# Preliminary Exam - Fall 1980

Problem 1 Define

$$
F ( x ) = \int _ { \sin x } ^ { \cos x } e ^ { ( t ^ { 2 } + x t ) } d t .
$$

Compute $F ^ { \prime } ( 0 )$

Problem 2 Are the matrices give below similar $\boldsymbol { \ell }$

$$
A = { \left( \begin{array} { l l l } { 1 } & { 0 } & { 0 } \\ { - 1 } & { 1 } & { 1 } \\ { - 1 } & { 0 } & { 2 } \end{array} \right) } \qquad a n d \qquad B = { \left( \begin{array} { l l l } { 1 } & { 1 } & { 0 } \\ { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 2 } \end{array} \right) }
$$

Problem 3 Do there exist functions $f ( z )$ and $g ( z )$ that are analytic at $z = 0$ and that satisfy

$$
1 . \ f \left( 1 / n \right) = f \left( - 1 / n \right) = 1 / n ^ { 2 } , \ n = 1 , 2 , . . . ,
$$

$$
\mathcal { Q } . \ g \left( 1 / n \right) = g \left( - 1 / n \right) = 1 / n ^ { 3 } , \ n = 1 , 2 , \ldots \mathcal { Q }
$$

Problem 4 Let G be the group of orthogonal transformations of $\mathbb { R } ^ { 3 }$ to $\mathbb { R } ^ { 3 }$ with determinant 1. Let $v \in \mathbb { R } ^ { 3 } , \ | v | = 1$ , and let $H _ { v } = \{ T \in G \mid T v = v \}$

1. Show that $H _ { v }$ is a subgroup of G.

2. Let $S _ { v } = \{ T \in G \mid T$ is a rotation of $1 8 0 ^ { \circ }$ about a line orthogonal to $v \}$ . Show that $S _ { v }$ is a coset of $H _ { v }$ in G.

Problem 5 Evaluate

$$
\int _ { 0 } ^ { \infty } { \frac { x ^ { m - 1 } } { 1 + x ^ { n } } } d x
$$

where n and m are positive integers and $0 < m < n$

Problem 6 Let $M _ { 2 \times 2 }$ be the ring of real ${ \mathcal { Q } } \times { \mathcal { Q } }$ matrices and $S \subset M _ { 2 \times 2 }$ the subring of matrices of the form

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

1. Exhibit an isomorphism between S and $\mathbb { C }$

2. Prove that

$$
A = { \left( \begin{array} { l l } { 0 } & { 3 } \\ { - 4 } & { 1 } \end{array} \right) }
$$

lies in a subring isomorphic to $S _ { \star }$

3. Prove that there is an $X \in M _ { 2 \times 2 }$ such that $X ^ { 4 } + 1 3 X = A$

Problem 7 Let g be 2π-periodic, continuous on $[ - \pi , \pi ]$ and have Fourier series

$$
{ \frac { a _ { 0 } } { 2 } } + \sum _ { n = 1 } ^ { \infty } ( a _ { n } \cos n x + b _ { n } \sin n x ) .
$$

Let f be 2π-periodic and satisfy the differential equation

$$
f ^ { \prime \prime } ( x ) + k f ( x ) = g ( x )
$$

where k $\neq n ^ { 2 } , n = 1 , 2 , 3 , . . .$ . Find the Fourier series of f and prove that it converges everywhere.

Problem 8 Let $\mathbf { F } _ { 2 } = \{ 0 , 1 \}$ be the field with two elements. Let G be the group of invertible ${ \it 2 } \times { \it 2 }$ matrices with entries in $\mathbf { F } _ { 2 }$ . Show that $G$ is isomorphic to $S _ { 3 }$ , the group of permutations of three objects.

Problem 9 For a real ${ \mathcal { Q } } \times { \mathcal { Q } }$ matrix

$$
X = { \binom { x } { z } } \ y \sp { } \nonumber ,
$$

let $\| X \| = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + t ^ { 2 }$ , and define a metric by $d ( X , Y ) = \| X - Y \|$ . Let $\Sigma = \{ X \mid \operatorname* { d e t } ( X ) = 0 \}$ . Let

$$
A = { \binom { 1 } { 0 } } \ { \stackrel { 0 } { 2 } } ) .
$$

Find the minimum distance from A to Σ and exhibit an $S \in \Sigma$ that achieves this minimum.

Problem 10 Show that there is an $\varepsilon > 0$ such that if A is any real ${ \mathcal { Q } } \times { \mathcal { Q } }$ matrix satisfying $| a _ { i j } | \leqslant \varepsilon$ for all entries $a _ { i j }$ of A, then there is a real ${ \mathcal { Q } } \times { \mathcal { Q } }$ matrix X such that $X ^ { 2 } + X ^ { t } = A$ , where $X ^ { t }$ is the transpose of X. Is X unique?

Problem 11 Let $f ( z )$ be an analytic function defined for $| z | \leqslant 1$ and let

$$
u ( x , y ) = \Re f ( z ) , \quad z = x + i y .
$$

Prove that

$$
\int _ { C } { \frac { \partial u } { \partial y } } d x - { \frac { \partial u } { \partial x } } d y = 0
$$

where C is the unit circle, $x ^ { 2 } + y ^ { 2 } = 1$

Problem 12 Prove that any group of order 6 is isomorphic to either $\mathbb { Z } _ { 6 }$ or $S _ { 3 }$ (the group of permutations of three objects).

Problem 13 Let $\begin{array} { r } { f ( x ) = \frac { 1 } { 4 } + x - x ^ { 2 } } \end{array}$ . For any real number x, define a sequence $( x _ { n } )$ by $x _ { 0 } = x$ and $x _ { n + 1 } = f ( x _ { n } )$ . If the sequence converges, let $x _ { \infty }$ denote the limit.

1. For $x = 0$ , show that the sequence is bounded and nondecreasing and find $x _ { \infty } = \lambda$

2. Find all $y \in \mathbb { R }$ such that $y _ { \infty } = \lambda$

Problem 14 Exhibit a set of ${ \it 2 } \times { \it 2 }$ real matrices with the following property: A matrix A is similar to exactly one matrix in S provided A is a ${ \it 2 } \times { \it 2 }$ invertible matrix of integers with all the roots of its characteristic polynomial on the unit circle.

Problem 15 Consider the differential equation $x ^ { \prime \prime } + x ^ { \prime } + x ^ { 3 } = 0$ and the function $f ( x , x ^ { \prime } ) = ( x + x ^ { \prime } ) ^ { 2 } + ( x ^ { \prime } ) ^ { 2 } + x ^ { 4 }$

1. Show that f decreases along trajectories of the differential equation.

2. Show that if $x ( t )$ is any solution, then $( x ( t ) , x ^ { \prime } ( t ) )$ tends to (0, 0) as $t \to \infty$

Problem 16 Suppose that A and B are real matrices such that $A ^ { t } = A$ 2

$$
v ^ { t } A v \geqslant 0
$$

for all $v \in \mathbb { R } ^ { n }$ and

$$
A B + B A = 0 .
$$

Show that $A B = B A = 0$ and give an example where neither A nor B is zero.

Problem 17 Let $P _ { n }$ be a sequence of real polynomials of degree $\leqslant D$ , a fixed integer. Suppose that $P _ { n } ( x ) \to 0$ pointwise for $0 \leqslant x \leqslant 1$ . Prove that $P _ { n } \to 0$ uniformly on [0, 1].

Problem 18 Suppose that f is analytic inside and on the unit circle $| z | = 1$ and satisfies $| f ( z ) | < 1$ for $| z | = 1$ Show that the equation $f ( z ) = z ^ { 3 }$ has exactly three solutions (counting multiplicities) inside the unit circle.

Problem 19 Let X be a compact metric space and $f : X \to X$ an isometry. Show that $f ( X ) = X$

Problem 20 Let R be a ring with multiplicative identity 1. Call $x \in R$ a unit $i f x y = y x = 1$ for some $y \in R$ . Let $G ( R )$ denote the set of units.

1. Prove $G ( R )$ is a multiplicative group.

2. Let R be the ring of complex numbers $a + b i$ , where a and b are integers. Prove $G ( R )$ is isomorphic to $\mathbb { Z } _ { 4 }$ (the additive group of integers modulo 4).