Riemann surface

matlab

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1.1. Problems. #0: Write a program to generate Pascal’s triangle modulo 2. How far can you go? Can you use the symmetries to compute it quickly? You do not need to hand this in. From the textbook: #1: Exercise 1.7.4 (there are many trig tables online: see for example http://www.sosmath. com/tables/trigtable/trigtable. html), and read BUT DO NOT DO Exercise 1.7.5. #2: Exercise 1.7.7. #3: Exercise 1.7.18. #4: Exercise 1.7.34. #5: Exercise 1.7.26. #5: Exercise 1.7.36.

1.2. Solutions.

#1: Exercise 1.7.4. If we know the values of either $\sin (x)$ or $\cos (x)$ for $0 \leq x \leq \pi / 4$ (or from 0 to 45 degrees if you prefer not to work in radians) then we can find the values of all trig functions using basic relations (such as $\sin (x+\pi / 2)=\cos (x))$. Create a look-up table of $\sin (x)$ by finding its values for $x=\frac{k}{45} \frac{\pi}{4}$ with $k \in{0,1, \ldots, 45}$. Come up with at least two different ways to interpolate values of $\sin (x)$ for $x$ not in your list, and compare their accuracies. For which values of $x$ are your interpolations most accurate?

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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