a.
4 * 0.12 = 0.04 = 4% = 1/25
b.
\(2^d \cdot \epsilon^d\) where \(\epsilon \le 0.1\)
c.
- \((2 \cdot \epsilon) \le 0.2\)
- \((2\epsilon)^d \rightarrow 0\) as d trends toward \(\infty\)
d.
- \(V = 1 - (1-\epsilon)^d\)
- volume of unit hypercube is 1
- volume of inner hypercube is \((1-\epsilon)^d\)
- volume of inner cube subtracted from outer cube is the volume of the shell
e.
- \(1 - (1-\epsilon)^d \rightarrow 0\) as d trends toward \(\infty\) because \(\epsilon \le 0.1\)
2.
a.
b.
- Diagonal: \((0,0,0)\rightarrow (1, 1, 1)\)
- Anti-diagonal: \((1, 1, 0) \rightarrow (0,0,1)\)
c.
- \(\theta = \arccos(\frac{(v_1 \cdot v_2)}{\|v_1\| \cdot \|v_2\|})\)
- Change to standard vector notation
- \(v_1 = (1,1,1), v_2 = (0-1, 0-1, 1-0)\)
- \(\theta = \arccos(\frac{(-1 -1 + 1)}{\sqrt{3} \cdot \sqrt{3}}) = 109.47^\circ\)
e.
- magnitude is the same for both vectors, equal to d
- \(v_1 = (1, 1, ...)\) with dimension d
- \(v_2 = (-1, -1, ..., 1)\) with dimension d
- \(v_1 \cdot v_2 = v_2 \cdot v_2\)
- \(\theta_d = \arccos{(\frac{-(d-2)}{d})\)
e.
- Intuitively the angle goes toward 180 degrees
- Angle increases as dimension increases
- definition of cos \(\theta\) in higher dimensions goes to 1
- \(\frac{-(d-2)}{d} \rightarrow 1\) as d goes to \(\infty\)
- \(\arccos(1) = 180^\circ\)