Preliminary Exam - Fall 1998

Problem 1   Let $M$ be a metric space with metric $d$. Let $C$ be a compact subset of $M$, and let $\left(U_\alpha\right)_{\alpha \in I}$ be an open cover of $C$. Show that there exists $\epsilon > 0$ such that, for every $p \in C$, the open ball $B(p, \epsilon)$ is contained in at least one of the sets $U_\alpha$.

Problem 2   Find a function $y(x)$ such that $y^{(4)} + y = 0$ for $x \geq 0$, $y(0) = 0$, $y'(0) = 1$ and $${\lim_{x\rightarrow\infty} y(x) = \lim_{x\rightarrow\infty} y'(x) = 0}$$.

Problem 3   Prove that for any real $\alpha > 1$, $${\int_0^{\infty}\frac{dx}{1 + x^{\alpha}} = \frac{\pi/\alpha}{\sin(\pi/\alpha)\,\cdot}}$$

Problem 4   Let $f$ be analytic in the closed unit disc, with $f(-\log 2) = 0$ and $\vert f(z)\vert \leq \vert e^z\vert$ for all $z$ with $\vert z\vert = 1$. How large can $\vert f(\log 2)\vert$ be?

Problem 5   Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a function such that

1. the function $g$ defined by $g(x, y) = f(x + y) - f(x) - f(y)$ is bilinear,
2. for all $x \in \mathbb{R}^n$ and $t \in \mathbb{R}$, $f(tx) = t^2 f(x)$.

Show that there is a linear transformation $A : \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $f(x) = \langle x, Ax \rangle$ where $\langle \cdot, \cdot \rangle$ is the usual inner product on $\mathbb{R}^n$ (in other words, $f$ is a quadratic form).

Problem 6   Let $A$ and $B$ be linear transformations on a finite dimensional vector space $V$. Prove that $\dim \ker (AB) \leq \dim \ker A + \dim \ker B$.

Problem 7   A real symmetric $n \times n$ matrix $A$ is called positive semi-definite if $x^tAx \geq 0$ for all $x \in \mathbb{R}^n$. Prove that $A$ is positive semi-definite if and only if $\mathrm{tr}\,AB \geq 0$ for every real symmetric positive semi-definite n$\times$n matrix $B$.

Problem 8   Let $R$ be a finite ring with identity. Let $a$ be an element of $R$ which is not a zero divisor. Show that $a$ is invertible.

Problem 9   Suppose that $G$ is a finite group such that every Sylow subgroup is normal and abelian. Show that $G$ is abelian.

Problem 10   Let $f$ be a real function on $[a, b]$. Assume that $f$ is differentiable and that $f'$ is Riemann integrable. Prove that

$$\int_a^b f'(x)\,dx = f(b) - f(a).$$

Problem 11   Find the minimal value of the areas of hexagons circumscribing the unit circle in $\mathbb{R}^2$.

Note: See also Problem .

Problem 12   Let $\varphi(x, y)$ be a function with continuous second order partial derivatives such that

1. $\varphi_{x x} + \varphi_{y y} + \varphi_x = 0$ in the punctured plane $\mathbb{R}^2 \setminus \{0\}$,
2. $${ r\varphi_x \rightarrow \frac{x}{2\pi r}}$$ and $${ r\varphi_y \rightarrow \frac{y}{2\pi r}}$$ as $r = \sqrt{x^2 + y^2} \rightarrow 0$.

Let $C_R$ be the circle $x^2 + y^2 = R^2$. Show that the line integral

$$\int_{C_R}e^x(-\varphi_y\,dx + \varphi_x\,dy)$$

is independent of $R$, and evaluate it.

Problem 13   Let $f$ be an entire function. Define $\Omega = \mathbb{C}\setminus (-\infty, 0]$, the complex plane with the ray $(-\infty, 0]$ removed. Suppose that for all $z \in \Omega$, $\vert f(z)\vert \leq \vert\log z\vert$, where $\log z$ is the principal branch of the logarithm. What can you conclude about the function $f$?

Problem 14   Let $z_1, \dots, z_n$ be distinct complex numbers, and let
$a_1, \dots, a_n$ be nonzero complex numbers such that $S_p = \sum_{j = 1}^n a_j z_j^p = 0$ for $p = 0, 1, \dots, m - 1$ but $S_m \neq 0$. Here $1 \leq m \leq n - 1$. How many zeros does the rational function $f(z) = \sum_{j = 1}^n \frac{a_j}{z - z_j}$ have in $\mathbb{C}$? Explain why $m \geq n$ is impossible.

Problem 15   Let $A$ and $B$ be n$\times$n matrices. Show that the eigenvalues of $AB$ are the same as the eigenvalues of $BA$.

Problem 16   Let $B$ be a 3$\times$3 matrix whose null space is 2-dimensional, and let $\chi(\lambda)$ be the characteristic polynomial of $B$. For each assertion below, provide either a proof or a counterexample.

1. $\lambda^2$ is a factor of $\chi(\lambda)$.
2. The trace of $B$ is an eigenvalue of $B$.
3. $B$ is diagonalizable.

Problem 17   Let $\mbox{\bf {F}}$ be a finite field with $q$ elements. Denote by $GL_n(\mbox{\bf {F}})$ the group of invertible n$\times$n matrices with entries if $\mbox{\bf {F}}$. What is the order of this group?

Problem 18   Show that the field $\mathbb{Q}(t_1, \dots, t_n)$ of rational functions in $n$ variables over the rational numbers is isomorphic to a subfield of $\mathbb{R}$.