the discriminant of f.
  6. Consider a unit square {H}. Inscribe in {H} exactly {n} squares with no common interior point. (The squares do not need to cover all of {H}.) Denote by {e_1,\dots,e_n} the side lengths of these squares, and define \displaystyle  f(n)=\max\sum_{i=1}^ne_i. Show that {f(n)\le\sqrt n}, and that equality holds iff {n} is a perfect square. (An 80+ years old open problem of Erdös is to find all {n} for which {f(n)=f(n+1)}. Currently, it is only known that {f(n)<f(n+2)} for all {n}, {f(1)=f(2)=1}, {f(4)=f(5)=2}, and that if {f(n)=f(n+1)}, then {n} is a perfect square.)

403/503- Nim addition and multiplication

March 7, 2011

The notions of Nim addition and Nim multiplication that we discussed in the first homework set are due to John Conway, who studied them in the context of ordinal numbers. The ordinals extend the natural numbers, and what we did was to only consider “an initial segment”. Recently, the excellent blog neverendingbooks by Lieven Le Bruyn has discussed Conway’s construction in detail, in a series of (so far, ten) posts that you may enjoy reading and I highly recommend:

  1. On2 : transfinite number hacking
  2. On2 : Conway’s nim-arithmetics
  3. On2 : extending Lenstra’s list
  4. The odd knights of the round table
  5. Seating the first few thousand Knights
  6. Seating the first few billion Knights
  7. How to play Nimbers?
  8. n-dimensional and transfinite Nimbers
  9. How to win transfinite Nimbers?
  10. Aaron Siegel on transfinite number hacking

In particular, the posts have links to papers and talks on related subjects.