# Preliminary Exam - Fall 1987

Problem 1 Prove that $( \cos \theta ) ^ { p } \leqslant \cos ( p \theta )$ for $0 \leqslant \theta \leqslant \pi / 2$ and $0 < p < 1$

Problem 2 Suppose that $\left\{ f _ { n } \right\}$ is a sequence of nondecreasing functions which map the unit interval into itself. Suppose that

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

pointwise and that f is a continuous function. Prove that $f _ { n } ( x ) \to f ( x )$ uniformly as $n \to \infty , 0 \leq x \leqslant 1$ Note that the functions $f _ { n }$ are not necessarily continuous.

Problem 3 Show that the following limit exists and is finite:

$$
\operatorname* { l i m } _ { t \to 0 ^ { + } } \left( \int _ { 0 } ^ { 1 } { \frac { d x } { \left( x ^ { 4 } + t ^ { 4 } \right) ^ { 1 / 4 } } } + \log t \right) .
$$

Problem 4 Let u and v be two real valued $C ^ { 1 }$ functions on $\mathbb { R } ^ { 2 }$ such that the gradient $\nabla u$ is never 0, and such that, at each point, $\nabla \boldsymbol { v }$ and ∇u are linearly dependent vectors. Given $p _ { 0 } = ( x _ { 0 } , y _ { 0 } ) \in \mathbb { R } ^ { 2 }$ , show that there is $a C ^ { 1 }$ function F of one variable such that $v ( x , y ) = F \left( u ( x , y ) \right)$ in some neighborhood of $p _ { 0 }$

Problem 5 Calculate $A ^ { 1 0 0 }$ and $A ^ { - 7 }$ , where

$$
A = \left( \begin{array} { l l } { { 3 / 2 } } & { { 1 / 2 } } \\ { { - 1 / 2 } } & { { 1 / 2 } } \end{array} \right) .
$$

Problem 6 Let G and H be finite groups of relatively prime order. Show that ${ \mathrm { A u t } } ( G \times H )$ , the group of automorphisms of $G \times H$ , is isomorphic to the direct product of Aut(G) and Aut(H).

Problem 7 Let A and B be real n×n symmetric matrices with B positive definite. Consider the function defined for $\boldsymbol { v } \neq \boldsymbol { 0 }$ by

$$
G ( x ) = \frac { \langle A x , x \rangle } { \langle B x , x \rangle } \cdot
$$

1. Show that G attains its maximum value.

2. Show that any maximum point U for G is an eigenvector for a certain matrix related to A and B and show which matrix.

Problem 8 Let R be the set of ${ \it 2 } \times { \it 2 }$ matrices of the form

$$
\left( \begin{array} { l l } { a } & { - b } \\ { b } & { a } \end{array} \right)
$$

where a, b are elements of a given field F. Show that with the usual matrix operations, R is a commutative ring with identity. For which of the following fields F is R a field: $F = \mathbb { Q } , \mathbb { C } , \mathbb { Z } _ { 5 } , \mathbb { Z } _ { 7 } \ { \overset { \mathcal { Q } } { . } }$

Problem 9 Evaluate the integral

$$
I = \int _ { 0 } ^ { 2 \pi } { \frac { \cos ^ { 2 } 3 \theta } { 5 - 4 \cos 2 \theta } } d \theta .
$$

Problem 10 $I f f ( z )$ is analytic in the open disc $| z | < 1$ , and $| f ( z ) | \leqslant 1 / ( 1 -$ $| z | )$ , show that

$$
| a _ { n } | = \left| \frac { f ^ { ( n ) } ( 0 ) } { n ! } \right| \leqslant ( n + 1 ) \left( 1 + \frac { 1 } { n } \right) ^ { n } < e ( n + 1 ) .
$$

Problem 11 Let V be a finite-dimensional vector space and $T { : } V \to V$ a diagonalizable linear transformation. Let $W \subset V$ be a linear subspace which is mapped into itself by T . Show that the restriction of T to W is diagonalizable.

Problem 12 Given two real $n \times n$ matrices A and B, suppose that there is a nonsingular complex matrix C such that $C A C ^ { - 1 } = B$ Show that there exists a real nonsingular $n \times n$ matrix C with this property.

Problem 13 Let A be the group of rational numbers under addition, and let M be the group of positive rational numbers under multiplication. Determine all homomorphisms $\varphi : A  M$

Problem 14 Show that $M _ { n \times n } ( \mathbf { F } )$ , the ring of all $n \times n$ matrices over the field F, has no proper two sided ideals.

Problem 15 Let $f ( z )$ be analytic for $z \neq 0$ , and suppose $t h a t f ( 1 / z ) = f ( z )$ Suppose also that $f ( z )$ is real for all z on the unit circle $| z | = 1$ . Prove that $f ( z )$ is real for all real $z \neq 0$

Problem 16 How many zeros (counting multiplicities) does the polynomial

$$
2 z ^ { 5 } - 6 z ^ { 3 } + z + 1
$$

have in the annular region $1 \leqslant | z | \leqslant 2 \leqslant$

Problem 17 Let u be a positive harmonic function on $\mathbb { R } ^ { 2 }$ ; that $i s ,$

$$
{ \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } } + { \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } } = 0 .
$$

Show that u is constant.

Problem 18 Find a curve C in $\mathbb { R } ^ { 2 }$ , passing through the point $( 3 , 2 )$ , with the following property: Let $L ( x _ { 0 } , y _ { 0 } )$ be the segment of the tangent line to C at $( x _ { 0 } , y _ { 0 } )$ which lies in the first quadrant. Then each point $( x _ { 0 } , y _ { 0 } )$ of C is the midpoint of $L ( x _ { 0 } , y _ { 0 } )$

$$
\cdot \cdot \cdot \operatorname { \mathrm { ~ ‰ ~ } }
$$

Problem 19 Define a sequence of positive numbers as follows. $L e t x _ { 0 } > 0$ be any positive number, and let $x _ { n + 1 } = ( 1 + x _ { n } ) ^ { - 1 }$ . Prove that this sequence converges, and find its limit.

Problem 20 Let S be the set of all real $C ^ { 1 }$ functions f on [0, 1] such that $f ( 0 ) = 0$ and

$$
\int _ { 0 } ^ { 1 } f ^ { \prime } ( x ) ^ { 2 } d x \leqslant 1 .
$$

Define

$$
J ( f ) = \int _ { 0 } ^ { 1 } f ( x ) d x .
$$

Show that the function J is bounded on S, and compute its supremum. Is there a function $f _ { 0 } \in S$ at which J attains its maximum value? If so, what is $f _ { 0 } \ell$