Preliminary Exam - Spring 1994

Problem 1   Let the collection $\cal U$ of open subsets of $\mathbb{R}$ cover the interval $[0,1]$. Prove that there is a positive number $\delta$ such that any two points $x$ and $y$ of $[0,1]$ satisfying $\vert x-y\vert < \delta$ belong together to some member of the cover $\cal U$.

Problem 2   Let $A$ be a real $n\times n$ matrix. Let $M$ denote the maximum of the absolute values of the eigenvalues of $A$.

1. Prove that if $A$ is symmetric, then $\Vert Ax\Vert \leq M\Vert x\Vert$ for all $x$ in $\mathbb{R}^n$. (Here, $\Vert \cdot \Vert$ denotes the Euclidean norm.)
2. Prove that the preceding inequality can fail if $A$ is not symmetric.

Problem 3   Evaluate

$$\int^\pi_{-\pi} \, \frac{d\theta}{3 - \cos \theta}\cdot$$

Problem 4   Let $G$ be a group having a subgroup $A$ of finite index. Prove that there is a normal subgroup $N$ of $G$ contained in $A$ such that $N$ is of finite index in $G$.

Problem 5  

1. Suppose the functions $\sin t$ and $\sin 2t$ are both solutions of the differential equation
   
   $$\sum^n_{k=0} \ c_k \, \frac{d^{k}x}{dt^{k}} = 0 \ ,$$
   
   
   where $c_{0}, \ldots , c_n$ are real constants. What is the smallest possible order of the equation? Explain. Write down an equation of minimum order having the given functions as solutions.
2. Will the answers to Part 1 be different if the constants $c_{0}, \ldots , c_n$ are allowed to be complex? Explain.

Problem 6   Prove or disprove: A square complex matrix, $A$, is similar to its transpose, $A^{t}$.

Problem 7   Let $a_{1}, a_{2}, \ldots , a_n$ be complex numbers. Prove that there is a point $x$ in $[0,1]$ such that

$$\left\vert 1 - \sum^n_{k=1} \ a_{k}e^{2 \pi ikx}\right\vert \geq 1.$$

Problem 8   Find all automorphisms of $\mathbb{Z}[x]$, the ring of polynomials over $\mathbb{Z}$.

Problem 9  

1. Let $U$ and $V$ be open connected subsets of the complex plane, and let $f$ be an analytic function in $U$ such that $f(U) \subset V$. Assume $f^{-1}(K)$ is compact whenever $K$ is a compact subset of $V$. Prove that $f(U) = V$.
2. Prove that the last equality can fail if analytic is replaced by continuous in the preceding statement.

Problem 10   Let f be a continuous real valued function on $\mathbb{R}$ such that the improper Riemann integral $\int^{\infty}_{-\infty} \, \vert f(x)\vert \, dx$ converges. Define the function g on $\mathbb{R}$ by

$$g(y) = \int^{\infty}_{-\infty} \, f(x) \cos (xy) \, dx\, .$$

Prove that g is continuous.

Problem 11   Let $T: \mathbb{R}^{n} \to \mathbb{R}^n$ be a diagonalizable linear transformation. Prove that there is an orthonormal basis for $\mathbb{R}^n$ with respect to which $T$ has an upper-triangular matrix.

Problem 12   Let $f = u + iv$ and $g = p + iq$ be analytic functions defined in a neighborhood of the origin in the complex plane. Assume
$\vert g^{\prime}(0)\vert \, < \, \vert f^{\prime}(0)\vert$. Prove that there is a neighborhood of the origin in which the function $h = f + \overline{g}$ is one-to-one.

Problem 13   Let $\mbox{\bf {F}}$ be a finite field with q elements. Say that a function $f: \mbox{\bf {F}} \to \mbox{\bf {F}}$ is a polynomial function if there are elements $a_{0},a_{1}, \ldots ,a_n$ of $\mbox{\bf {F}}$ such that $f(x) = a_{0} + a_{1}x + \cdots + a_{n}x^n$ for all $x \in \mbox{\bf {F}}$. How many polynomial functions are there? _function,>polynomial

Problem 14   Let $W$ be a real 3$\times$3 antisymmetric matrix, i.e.,
$W^{t} = - W$. Let the function

$$X(t) = \left( \begin{array}{c} x_{1}(t) \\ x_{2}(t) \\ x_{3}(t) \end{array} \right)$$

be a real solution of the vector differential equation ${dX}/{dt} = WX$.

1. Prove that $\Vert X(t)\Vert$, the Euclidean norm of $X(t)$, is independent of $t$.

2. Prove that if $v$ is a vector in the null space of $W$, then $X(t) \cdot v$ is independent of $t$.

3. Prove that the values $X(t)$ all lie on a fixed circle in $\mathbb{R}^3$.

Problem 15   Let $\alpha$ be a number in $(0,1)$. Prove that any sequence $(x_{n})^{\infty}_{0}$ of real numbers satisfying the recurrence relation

$$x_{n+1} = \alpha x_{n} + (1 - \alpha )x_{n-1}$$

has a limit, and find an expression for the limit in terms of $\alpha$, $x_0$, and $x_1$.

Problem 16   For which numbers $a$ in $(1, \infty)$ is it true that $x^{a} \leq a^x$ for all $x$ in $(1, \infty)$?

Problem 17   Prove that there are at least two nonisomorphic nonabelian groups of order $30$.

Problem 18   Let u be a real valued harmonic function in the complex plane such that

$$u(z) \leq a \left\vert \log\vert z\vert \right\vert + b$$

for all z, where a and b are positive constants. Prove that u is constant.