# Preliminary Exam - Spring 1995

Problem 1 For each positive integer n, define $f _ { n } : \mathbb { R } \to \mathbb { R }$ by $f _ { n } ( x ) \ =$ cos nx. Prove that the sequence of functions $\{ f _ { n } \}$ has no uniformly convergent subsequence.

Problem 2 Let A be the $\mathcal { B } \times \mathcal { B }$ matrix

$$
\left( { \begin{array} { r r r } { 1 } & { - 1 } & { 0 } \\ { - 1 } & { 2 } & { - 1 } \\ { 0 } & { - 1 } & { 1 } \end{array} } \right)
$$

Determine all real numbers a for which the limit lim $a ^ { n } A ^ { n }$ exists and is n→∞   
nonzero (as a matrix).

Problem 3 Let n be a positive integer and $0 < \theta < \pi$ . Prove that

$$
\frac { 1 } { 2 \pi i } \int _ { | z | = 2 } \frac { z ^ { n } } { 1 - 2 z \cos \theta + z ^ { 2 } } d z = \frac { \sin n \theta } { \sin \theta }
$$

where the circle $| z | = 2$ is oriented counterclockwise.

Problem 4 Let F be a finite field of cardinality $p ^ { n }$ , with p prime and $n > 0$ and let G be the group of invertible 2×2 matrices with coefficients in F.

1. Prove that G has order $( p ^ { 2 n } - 1 ) ( p ^ { 2 n } - p ^ { n } )$

2. Show that any p-Sylow subgroup of G is isomorphic to the additive group of F.

Problem 5 Let $f : \mathbb { R } \to \mathbb { R }$ be a bounded continuously differentiable function.   
Show that every solution of $y ^ { \prime } ( x ) = f \left( y ( x ) \right)$ is monotone.

Problem 6 Suppose that R is a subring of a commutative ring S and that R is of finite index n in S. Let m be an integer that is relatively prime to n. Prove that the natural map $R / m R  S / m S$ is a ring isomorphism.

Problem 7 Let f, $g \colon [ 0 , 1 ]  [ 0 , \infty )$ be continuous functions satisfying

$$
\operatorname* { s u p } _ { 0 \leqslant x \leqslant 1 } f ( x ) = \operatorname* { s u p } _ { 0 \leqslant x \leqslant 1 } g ( x ) .
$$

Prove that there exists $t \in [ 0 , 1 ]$ with $f ( t ) ^ { 2 } + 3 f ( t ) = g ( t ) ^ { 2 } + 3 g ( t )$

Problem 8 Suppose that $W \subset V$ are finite-dimensional vector spaces over $a \ { \mathrm { ~ } } f e l d ,$ , and let $L \colon V \to V$ be a linear transformation with $L ( V ) \subset W$ . Denote the restriction of L to W by $L _ { W }$ . Prove that det $( 1 - t L ) = \operatorname* { d e t } ( 1 - t L _ { W } )$

Problem 9 Let $P ( x )$ be a polynomial with real coefficients and with leading coefficient 1. Suppose that $P ( 0 ) = - 1$ and that $P ( x )$ has no complex zeros inside the unit circle. Prove that $P ( 1 ) = 0$

Problem 10 Let $f _ { n } \colon [ 0 , 1 ] \to [ 0 , \infty )$ be a continuous function, $f o r n = 1 , 2 , \dots$ Suppose that one has

$$
( * ) \qquad f _ { 1 } ( x ) \geqslant f _ { 2 } ( x ) \geqslant f _ { 3 } ( x ) \geqslant \dotsb \mathrm { ~ } f o r \mathrm { ~ } a l l \mathrm { ~ } x \in [ 0 , 1 ] .
$$

Let $f ( x ) = \operatorname* { l i m } _ { n \to \infty } f _ { n } ( x )$ and $M = \operatorname* { s u p } _ { 0 \leqslant x \leqslant 1 } f ( x )$

1. Prove that there exists $t \in [ 0 , 1 ]$ with $f ( t ) = M$

2. Show by example that the conclusion of Part 1 need not hold if instead $o f \left( * \right)$ we merely know that for each $x \in [ 0 , 1 ]$ there exists $n _ { x }$ such that for all $n \geqslant n _ { x }$ one has $f _ { n } ( x ) \geqslant f _ { n + 1 } ( x )$

Problem 11 Let n be a positive integer, and let $S \subset \mathbb { R } ^ { n }$ a finite subset with $0 \in S$ . Suppose that $\varphi : S  S$ is a map satisfying

$$
\begin{array} { r l } { \varphi ( 0 ) = 0 , } & { { } } \\ { d ( \varphi ( s ) , \varphi ( t ) ) = d ( s , t ) } & { { } \quad f o r \ a l l \quad s , t \in S , } \end{array}
$$

where $d ( \mathbf { \theta } , \mathbf { \alpha } )$ denotes Euclidean metric. Prove that there is a linear mapf : $\mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ whose restriction to $S$ is $\varphi$ .

Problem 12 Let n be a positive integer. Compute

$$
\int _ { 0 } ^ { 2 \pi } \frac { 1 - \cos n \theta } { 1 - \cos \theta } d \theta .
$$

Problem 13 Let n be an odd positive integer, and denote by $S _ { n }$ the group of all permutations of $\{ 1 , 2 , \ldots , n \}$ . Suppose that G is a subgroup of $S _ { n }$ of 2- power order. Prove that there exists $i \in \{ 1 , 2 , \ldots , n \}$ such that for all $\sigma \in G ,$ one has $\sigma ( i ) = i$

Problem 14 Let $y : \mathbb { R } \to \mathbb { R }$ be a three times differentiable function satisfying the differential equation $y ^ { \prime \prime \prime } - y = 0$ Suppose that lim $\scriptstyle 1 _ { x \to \infty } y ( x ) = 0$ . Find real numbers $a , b , c ,$ and d, not all zero, such that $a y ( 0 ) + y ^ { \prime } ( 0 ) + c y ^ { \prime \prime } ( 0 ) = d$

Problem 15 Let F be a finite field, and suppose that the subfield of F generated by $\{ x ^ { 3 } \mid x \in \mathbf { F } \}$ is different from F. Show that F has cardinality 4.

Problem 16 Let K be a nonempty compact set in a metric space with distance function d. Suppose that $\varphi \colon K \to K$ satisfies

$$
d ( \varphi ( x ) , \varphi ( y ) ) < d ( x , y )
$$

for all $x \neq y$ in K. Show there exists precisely one point $x \in K$ such that $x = \varphi ( x )$

Problem 17 Let V be a finite-dimensional vector space over a field F, and let $L : V \to V$ be a linear transformation. Suppose that the characteristic polynomial χ of L is written as $\chi = \chi _ { 1 } \chi _ { 2 }$ , where $\chi _ { 1 }$ and $\chi _ { 2 }$ are two relatively prime polynomials with coefficients in F. Show that V can be written as the direct sum of two subspaces $V _ { 1 }$ and $V _ { 2 }$ with the property that $\chi _ { i } ( L ) V _ { i } = 0$ for $i = 1 , 2$

Problem 18 Prove that there is no one-to-one conformal mapof the punctured disc $G = \left\{ z \in \mathbb { C } \mid 0 < | z | < 1 \right\}$ onto the annulusA $= \{ z \in \mathbb { C } \mid 1 <$ $| z | < 2 \}$