Preliminary Exam - Summer 1983

Problem 1   The number $21982145917308330487013369$ is the thirteenth power of a positive integer. Which positive integer?

Problem 2   Let $f:\mbox{$\mathbb{C}\,^{}$} \to \mbox{$\mathbb{C}\,^{}$}$ be an analytic function such that

$$\left(1+\vert z\vert^k\right)^{-1} \frac{d^mf}{dz^m}$$

is bounded for some k and m. Prove that $d^nf/dz^n$ is identically zero for sufficiently large n. How large must n be, in terms of k and m?

Problem 3   Let $A$ be an $n\times n$ complex matrix, and let $\chi$ and $\mu$ be the characteristic and minimal polynomials of $A$. Suppose that

$$\chi(x) = \mu(x)(x-i),$$

$$\mu(x)^2 = \chi(x)(x^2+1).$$

Determine the Jordan Canonical Form of $A$.

Problem 4   Outline a proof, starting from basic properties of the real numbers, of the following theorem: Let $f:[a,b] \to \mbox{$\mathbb{R}^{}$}$ be a continuous function such that $f'(x)=0$ for all $x\in (a,b)$. Then $f(b)=f(a)$.

Problem 5   Let $b_1,b_2,\ldots$ be positive real numbers with

$$\lim_{n\to\infty}b_n=\infty \;\; and\;\; \lim_{n\to\infty}(b_n/b_{n+1})=1.$$

Assume also that $b_1<b_2<b_3<\cdots$. Show that the set of quotients $(b_m/b_n)_{1\leq n<m}$ is dense in $(1,\infty)$.

Problem 6   Let $V$ be a real vector space of dimension $n$ with a positive definite inner product. We say that two bases $(a_i)$ and $(b_i)$ have the same orientation if the matrix of the change of basis from $(a_i)$ to $(b_i)$ has a positive determinant. Suppose now that $(a_i)$ and $(b_i)$ are orthonormal bases with the same orientation. Show that $(a_i+2b_i)$ is again a basis of $V$ with the same orientation as $(a_i)$.

Problem 7   Compute

$$\int_0^{\infty}\frac{\log x}{x^2+a^2}\,dx$$

where $a>0$ is a constant.

Problem 8   Let $G_1$, $G_2$, and $G_3$ be finite groups, each of which is generated by its commutators (elements of the form $xyx^{-1}y^{-1}$). Let $A$ be a subgroup of $G_1 \times G_2 \times G_3$, which maps surjectively, by the natural projection map, to the partial products $G_1 \times G_2$, $G_1 \times G_3$ and $G_2 \times G_3$. Show that $A$ is equal to $G_1 \times G_2 \times G_3$.

Problem 9   Suppose $\Omega$ is a bounded domain in $\mathbb{C}\,^{}$ with a boundary consisting of a smooth Jordan curve $\gamma$. Let $f$ be holomorphic in a neighborhood of the closure of $\Omega$, and suppose that $f(z) \neq 0$ for $z \in \gamma$. Let $z_1,\ldots,z_k$ be the zeros of $f$ in $\Omega$, and let $n_j$ be the order of the zero of $f$ at $z_j$ (for $j=1,\ldots,k$).

1. Use Cauchy's integral formula to show that
   
   $$\frac{1}{2\pi i}\int_{\gamma}\frac{f'(z)}{f(z)}\,dz = \sum_{j=1}^{k}n_j.$$
   
   

2. Suppose that $f$ has only one zero $z_1$ in $\Omega$ with multiplicity $n_1=1$. Find a boundary integral involving $f$ whose value is the point $z_1$.

Problem 10   Let $f$ be a twice differentiable real valued function on $[0,2\pi]$, with $\int_{0}^{2\pi}f(x)dx=0=f(2\pi)-f(0)$. Show that

$$\int_0^{2\pi}\left(f(x)\right)^2\,dx \leq \int_0^{2\pi}\left(f'(x)\right)^2\,dx\, .$$

Problem 11   Find the eigenvalues, eigenvectors, and the Jordan Canonical Form of

$$A = \left( \begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array} \right),$$

considered as a matrix with entries in $\mbox{\bf {F}}_3 = \mbox{$\mathbb{Z}^{}$}/3\mbox{$\mathbb{Z}^{}$}$.

Problem 12   Prove that a continuous function from $\mathbb{R}^{}$ to $\mathbb{R}^{}$ which maps open sets to open sets must be monotonic.

Problem 13   Let $A$ be an n$\times$n complex matrix, all of whose eigenvalues are equal to $1$. Suppose that the set $\{A^n \;\vert\;n=1,2,\ldots\}$ is bounded. Show that $A$ is the identity matrix.

Problem 14   Let $G$ be a transitive subgroup of the group $S_n$ of permutations of the set $\{1,\ldots,n\}$. Suppose that $G$ is a simple group and that $\sim$ is an equivalence relation on $\{1,\ldots,n\}$ such that $i\sim j$ implies that $\sigma(i)\sim \sigma(j)$ for all $\sigma \in G$. What can you conclude about the relation $\sim$?

Problem 15   Let $f$ be analytic on and inside the unit circle
$C = \{z \;\vert\; \vert z\vert = 1\}$. Let $L$ be the length of the image of $C$ under $f$. Show that $L \geq 2\pi \vert f'(0)\vert$.

Problem 16   Let $\Omega$ be an open subset of $\mathbb{R}^{2}$, and let $f:\Omega \to \mbox{$\mathbb{R}^{2}$}$ be a smooth map. Assume that $f$ preserves orientation and maps any pair of orthogonal curves to a pair of orthogonal curves. Show that $f$ is holomorphic.

Note: Here we identify $\mathbb{R}^{2}$ with $\mathbb{C}\,^{}$.

Problem 17   Let $A$ be an $n\times n$ Hermitian matrix satisfying the condition

$$A^5 + A^3 + A = 3 I \,.$$

Show that $A = I$.

Problem 18   Find all real valued $C^1$ solutions $u$ of the differential equation

$$x\frac{du}{dx} + u = x \hspace{.2in} (-1<x<1).$$

Problem 19   Compute the area of the image of the unit disc $\{z \;\vert\; \vert z\vert < 1\}$ under the map $f(z)=z+z^2/2$.

Problem 20   Let $f:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$ be continuously differentiable, periodic of period $1$, and nonnegative. Show that

$$\frac{d }{dx}\left(\frac{f(x)}{1+cf(x)}\right)\to 0 \quad (as\; c \to \infty)$$

uniformly in $x$.