# Preliminary Exam - Fall 1986

Problem 1 The Arzel\`a–Ascoli Theorem asserts that the sequence $\left\{ f _ { n } \right\}$ of continuous real valued functions on a metric space Ω is precompact (i.e., has a uniformly convergent subsequence) if

(i) Ω is compact,

(ii) sup $\| f _ { n } \| < \infty$ (where $\Vert f _ { n } \Vert = \operatorname* { s u p } \{ | f _ { n } ( x ) | | x \in \Omega \} \bigr )$

(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 Ω 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

$$
\int \int _ { \mathcal { R } } ( x ^ { 3 } - 3 x y ^ { 2 } ) d x d y ,
$$

where

$$
\begin{array} { r } { \mathcal { R } = \{ ( x , y ) \in \mathbb { R } ^ { 2 } \mid ( x + 1 ) ^ { 2 } + y ^ { 2 } \leqslant 9 , ( x - 1 ) ^ { 2 } + y ^ { 2 } \geqslant 1 \} . } \end{array}
$$

Problem 4 Show that the polynomial $p ( z ) = z ^ { 5 } - 6 z + 3$ 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 2×2 matrices. Let

$$
A = { \binom { 0 } { 0 } } \ 1
$$

and let the linear transformation $T : M _ { 2 \times 2 } \to M _ { 2 \times 2 }$ be defined $b y T ( X ) =$ $X A - A X$ . 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 R such that $| q ( x ) | \leqslant | p ( x ) |$ for all x, and $i f$ every solution f of the differential equation

$$
f ^ { \prime } + q f = 0
$$

satisfies lim f (x) = 0, then every solution f of the differential equation x→+∞

$$
f ^ { \prime } + p f = 0
$$

satisfies $\operatorname* { l i m } _ { x \to + \infty } f ( x ) = 0$

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

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

$$
\operatorname* { l i m } _ { x \to x _ { 0 } ^ { + } } { \frac { f ( x ) - f ( x _ { 0 } ) } { x - x _ { 0 } } } \geqslant 0 .
$$

Prove that f is nondecreasing.

Problem 9 Evaluate

$$
\int _ { 0 } ^ { \infty } { \frac { \log x } { ( x ^ { 2 } + 1 ) ( x ^ { 2 } + 4 ) } } d x .
$$

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 $n \geqslant 2$ , define $\triangle ^ { n } f$ recursively by $\triangle ^ { n } f = \triangle ( \triangle ^ { n - 1 } f )$ . Prove that $\triangle ^ { n } f = 0$ if and only if f has the $f o r m f ( x ) =$ $a _ { 0 } ( x ) + a _ { 1 } ( x ) x + \cdot \cdot \cdot + a _ { n - 1 } ( x ) x ^ { n - 1 }$ where $a _ { 0 } , a _ { 1 } , \ldots , a _ { n - 1 }$ are periodic functions of period 1.

Problem 11 Let A be an m × n matrix with entries in a field 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 $f : \mathbb { R } ^ { 2 } \to \mathbb { R }$ be defined by:

$$
f ( x , y ) = { \left\{ \begin{array} { l l } { x ^ { 4 / 3 } \sin ( y / x ) } & { i f \quad x \neq 0 } \\ { 0 } & { i f \quad x = 0 . } \end{array} \right. }
$$

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 $\boldsymbol { v } = ( v _ { 1 } , v _ { 2 } , b )$ if and only if $a ^ { 2 } + b ^ { 2 } \leqslant 1$

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

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

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

Problem 16 Discuss the solvability of the differential equation

$$
( e ^ { x } \sin y ) ( y ^ { \prime } ) ^ { 3 } + ( e ^ { x } \cos y ) y ^ { \prime } + e ^ { y } \tan x = 0
$$

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 of five objects Prove that if G contains a 5-cycle and a 2-cycle, then $G = S _ { 5 }$

Problem 18 Evaluate

$$
\frac { 1 } { 2 \pi i } \int _ { | z | = 1 } \frac { z ^ { 1 1 } } { 1 2 z ^ { 1 2 } - 4 z ^ { 9 } + 2 z ^ { 6 } - 4 z ^ { 3 } + 1 } d z
$$

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

$$
\operatorname* { l i m } _ { x \to \infty } { \biggl ( } f ( x ) + \int _ { 0 } ^ { x } f ( t ) d t { \biggr ) }
$$

exists. Prove that

$$
\operatorname* { l i m } _ { x \to \infty } f ( x ) = 0 .
$$