# Lecture 10: Group Work
Form groups of three people, ideally strangers from different departments, and solve the following problems in order.
1. Recall the Volterra operator:
$$(Vf)(x):=\int_0^x f(y)dy$$
on $L^2[0,1]$.
Compute the singular value decomposition of $V$. What is its norm? Is it trace-class? Does it have a bounded inverse?
2. Show that a general second order ODE
$$ a_2(x)u''(x)+a_1(x)u'(x)+a_0(x)u(x)=f(x)$$
with $a_2(x)\neq 0, u\in C^2[a,b]$
has the same solution space as one of the form
$$ -(p(x)u'(x))' + q(x)u(x) = F(x)$$
This is called *Sturm-Liouville Form*. How does this transformation affect the boundary conditions?
3. Show that a general Sturm-Liouville eigenvalue equation
$$ -(p(x)u'(x))' +q(x)u(x) = \lambda u(x),$$
$p(x)\neq 0$, can be converted to *Schrodinger form* (which we studied in the previous lecture)
$$ -U''(s)+Q(s)U(s)=\lambda U(s)$$
via the change of variables (i.e., diffeomorphism):
$$\frac{dx}{ds}=\sqrt{p(x)},\quad f(s):=(p(x(s)))^{1/4},\quad Q(s)=q(x(s))+\frac{f''(s)}{f(s)}.$$
How does this affect the boundary conditions and domain of definition of U?
Put the Kimura equation
$$-x(1-x)u''(x)+\lambda u(x)$$
in Schrodinger form.
After the class, get one person from each group to (1) write down the names of the group members (2) comment on anything you thought was interesting / were confused about.
## Group Summaries
1. Dipti Jasrasaria, Paul Wrona, Sam Olivier, Nicole Farias, Sarvesh Sadana, Jack Spilecki, Gang Yang, Jonathan Liu, Mathias Palmstroem, Shizhe Liu (we all worked together) - we spent most of our time figuring out problem one
2.
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7.

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: JosuĂ© 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
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