# Preliminary Exam - Spring 1980

Problem 1 Let $f : \mathbb { R }  \mathbb { R }$ be the unique function such $t h a t f ( x ) = x$ if $- \pi \leqslant x < \pi$ and $f ( x + 2 n \pi ) = f ( x )$ for all $n \in \mathbb { Z }$

1. Prove that the Fourier series of f is

$$
\sum _ { n = 1 } ^ { \infty } { \frac { ( - 1 ) ^ { n + 1 } 2 \sin n x } { n } } \cdotp
$$

2. Prove that the series does not converge uniformly.

3. For each $x \in \mathbb { R }$ , find the sum of the series.

Problem 2 Let $f _ { n } : \mathbb { R } \to \mathbb { R }$ be differentiable for each $n = 1 , \ 2 , \ \ldots .$ . with $| f _ { n } ^ { \prime } ( x ) | \leqslant 1$ for all $n , \ x .$ Assume

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

for all x. Prove that $g : \mathbb { R } \to \mathbb { R }$ is continuous.

Problem 3 Let $P _ { 2 }$ denote the set of real polynomials of degree $\leqslant 2$ . Define the map $J : P _ { 2 } \to \mathbb { R }$ by

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

Let $Q = \{ f \in P _ { 2 } \mid f ( 1 ) = 1 \}$ . Show that J attains a minimum value on $Q$ and determine where the minimum occurs.

Problem 4 Let $a ~ > ~ 0$ be a constant $\neq ~ 2$ . Let $C _ { a }$ denote the positively oriented circle of radius a centered at the origin. Evaluate

$$
\int _ { C _ { a } } { \frac { z ^ { 2 } + e ^ { z } } { z ^ { 2 } ( z - 2 ) } } d z .
$$

Problem 5 Let

$$
f ( z ) = \sum _ { n = 0 } ^ { \infty } a _ { n } z ^ { n }
$$

be an analytic function in the open unit disc D. Assume that

$$
\sum _ { n = 2 } ^ { \infty } n | a _ { n } | \leqslant | a _ { 1 } | \quad w i t h \quad a _ { 1 } \neq 0 .
$$

Prove that f is injective.

Problem 6 G is a group of order n, H a proper subgroup of order m, and $( n / m ) ! < 2 n$ Prove G has a proper normal subgroup different from the identity.

Problem 7 Let $n \geqslant 2$ be an integer such that $2 ^ { n } + n ^ { 2 }$ is prime. Prove that

$$
n \equiv 3 { \pmod { 6 } } .
$$

Problem 8 Let A and B be $n \times n$ complex matrices. Prove or disprove each of the following statements:

1. If A and B are diagonalizable, so is $A + B$

2. If A and B are diagonalizable, so is $A B$

3. $I f A ^ { 2 } = A$ , then A is diagonalizable.

4. If A is invertible and $A ^ { 2 }$ is diagonalizable, then A is diagonalizable.

Problem 9 Let

$$
A = { \binom { 1 } { 3 } } \ { 2 } ) .
$$

Show that every real matrix B such that $A B = B A$ has the form $s I + t A$ where s, $t \in \mathbb { R }$

Problem 10 Consider the differential equation

$$
x ^ { \prime } = \frac { x ^ { 3 } - x } { 1 + e ^ { x } } .
$$

1. Find all its constant solutions.

2. Discuss $\operatorname* { l i m } _ { t \to \infty } x ( t )$ , where $x ( t )$ is the solution such that $x ( 0 ) = \frac { 1 } { 2 }$

Problem 11 Let $\mathcal { S } = \{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 \}$ denote the unit sphere in $\mathbb { R } ^ { 3 }$ . Evaluate the surface integral over $s$ :

$$
\int \int _ { S } ( x ^ { 2 } + y + z ) d A .
$$

Problem 12 Let $M _ { 3 \times 3 }$ denote the vector space of real $\mathcal { B } { \times } \mathcal { B }$ matrices. For any matrix $A \in M _ { 3 \times 3 }$ , define the linear operator $L _ { A } : M _ { 3 \times 3 } \longrightarrow M _ { 3 \times 3 } , L _ { A } ( B ) =$ $A B$ . Suppose that the determinant of A is 32 and the minimal polynomial is $( t - 4 ) ( t - 2 )$ . What is the trace of $L _ { A } \mathcal { Z }$

Problem 13 Let G be a subgroup of $S _ { n }$ the group of permutations of n objects. Assume G is transitive; that is, for any x and y in S, there is some $\sigma \in G$ with $\sigma ( x ) = y$

1. Prove that n divides the order of G.

2. Suppose $n = 4$ . For which integers $k \geqslant 1$ can such a G have order 4k?

Problem 14 Find a real matrix B such that

$$
B ^ { 4 } = \left( \begin{array} { c c c } { { 2 } } & { { 0 } } & { { 0 } } \\ { { 0 } } & { { 2 } } & { { 0 } } \\ { { 0 } } & { { - 1 } } & { { 1 } } \end{array} \right) .
$$

Problem 15 Show that a vector space over an infinite field cannot be the union of a finite number of proper subspaces.

Problem 16 Let $f : \mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ be continuously differentiable. Assume the Jacobian matrix $( \partial f _ { i } / \partial x _ { j } )$ has rank n everywhere. Suppose f is proper; that $i s , f ^ { - 1 } ( K )$ is compact whenever K is compact. Prove $f ( \mathbb { R } ^ { n } ) = \mathbb { R } ^ { n }$

Problem 17 S9 is the group of permutations of 9 objects.

1. Exhibit an element of $S _ { 9 }$ of order 20.

2. Prove that no element of $S _ { 9 }$ has order 18.

Problem 18 For each $t \in \mathbb { R }$ , let $P ( t )$ be a symmetric real $n \times n$ matrix whose entries are continuous functions of t. Suppose for all t that the eigenvalues of $P ( t )$ are all $\leqslant - 1$ . Let $x ( t ) = ( x _ { 1 } ( t ) , \ldots , x _ { n } ( t ) )$ be a solution of the vector differential equation

$$
\frac { d x } { d t } = P ( t ) x .
$$

Prove that

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

Problem 19 Let

$$
f ( z ) = \sum _ { n = 0 } ^ { \infty } c _ { n } z ^ { n }
$$

be analytic in the disc $\mathbb { D } = \{ z \in \mathbb { C } \mid | z | < 1 \}$ . Assume f maps D one-to-one onto a domain G having area A. Prove

$$
A = \pi \sum _ { n = 1 } ^ { \infty } n | c _ { n } | ^ { 2 } .
$$

Problem 20 Does there exist an analytic function mapping the annulus

$$
A = \left\{ z \mid 1 \leqslant | z | \leqslant 4 \right\}
$$

onto the annulus

$$
B = \{ z \mid 1 \leqslant | z | \leqslant 2 \}
$$

and taking $C _ { 1 }  C _ { 1 } , C _ { 4 }  C _ { 2 }$ , where $C _ { r }$ is the circle of radius $r _ { \cdot } ^ { \it 2 }$