# Homework 6
### Due 12/17 (Tuesday of Finals week)
1. Prove that the Legendre polynomials normalized to have $P_n(1)=1$ satisfy the *Rodridgues Formula* $$p_n(x)=\frac{1}{2^nn!}\frac{d^n}{dx^n} (x^2-1)^n.$$
Use this to show that they satisfy the *Legendre Differential Equation* $$[(x^2-1)u']'=\lambda u,$$
with $\lambda=n(n+1)$.
2. Suppose $p_n(x)$ are monic OPRL on a finite interval $(-a,a)$ with even weight function $w(x)=w(-x)$. Show that the Jacobi coefficients $a_n=0$ for all $n$.
3. (Exercise 8.7, Trefethen) The function $|x−i|$ is real analytic for $x∈[−1,1]$. This means it can be analytically continued to an analytic function $f(x)$ in a neighborhood of $[−1,1]$ in the complex $x-$plane. The formula $|x−i|$ itself does not define an analytic function in any complex neighborhood. Find another formula for $f$ that does, and use it to explain what singularities $f$ has in the complex plane.
4. (Exercise 8.10, Trefethen) Suppose we wish to approximate $f(x) =1/x$ on the interval $[m,M]$ with $0< m < M$. Show that for any $κ < M/m$, there exist polynomials $p_n∈P_n$ such that $‖f−p_n‖=O((1 + 2/√κ)^{−n})$ as $n→∞$, where $‖·‖$ is the ∞-norm on $[m,M]$. This result is famous in numerical linear algebra as providing an upper bound for the convergence of the conjugate gradient iteration applied to a symmetric positive definite system of equations $Ax=b$ with condition number $κ$.
5. Exercise 4.7, Trefethen http://www.chebfun.org/ATAP/atap-first6chapters.pdf
6. Make a 2-3 page typed cheat sheet summarizing what you consider the important topics and ideas covered in this class, and the connections between them. Keep a copy for future reference.

The official schedule is posted at https://www.jointmathematicsmeetings.org/meetings/national/jmm2023/2270_program_ss94.html#title The Zoom link for virtual attendance/virtual talks is: https://berkeley.zoom.us/j/94982941411?pwd=RWRXeVdoZUNVTmdqK1FIMksrZFllZz09 Friday, January 6 (101 Hynes Convention Center) 1-2pm: Matt Colbrook, The foundations of infinite-dimensional spectral computations (Zoom) 2-3pm: Josué Tonelli-Cueto, Condition-based Low-Degree Approximation of Real Polynomial Systems

1/3/2023Lecture 1: Propositional Logic A proposition is a declarative sentence that is either true or false but not both. A proposition has a truth value which is either $T$ or $F$. A letter used to denote a proposition is called a propositional variable. If $p$ is a proposition, the negation of $p$, denoted $\lnot p$, is the proposition "it is not the case that $p$". The truth value of $\lnot p$ is the opposite of that of $p$. If $p$ and $q$ are propositions, the conjunction of $p$ and $q$ (denoted $p\land q$) is the proposition "p and q." Its truth value is $T$ when both $p$ and $q$ are $T$, and it is $F$ otherwise. If $p$ and $q$ are propositions,he disjunction of $p$ and $q$ is the proposition "p or q". Its truth value is $T$ when at least one of $p$ or $q$ is $T$, and $F$ if both $p$ and $q$ are $F$. If $p$ and $q$ are propositions, the conditional $p\rightarrow q$ is the proposition "if $p$ then $q$". Its truth values are given by the following truth table (which lists its truth values in terms of those of $p$ and $q$): [see Table 5 in Rosen] The converse of $p\rightarrow q$ is $q\rightarrow p$. A compound proposition consists of logical operations (the four listed above) applied to propositions or propositional variables, possibly with parentheses. The truth value of a compound proposition can be mechanically determined given the truth values of its constituents.

12/12/2022Lecture 14 Basic Terminology, Handshaking Theorem A graph is a pair $G=(V,E)$ where $V$ is a finite set of vertices and $E$ is a finite multiset of $2-$element subsets of $V$, called edges. If $E$ has repeated elements $G$ is called a multigraph, otherwise it is called a simple graph. We will not consider directed graphs or graphs with loops (edges from a vertex to itself). Two vertices $x,y\in V$ are adjacent if ${x,y}\in E$. An edge $e={x,y}$ is said to be incident with $x$ and $y$. The degree of a vertex $x$ is the number of edges incident with it. Theorem. In any graph $G=(V,E)$, $$ \sum_{v\in V} deg(v) = 2|E|.$$ Proof. Let $I={(e,v):e\in E,v\in V,\textrm{$e$ is incident with $v$ in $G$}}$ be the set of edge-vertex incidences in $G$. We will count the number of incidences in two ways: (i) Observe that every edge participates in exactly two incidences, and no incidence participates in more than one edge. Thus, the total number of incidences is $|I|=2|E|$.

12/12/2022Prove that every simple graph with $n$ vertices and $k$ edges has at least $n − k$ connected components (hint: induction, or contradiction). Let $p\ge 3$ be a prime. Consider the graph $G=(V,E)$ with $V={0,1,2,\ldots, (p-1)}$ and $$E={{x,y}:x-y\equiv 2(mod p)\lor x-y\equiv -2 (mod p)}$$ Show that $G$ is connected. Does $G$ have an Euler circuit? Prove your answer. Let $n\ge 1$ be an integer and let $k\le n/2$. Consider the graph $G=(V,E)$ with $V={S\subseteq {1,2,\ldots,n}: |S|=k}$ and $$E = {{S,T}:|(S-T)\cup (T-S)|=1}.$$ What are the degrees of the vertices in $G$? Is $G$ connected? For which values of $n$ and $k$ does $G$ have an Eulerian circuit? For which values of $n$ and $k$ is $G$ $2-$colorable? Prove your answers.

11/8/2022
Published on ** HackMD**

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