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%==================== END PREAMBLE ==============
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\vspace*{-50pt}
{\bf  Math 361, Fall 2015\hfill Jerry L. Kazdan}
\medskip

\begin{center}
{\large\bf Problem Set 4}
\smallskip

%\fbox{{\sc \large Draft}}
\end{center}

%\comments{
{\sc Due:} Thurs. Sept.~24 in class. [Late papers will be accepted (without penalty) until 1:00 PM Monday.]
%}
\medskip

 Please carefully read all of Chapter 6 in the Marsden-Hoffman text. 
\medskip

The following short answer problems from Marsden-Hoffman are not
assigned, but you should know how to do them.  Some short answer
problems will be on our exams.

\medskip
p.~330 \#4, 5,\qquad p.~334 \#2, 3, 4,\qquad p.~344 \#1, 4,\qquad
p. 348 \#2, 5,\qquad p.~352 \#3, 4, p.~384~\#4

\bigskip

\begin{problems}

\item If the (possibly complex) sequence $\{a_n\}$ is bounded and $c>1$,
show that the series $\displaystyle\sum_{n=1}^{\infty} \frac{a_n}
{n^z}$ converges absolutely and uniformly for all complex $z=x+iy$ in
the closed half-plane $c\le x < \infty$.

\item Let $f\in C([0.\,2\pi])$.  Show that 
$\displaystyle \lim_{n\to\infty}\int_0^\pi f(x)\sin nx\,dx =0$.  

[There are several ways to do this.  One is with your bare hands.
Another is to approximate $f$ uniformly by a ``simpler'' function.]

\item  Say you have $k$ linear algebraic equations
in $n$ variables;  in matrix form we write $AX=Y$.  Give a proof or
counterexample for each of the following.
 \begin{parts}
\item If $n=k$ there is always {\it at most one\/} solution.
\item If $n>k$ you can {\it always\/} solve $AX=Y$.
\item If $n>k$ the nullspace of $A$ has dimension greater than zero.
\item If $n<k$ then for {\it some\/} $Y$ there is \emph{no} solution of $AX=Y$.
\item If $n<k$ the {\it only\/} solution of $AX=0$ is $X=0$.
\end{parts}

\item Let $L$, $M$, and $N$ be linear maps from the (two dimensional) plane
to the plane given in terms of the standard ${\bf i,\,j}$ basis
vectors by:

{\parindent 30pt $L{\bf i} = {\bf j}$,\quad  $L{\bf j} = -{\bf i}$ \
(rotation by 90 degrees counterclockwise)

$M{\bf i} = -{\bf i}$,\quad  $M{\bf j} = {\bf j}$ \
(reflection across the vertical axis)

$NV = - V$\   (reflection across the origin)
}
\begin{parts}
\item Draw pictures describing the actions of the maps $L, M$, and $N$
and the compositions: $LM,\ ML,\ LN,\ NL,\ MN$, and $NM$.
\item Which pairs of these maps commute?
\item Which of the following identities are correct---and why?

{\parindent 10pt
\begin{tabular}{lclclclc}
1) & $L^2  =  N$ \qquad      & 2) & $N^2  =  I$  \qquad     &
3) & $L^4  =  I$  \qquad     & 4) & $L^5  =  L$  \qquad  \\
5) & $M^2  =  I$  \qquad     & 6) & $M^3  =  M$  \qquad     &
7) & $MNM  =  N$  \qquad     & 8) & $NMN  =  L$  \qquad
 \end{tabular}
}
\item Find matrices representing each of the linear maps $L,\, M$, and $N$.
\end{parts}

\item Linear maps $F(X) = AX$, where $A$ is a matrix, have the
property that
$F(0) = A0 = 0$, so they necessarily leave the origin fixed.  It is
simple to extend this to include a translation,
$$
F(X) = V + AX,
$$ 
where $V$ is a vector. These are called \emph{affine maps}.  Note
that $F(0) = V$.

Find the vector $V$ and the matrix $A$ that describe each of the
following mappings [here the light blue $F$ is mapped to the dark red $F$].
%\kern 0.1in \raise-3.0cm

\qquad a). \vbox{\scalebox{0.9}{\includegraphics{affine1}}
\qquad b). \hbox{\scalebox{0.9}{\includegraphics{affine2}}}}
\medskip

\quad c).\vbox{\scalebox{0.9}{\includegraphics{affine3}}
\qquad \qquad d).\hbox{\scalebox{0.9}{\includegraphics{affine4}}}}

\newpage
\item Let $A(t)$ be an invertible $n\times n$ matrix whose elements
  are smooth functions of $t\in \R$.  Using the definition of the
  derivative as
$$
\lim_{t\to t_0}\frac{A(t)-A(t_0)}{t-t_0},
$$
show that the inverse, $A^{-1}$, is also differentiable and give a
formula for it in terms of $A^{-1}(t)$ and $A'(t)$. 

{\sc Suggestion:}
First do the special case $n=1$.  Also, don't assume that $A(t)$ and
$A(t_0)$ commute.  Key step: for invertible matrices $A$, $B$, verify
and use
$$
A^{-1} - B^{-1}= A^{-1}(B - A)B^{-1}.
$$


\item Let $f(t):\R\to \R$ be any smooth function.  For any real 
numbers $a$ and $b$ let $u(x,y)=f(ax+by)$.  Show that \begin{parts}
\item $bu_x(x,y) -au_y(x,y)=0$.
\item 
$u_{xx}u_{yy} - (u_{xy})^2 = 0$.
\end{parts}


\item (Euler) [Marsden-Hoffman, p.~385 \#12] Let $f(x)$ be a
  differentiable function of $x\in\R^n$.  If $f$ is {\it homogeneous
    of degree k} in the sense that $f(cx) = c^kf(x)$ for all $c>0$,
  show that $x\cdot \nabla f(x) = kf(x)$.

\item Let $p>1$ and define $q$ by $1/p + 1/q = 1$.  Prove {\bf Young's
    inequality}: for all real $x\ge 0$ and $y\ge 0$:
$$
xy \le \frac{x^p}{p} + \frac{y^q}{q}.
$$
[{\sc Remark:} There are many (elementary) ways to show this; none of
them are completely obvious. In one approach, fix $y=a>0$ and find the
maximum of $g(x):=ax-x^p/p$].

\item Let $x=(x_1,\ldots,x_n)\in\R^n$ and assume that $u(x)$ depends
only on the distance to the origin,
$r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}$, so $u(x)=f(r)$ for some function
$f$ depending only on $r$.
\begin{parts}
\item Show that
$\displaystyle \frac{\partial u}{\partial x_i}=\frac{x_i}{r}\frac{df}{dr}$.
\item Show that $\displaystyle \frac{\partial^2 u}{\partial
x_i^2}=\frac{d^2f}{dr^2}\left(\frac{x_i^2}{r^2}\right) +
\frac{df}{dr}\left(\frac{1}{r}-\frac{x_i^2}{r^3}\right)$.
\item Compute $\displaystyle \Delta u :=
\frac{\partial^2u}{\partial x_1^2} + \frac{\partial^2u}{\partial x_2^2}+
\ldots+\frac{\partial^2u}{\partial x_n^2}
$
\quad in terms of $f$ and its derivatives.
\item If $n=3$, use this to find all functions $u(x)=f(r)$ that satisfy
$\Delta u=0$ for all $x\ne 0$.
\end{parts}


\end{problems}

\comments{
\vskip 20pt

\begin{center}{\large \bf Bonus Problem}
			
[Please give this directly to Professor Kazdan]
\end{center}

\begin{problems}

\item [B-1] Let $f(x)$ be a continuous function for $0\le x \le 1$.
Evaluate \ $\ \displaystyle  \lim_{n\to \infty} \int_0^1
nf(x)x^n\,dx.$  (Justify your assertions.)


 \end{problems}
}


\medskip
[Last revised: \today]
\end{document}

