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%\begin{document}

\title{Integration workshop Projects}

%\author{Shankar C. Venkataramani}

\date{\today}

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\maketitle

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 \section{Lie Brackets and parallel parking a car} 
  
   $U \subseteq \mathbb{R}^n$ is an open set. A {\em smooth vector
     field} on $U$ is a smooth mapping $v: U \rightarrow \mathbb{R}^n$. 
  
   A {\em smooth derivation} on $C^{\infty}(U)$ is a mapping $\delta:
   C^{\infty}(U) \rightarrow C^{\infty}(U)$ that
   satisfies the following properties:
 \begin{align*}
 \delta(f + \lambda g) & = \delta(f) + \lambda \delta(g) \\
  \delta( f g) & = f \delta(g) + g \delta(f)
 \end{align*}
 for all $f,g \in C^{\infty}(U), \lambda \in \mathbb{R}$.
 
 (a) Show that there is a one-to-one correspondence between smooth
 vector fields and smooth derivations. (Hint: Consider $\delta_v(f) = v
 \cdot \nabla(f)$. What is the identification in the opposite direction?)

 (b) If $\delta_1$ and $\delta_2$ are smooth derivations, show that
 their commutator, that is the map
 $$
 f \mapsto \delta_1(\delta_2(f)) - \delta_2(\delta_1(f))
 $$
 is also a smooth derivation.

 (c) Given two smooth vector fields $u,v$, using the identification
 from part (a), and the result from (b), we can construct a new smooth
 vector field that corresponds to the commutator. This operation is
 called the {\em Lie Bracket} and is conventionally denoted by $[u,v]$,
 {\em i.e.}
 $$
 \delta_{[u,v]}(f) = \delta_u (\delta_v(f)) - \delta_v(\delta_u(f))
 $$
 Express the vector $[u,v]$ directly in terms of the vectors $u$ and
 $v$.

 (d) Associated with every vector field $v$ is a dynamical system given
 by
 $$
 \frac{d x}{dt} = v(x)
 $$
 What is the interpretation for the dynamical system corresponding
 to the Lie bracket of two vector fields.

 (optional) A simple model for the motion of a car is as follows:
 \begin{enumerate}
 \item The state of the car is described by a triple $(x,y,\theta)$
   where $(x,y) \in \mathbb{R}^2$ give the location of the center of
   the car, and $\theta \in S^1$ is an angle that describes the
   orientation of the car.
 \item For simplicity, we assume that the car can only turn at a fixed
   rate. The trajectory of the car in $(x,y)$ is differentiable, and
   piecewise smooth. Each smooth piece being either a straight line
   along the orientation of the car (no turning), or circular arcs with
   a fixed radius $a$ (turning to the right or the left at a constant
   rate). The trajectory $\theta(t)$ is piecewise linear, with $\theta'
   = 0$ for no turning, or $\theta' = \pm \beta$ otherwise.
 \end{enumerate}

 (e) Determine the vector fields that correspond to the three possible
 motions of the car. Show that, it is indeed possible to parallel park
 a car with only these three allowed motions. 
 
 (f) Assuming that the car has only two allowed motions, namely turning to the left, or turning to the right with a fixed turning radius $a$, is it still possible to move in (roughly) a straight line in a given direction? Is it possible to parallel park this car?


%% \section{Solving elliptic PDEs} 

%% A Hilbert space $H$ is a complete, normed linear space whose norm is
%% induced by an inner product $\langle \cdot, \cdot \rangle$. We shall
%% denote the norm by $\|u\| = \sqrt{\langle u, u \rangle}$.

%% (a) If $f : H \rightarrow \mathbb{R}$ is a continuous linear
%% functional, show that there is a unique element $u \in H$ such that
%% $$
%% f(v) = \langle u, v\rangle
%% $$
%% for all $v \in H$. (This is the Riesz representation theorem for
%% Hilbert spaces, and you don't need any measure theory to prove this
%% special case)

%% A bilinear form on $H$ is a real valued function that takes two
%% arguments (inputs) from $H$ and is separately linear in each argument.
%% Clearly, the inner product is a bilinear form, but there are many
%% other bilinear forms. A natural question is whether we can extend the
%% above conclusion to general bilinear forms. As we see below, this idea
%% is particularly fruitful for solving elliptic PDEs.

%% A bilinear form $a(\cdot,\cdot)$ on a Hilbert space $H$ is {\em
%%   continuous} (or bounded) if there exists a $C < \infty$ such that
%% $$|a(u,v)| \leq C \|u\| \|v\|.$$ 
%% The bilinear form is {\em coercive} on a subspace $V \subseteq H$ if
%% $\exists \alpha > 0$ such that
%% $$
%% a(v,v) \geq \alpha \|v\|^2 \quad \forall v \in V
%% $$

%% The inner product is clearly both continuous and coercive on $H$.

%% (b) If $a(\cdot,\cdot)$ is a {\em symmetric} bilinear form that is
%% continuous and coercive on a closed subspace $V \subseteq H$ and $f :
%% H \rightarrow \mathbb{R}$ is a continuous linear functional, show
%% that, there exists a $u \in V$ (note $V$ and not $H$) such that
%% $$
%% f(v) = a(u,v) \quad \forall v \in V
%% $$
%% (Hint: Show that $(V,a(\cdot,\cdot))$ is a Hilbert space.)

%% By choosing $V$ as a {\em finite dimensional} subspace of a
%% potentially infinite dimensional space $H$, this idea allows one to
%% approximate the solutions of a PDE by solutions of matrix equations,
%% {\em i.e.} linear algebra. This is the idea that underlies {\em Finite
%%   Element methods} for numerically solving PDEs on a computer.

%% (c) Show that for all smooth functions $g: [0,1] \rightarrow
%% \mathbb{R}$, the mixed boundary value problem
%% $$
%% -u'' + u = g, \quad \quad u(0) = 0, u_x(0) = 0
%% $$
%% has a unique solution. 

%% You can use the fact that the completion of $C^\infty{[0,1]}$ in the
%% norm given by $\|v\|^2 = \int_0^1 [v^2 + (v')^2]$ is a Hilbert space,
%% and all the elements of this space are continuous functions.

%% (d) Can you prove the existence of a solution using an appropriate
%% Fourier type basis? 

%% %\item{Vitali's covering lemma and the Hardy-Littlewood Maximal function}

%% \section{Partition of unity}
  
%% (a) Show that there is a smooth function $f : \mathbb{R} \rightarrow
%%   \mathbb{R}$ that satisfies:
%% \begin{enumerate}
%% \item $f(x) = 1$ for $x \in [-1,1]$.
%% \item $0 \leq f(x) \leq 1$ for all $x \in \mathbb{R}$.
%% \item $f(x) = 0$ for $|x| \geq 2$.
%% \end{enumerate}

%% (b) $W \subseteq \mathbb{R}^n$ is an open set and $\{U_{\alpha}\}$ is
%% a cover of $W$. Show that there is a {\em countable} collection of
%% smooth, non-negative functions $\psi_k$ such that:
%% \begin{enumerate}
%% \item For each $k$, there is an $\alpha$ such that the {\em support}
%%   of $\psi_k$ is contained in $U_{\alpha}$.
%% \item For each point $x \in W$, only finitely many $\psi_k(x)$ are
%%   non-zero, and 
%% $$
%% \sum_k \psi_k(x) = 1 \quad \quad \mbox{ for all } x \in W.
%% $$
%% \end{enumerate}

%% (Optional) I am in the process of making a problem that uses this construction.

%% %\item{Representations of SL2 and Invariants}

%% \section{Construction of Brownian motion, continuity of sample paths}
  
%%   The {\em Wiener process} $W(t)$ is a mathematical idealization of
%%   Brownian motion and is defined by the following requirements.
 
%% \begin{enumerate}
%% \item $W(0) = 1$ almost surely.
%% \item For $t > s$, $W(t) - W(s)$ is
%%   Normally distributed with variance $t - s$.
%% \item If $0 < t_1 < t_2 < \ldots < t_n$, the {\em increments}
%%   $W(t_1)$, $W(t_2) - W(t_1), \ldots, W(t_n) - W(t_{n-1})$ are
%%   independent.
%% \end{enumerate}

%% (a) Show that $E[W(t)] = 0$ and $E[W^2(t)] = t$.

%% The {\em Haar functions} $h_k$ are defined by
%% \begin{align*}
%%  h_0(t) & = 1 \quad  \mbox{ for } 0 \leq t \leq 1 \\
%% h_1(t) & = \begin{cases} 1 & \mbox{ for } 0 \leq t \leq 1/2 \\ 
%% -1 & \mbox{ for } 1/2 < t \leq 1 \end{cases}\\
%% \mbox{ If $2^n \leq k < 2^{n+1}$ } \quad  h_k(t) & =  \begin{cases} 2^{n/2} & \mbox{ for } 2^{-n} k -1  \leq t \leq 2^{-n} k + 2^{-n -1} -1 \\ 
%% -2^{n/2} & \mbox{ for } 2^{-n} k + 2^{-n -1} -1  <  t \leq 2^{-n} k + 2^{-n} -1 \\ 
%% 0 & \mbox{ otherwise } \end{cases}
%% \end{align*}

%% (b) Show that the Haar functions are an orthonormal family for the
%% inner product
%% $$
%% (f,g) = \int_0^1 f(x) g(x) dx
%% $$

%% (c) The {\em Schauder functions} $s_k$ are defined by
%% $$
%% s_k(t) = \int_0^t h_k(\tau) d \tau \quad \quad \mbox{ for } 0 \leq t \leq 1
%% $$
%% Show that each $s_k$ is non-negative and is bounded by
%% $\min(1,\sqrt{2 k^{-1}})$.

%% (d) $\phi$ is a smooth function defined on $[0,1]$. If $(\phi,h_k) =
%% 0$ for all $k$, show that $\phi \equiv 0$ on $[0,1]$. This implies
%% that the $h_k$ form a {\em complete orthonormal basis} of $L^2([0,1])$.
%% (Hint: Integrate by parts for $k \geq 1$, and use the positivity of
%% $s_k$)

%% (e) If $a_k$ is a real valued sequence such that $|a_k| \leq C_1 + C_2
%% k^\delta$ for some constants $C_1,C_2$ and for some $0 \leq \delta <
%% 1/2$, show that the series 
%% $$
%% \sum_{k = 0}^\infty a_k s_k(t)
%% $$
%% converges uniformly to a continuous function of $[0,1]$.

%% (f) If $A_k$ is a sequence of independent normal variables with unit
%% variance, show that the ``Random'' series $\sum_{k = 0}^{\infty} A_k
%% s_k(t)$ converges to a Wiener process $W(t)$.

%% This procedure is the {\em Levy -- Ciesielski} construction of the
%% Wiener process.

%% In your proof, you can use the following fact:

%% If $A_k$ is a sequence of independent normal variables with unit
%% variance, then with probability 1, there exists a $K$ such that for
%% all $k \geq K$, $|A_k| \leq 4 \sqrt{\log k}$. 

%% (Optional) Prove this fact. (Hint: Borel-Cantelli lemma).
\end{document}
