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Date
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Assignment |
| 7 May | Some game theory, and some more game theory |
| Material for Test 2, from here down |
| 21 Apr | Exercises 11.56, 11.57, 11.59, 11.60, 11.66
Use the techniques described in class to analyze the roots of the system in 11.56. |
| 14 Apr | Read §§11.2, 11.4 of notes on Gröbner bases.
Do Exercises 11.33, 11.34, 11.36, 11.43, 11.44 |
| Material for Test 1, from here down |
| study | p. 278 #1, 4-6 |
| 2628 March |
p. 264 #2-5, 10, 12, 13
Hints:
- For #3, you can find $a,b\in\mathbb Z$ such that $ae_A+be_B=1$. (Why? this expression should look familiar!)
Now use properties of exponents on $m^{ae_A+be_B}$.
- For #10, you need to think modularly:
- You can solve $x^2+\cdots\equiv a(\textrm{mod }m)$ by solving $x_1^2+\cdots\equiv a(\textrm{mod }p)$ and $x_2^2+\cdots\equiv a(\textrm{mod }q)$, then find $n_1, n_2$ such that $n_1\equiv1(\textrm{mod }p)$, $n_1\equiv0(\textrm{mod }q)$, $n_2\equiv0(\textrm{mod }p)$, $n_2\equiv1(\textrm{mod }q)$, and show that $x=n_1x_1+n_2x_2$ is a solution modulo $m$. If you haven't had Number Theory, this will be really, really hard.
- Each equation $x^2+\cdots\equiv a(\textrm{mod }p)$ has two solutions when $p$ is prime. (Technically, you could prove this, too, but don't.)
- For #12, the authors omitted an important point: $n$ must be odd!.
- For #13, the solutions manual tries “$n=13$ since this is Example 13.” (I think they meant Exercise 13, but the number's cursed, so you can't blame them.)
That seems a curious choice, since they say “to show that a given number $n$ is not prime,”
I suspect they're referring to the Solovay-Strassen test there, rather than what they want you to do.
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| 19 2124 March |
p. 253 #1-4, 7, 12 |
| 5 March |
p. 203 #1-6, 11, 16, 17, 20 |
| 17 February |
The Luhn algorithm |
| 10 February |
p. 191 #1-4, 10
Hints on #2: The number of ways to get \(t\) errors in \(m\) signals
is \(\binom{m}{t}\), so probability \(P_t\) of getting \(t\) errors
in \(m\) signals is \(\binom{m}{t}p^{m-t}(1-p)^t\). You can solve
the problem by showing that \[\frac{P_t}{P_{t-1}}<1.\]
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| 27 January |
Handout on finite fields
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