Preliminary Exam - Fall 1986

Problem 1   The Arzelà-Ascoli Theorem asserts that the sequence $\left\{f_n\right\}$ of continuous real valued functions on a metric space $\Omega$ is precompact (i.e., has a uniformly convergent subsequence) if

(i)

$\Omega$ is compact,

(ii)

$\sup\Vert f_n\Vert < \infty$ (where $\Vert f_n\Vert = \sup\{\vert f_n(x)\vert \;\vert\; x \in \Omega\}$),

(iii)

the sequence is equicontinuous.

Give examples of sequences which are not precompact such that: (i) and (ii) hold but (iii) fails; (i) and (iii) hold but (ii) fails; (ii) and (iii) hold but (i) fails. Take $\Omega$ to be a subset of the real line. Sketch the graph of a typical member of the sequence in each case.

Problem 2   Let the points $a$, $b$, and $c$ lie on the unit circle of the complex plane and satisfy $a+b+c=0$. Prove that $a$, $b$, and $c$ form the vertices of an equilateral triangle.

Problem 3   Evaluate

$$\int\!\!\int_{\cal R}(x^3-3xy^2)\,dxdy\,,$$

where

$$% latex2html id marker 707 {\cal R} = \{(x,y) \in \mbox{$\m... ...; (x+1)^2 + y^2 \leq 9, \hspace{.1in} (x-1)^2 + y^2 \geq 1\}.$$

Problem 4   Show that the polynomial $p(z) = z^5-6z+3$ has five distinct complex roots, of which exactly three (and not five) are real.

Problem 5   Let $M_{2\times 2}$ denote the vector space of complex 2$\times$2 matrices. Let

$$A = \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right)$$

and let the linear transformation $T: M_{2\times 2} \to M_{2\times 2}$ be defined by
$T(X) = XA-AX$. Find the Jordan Canonical Form for $T$.

Problem 6   Prove the following theorem, or find a counterexample: If $p$ and $q$ are continuous real valued functions on $\mathbb{R}^{}$ such that $\vert q(x)\vert\leq \vert p(x)\vert$ for all $x$, and if every solution $f$ of the differential equation

$$f' + qf = 0$$

satisfies $${\lim_{x\to +\infty}f(x)=0}$$, then every solution $f$ of the differential equation

$$f' + pf = 0$$

satisfies $${\lim_{x\to +\infty}f(x)=0}$$.

Problem 7   Let $\mbox{\bf {F}}$ be a field containing $\mathbb{Q}\,^{}$ such that $[\mbox{\bf {F}}:\mbox{$\mathbb{Q}\,^{}$}] = 2$. Prove that there exists a unique integer $m$ such that $m$ has no multiple prime factors and $\mbox{\bf {F}}$ is isomorphic to $\mbox{$\mathbb{Q}\,^{}$}(\sqrt{m}\,)$.

Problem 8   Let $f$ be a continuous real valued function on $[0,1]$ such that, for each $x_0 \in [0,1)$,

$$\limsup_{x \to x_0^+} \frac{f(x)-f(x_0)}{x-x_0} \geq 0.$$

Prove that $f$ is nondecreasing.

Problem 9   Evaluate

$$\int_0^{\infty}\frac{\log x}{(x^2+1)(x^2+4)}\,dx\, .$$

Problem 10   For $f$ a real valued function on the real line, define the function $\triangle f$ by $\triangle f(x) = f(x+1)-f(x)$. For $n\geq2$, define $\triangle^nf$ recursively by $\triangle^nf = \triangle(\triangle^{n-1}f)$. Prove that $\triangle^nf=0$ if and only if $f$ has the form $f(x)=a_0(x)+a_1(x)x+\cdots+a_{n-1}(x)x^{n-1}$ where $a_0,a_1,\ldots,a_{n-1}$ are periodic functions of period $1$.

Problem 11   Let $A$ be an $m\times n$ matrix with entries in a field $\mbox{\bf {F}}$. Define the row rank and the column rank of $A$ and show from first principles that they are equal.

Problem 12   Let $\{U_1, U_2, \ldots \}$ be a cover of $\mathbb{R}^{n}$ by open sets. Prove that there is a cover $\{V_1, V_2, \ldots \}$ such that

1. $V_j \subset U_j$ for each $j$;

2. each compact subset of $\mathbb{R}^{n}$ is disjoint from all but finitely many of the $V_j$.

Problem 13   Let $f: \mbox{$\mathbb{R}^{2}$} \to \mbox{$\mathbb{R}^{}$}$ be defined by:

$$% latex2html id marker 968 f(x,y) = \left\{ \begin{array}{l... ... (y/x) & if & x\neq 0 \\ 0 & if & x=0\, . \end{array} \right.$$

Determine all points at which $f$ is differentiable.

Problem 14   Let $a$ and $b$ be real numbers. Prove that there are two orthogonal unit vectors $u$ and $v$ in $\mathbb{R}^{3}$ such that $u = (u_1,u_2,a)$ and $v = (v_1,v_2,b)$ if and only if $a^2+b^2\leq 1$.

Problem 15   Prove that if $p$ is a prime number $(>0)$ then the polynomial

$$f(x) = x^{p-1}+x^{p-2}+\cdots+1$$

is irreducible in $\mathbb{Q}[x]$.

Problem 16   Discuss the solvability of the differential equation

$$(e^x\sin y)(y')^3 + (e^x \cos y)y' + e^y\tan x = 0$$

with the initial condition $y(0)=0$. Does a solution exist in some interval about $0$? If so, is it unique?

Problem 17   Let $G$ be a subgroup of $S_5$, the group of all permutations on the set $\{1, 2, 3, 4, 5\}$. Prove that if $G$ contains a $5$-cycle and a $2$-cycle, then $G = S_5$.

Problem 18   Evaluate

$$\frac{1}{2\pi i}\int_{\vert z\vert=1}\frac{z^{11}}{12z^{12}-4z^9+2z^6-4z^3+1}\,dz$$

where the direction of integration is counterclockwise.

Problem 19   Prove that if six people are riding together in an Evans Hall elevator, there is either a three-person subset of mutual friends (each knows the other two) or a three-person subset of mutual strangers (each knows neither of the other two).

Problem 20   Let $f$ be a real valued continuous function on $[0,\infty)$ such that

$$\lim_{x\to\infty}\left( f(x) + \int_0^xf(t)\,dt \right)$$

exists. Prove that

$$\lim_{x\to\infty}f(x) = 0.$$