403/503- Homework 3

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.)

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