Preliminary Exam - Fall 1978

Problem 1   Let $f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ satisfy $f(x)\leq f(y)$ for $x\leq y$. Prove that the set where $f$ is not continuous is finite or countably infinite.

Problem 2   Let $\{g_n\}$ be a sequence of Riemann integrable functions from $[0,1]$ into $\mathbb{R}^{}$ such that $\vert g_n(x)\vert\leq 1$ for all $n,x$. Define

$$G_n(x) = \int_0^xg_n(t)\,dt\,.$$

Prove that a subsequence of $\{G_n\}$ converges uniformly.

Problem 3   Let $M_{n\times n}$ denote the vector space of $n\times n$ real matrices (identified with $\mathbb{R}^{n^2}$). Prove that there are neighborhoods $U$ and $V$ in $M_{n\times n}$ of the identity matrix such that for every $A$ in $U$, there is a unique $X$ in $V$ such that $X^4 = A$.

Problem 4   Evaluate

$$\int_0^{2\pi}\frac{d\theta}{1-2r\cos\theta + r^2}$$

where $r^2 \neq 1$.

Problem 5   Let $f(z) = a_0 + a_1z + \cdots + a_nz^n$ be a complex polynomial of degree $n > 0$. Prove

$$\frac{1}{2\pi i}\int_{\vert z\vert = R}z^{n-1}\vert f(z)\vert^2\,dz = a_0 \overline{a}_nR^{2n}.$$

Problem 6   Solve the differential equation

$$\frac{dy}{dx} = x^2y-3x^2, \hspace{.2in} y(0)=1.$$

Problem 7   Let $H$ be a subgroup of a finite group $G$.

1. Show that $H$ has the same number of left cosets as right cosets.

2. Let $G$ be the group of symmetries of the square. Find a subgroup $H$ such that xH $\neq$ Hx for some x.

Problem 8   Let $M$ be the $n\times n$ matrix over a field $\mbox{\bf {F}}$, all of whose entries are equal to $1$.

1. Find the characteristic polynomial of $M$.
2. Is $M$ diagonalizable?
3. Find the Jordan Canonical Form of $M$ and discuss the extent to which the Jordan form depends on the characteristic of the field $\mbox{\bf {F}}$.

Problem 9   For $x, y \in \mbox{$\mathbb{C}\,^{n}$}$, let $\langle x, y \rangle$ be the Hermitian inner product $\sum_j x_j \overline{y}_j$. Let $T$ be a linear operator on $\mathbb{C}\,^{n}$ such that $\langle Tx, Ty \rangle = 0$ if $\langle x, y \rangle = 0$. Prove that $T = kS$ for some scalar $k$ and some operator $S$ which is unitary: $\langle Sx, Sy \rangle = \langle x, y \rangle$ for all $x$ and $y$. _matrix,>unitary

Problem 10   How many homomorphisms are there from the group
$\mbox{$\mathbb{Z}^{}$}_2 \times \mbox{$\mathbb{Z}^{}$}_2$ to the symmetric group on three letters?

Problem 11   Let $W\subset \mbox{$\mathbb{R}^{n}$}$ be an open connected set and $f$ a real valued function on $W$ such that all partial derivatives of $f$ are $0$. Prove that $f$ is constant.

Problem 12   Let $f:\mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{}$}$ have the following properties: $f$ is differentiable on $\mbox{$\mathbb{R}^{n}$} \setminus \{0\}$, $f$ is continuous at $0$, and

$$\lim_{p\to 0}\frac{\partial f}{\partial x_i}(p) = 0$$

for $i = 1, \ldots, n$. Prove that $f$ is differentiable at $0$.

Problem 13  

1. Show that if $u,v:\mbox{$\mathbb{R}^{2}$}\to\mbox{$\mathbb{R}^{}$}$ are continuously differentiable and $\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$, then $u=\frac{\partial f}{\partial x}$, $v=\frac{\partial f}{\partial y}$ for some $f:\mbox{$\mathbb{R}^{2}$}\to\mbox{$\mathbb{R}^{}$}$.

2. Prove there is no $f:\mbox{$\mathbb{R}^{2}$}\setminus\{0\}\to\mbox{$\mathbb{R}^{}$}$ such that
   
   $$\frac{\partial f}{\partial x}= \frac{-y}{x^2+y^2}\quad and \quad \frac{\partial f}{\partial y}= \frac{x}{x^2+y^2} \cdot$$
   
   

Problem 14   Give examples of conformal maps as follows:

1. from $\{z \;\vert\; \vert z\vert < 1\}$ onto $\{z \; \vert\; \Re z < 0\}$,
2. from $\{z \; \vert\; \vert z\vert < 1\}$ onto itself, with $f(0)=0$ and $f(1/2)=i/2$,
3. from $\{z \;\vert\; z \neq 0, 0<\arg z <\frac{3\pi}{2}\}$ onto $\{z \; \vert\; z \neq 0, 0< \arg z<\frac{\pi}{2}\}$.

Problem 15   Suppose $h(z)$ is analytic in the whole plane, $h(0)=3+4i$, and $\vert h(z)\vert\leq 5$ if $\vert z\vert< 1$. What is $h'(0)$?

Problem 16   Which pairs of the following matrices are similar?

$$\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right),... ...\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right),$$

$$\left( \begin{array}{cc} 1 & 0 \\ 1 & -1 \end{array} \right)... ... \left( \begin{array}{cc} 1 & 5 \\ 0 & 1 \end{array} \right).$$

Problem 17   Find all automorphisms of the additive group of rational numbers.

Problem 18   Prove that every finite multiplicative group of complex numbers is cyclic.

Problem 19   Let

$$A=\left( \begin{array}{cc} 1 & 2 \\ 1 & -1 \end{array} \right).$$

Express $A^{-1}$ as a polynomial in $A$ with real coefficients.

Problem 20   Let $M_{n \times n}$ be the vector space of real $n\times n$ matrices, identified with $\mathbb{R}^{n^2}$. Let $X \subset \mbox{$M_{n \times n}$}$ be a compact set. Let $S \subset \mbox{$\mathbb{C}\,^{}$}$ be the set of all numbers that are eigenvalues of at least one element of $X$. Prove that $S$ is compact.