Riemann surface
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#2: Exercise 1.7.7. Prove the five log laws. For example, for the first we have $\log _b x_i=y_i$, so $x_i=b^{y_i}$. Thus $x_1 x_2=b^{y_1} b^{y_2}=b^{y_1+y_2}$. By definition, we now get $\log _b\left(x_1 x_2\right)=y_1+y_2$, which finally yields $\log _b\left(x_1 x_2\right)=$ $\log _b x_1+\log _b x_2$.

Solution: (2) If $\log _b x=y$ then $x=b^y$ so $x^r=b^{r y}$ and thus $\log _b\left(x^r\right)=r y=r \log _b x$. (3) Take the logarithm base $b$ of $b^{\log _b x}$ and use (2). (4) follows from (1) applied to $x_1 x_2^{-1}$ and then using (2) for $x_2^{-1}$. (5) Is the most interesting. Let $\log _c x=y$ so $x=c^y$. As $b=c^{\log _c b}$ by $(3)$, we find
$$b^{\log _b x}=c^{\log _b x \log _c b} \Rightarrow \log _c b \log _b x=\log _b x \log _c b \log _c c$$
by taking logs of both sides and using (3); the result follows from division and noting $\log _c c=1$.

3.1. Problems. Due Friday, Oct 3, 2016: #0: Make sure you can do, but do not submit, the problems in the “Real Analysis Review” of Chapter 4. #1: Exercise 4.7.7: Prove that any cubic $a x^3+b x^2+c x+d=0$ can be written as $x^3+p x+q=0$ (i.e., we can rewrite so that the coefficient of the $x^2$ term vanishes and the coefficient of the $x^3$ term is 1); this is called the depressed cubic associated to the original one. (For fun see the next problem on how to solve the cubic.) #2: Exercise 4.7.12. Can you construct a canonical linear programming problem that has exactly two feasible solutions? Exactly three? Exactly $k$ where $k$ is a fixed integer? #3: Exercise 4.7.14. Find a continuous function defined in the region $(x / 2)^2+(y / 3)^2<1$ (i.e., the interior of an ellipse) that has neither a maximum nor a minimum but is bounded. #4: Exercise 5.4.3. Imagine we want to place $n$ queens on an $n \times n$ board in such a way as to maximize the number of pawns which can safely be placed. Find the largest number of pawns for $n \leq 5$. #5: Exercise 5.4.4. Write a computer program to expand your result in the previous problem to as large of an $n$ as you can. Does the resulting sequence have any interesting problems? Try inputting it in the OEIS. #6: Consider the problem of placing $n$ queens on an $n \times n$ board with the goal of maximizing the number of pawns which may safely be placed. For each $n$, let that maximum number be $p(n)$. Find the best upper and lower bounds you can for $p(n)$. For example, trivially one has $0 \leq p(n) \leq n^2$; can you do better? #7: Exercise 5.4.26: Prove $\frac{1}{\sqrt{N}} \sum_{n=-\infty}^{\infty} e^{-\pi n^2 / N}=\sum_{n=-\infty}^{\infty} e^{-\pi n^2 N}$. As $N$ tends to infinity, bound the error in replacing the sum on the right hand side with the zeroth term (i.e., taking just $n=0)$. Hint: the Fourier transform of a Gaussian is another Gaussian; if $f(x)=e^{-a x^2}$ then $\hat{f}(y)=\sqrt{\pi / a e^{-\pi^2} y^2 / a}$.

Solution: Implicit in the above is that $a \neq 0$, as if it did we would not have a cubic but a quadratic; this is the old rectangle-square debate… If $a \neq 0$ we may divide both sides by $a$ and thus may assume the coefficient of $x^3$ is 1 . We now change variables and let $x=x-b / 3$. This sends $x^3$ to $x^3-b x^2+b^2 x / 3-b^3 / 27$, and $b x^2$ to $b x^2-2 b^2 x / 3+b^3 / 9$ (the other terms are lower order and don’t involve $x^2$ ); note the coefficient of the $x^2$ term is now zero.
#2: Exercise 4.7.12. Can you construct a canonical linear programming problem that has exactly two feasible solutions? Exactly three? Exactly $k$ where $k$ is a fixed integer?

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