# Preliminary Exam - Summer 1982

Problem 1 Determine the Jordan Canonical Form of the matrix

$$
A = { \left( \begin{array} { l l l } { 1 } & { 2 } & { 3 } \\ { 0 } & { 4 } & { 5 } \\ { 0 } & { 0 } & { 4 } \end{array} \right) } ~ .
$$

Problem 2 Compute the integral

$$
\int _ { 0 } ^ { \infty } { \frac { x ^ { 5 0 } } { x ^ { 1 0 0 } + 1 } } d x .
$$

Problem 3 Let K be a nonempty compact set in a metric space with distance function d. Suppose that ϕ: $K  K$ 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 $t h a t x = \varphi ( x )$

Problem 4 Let G be a group with generators a and b satisfying

$$
a ^ { - 1 } b ^ { 2 } a = b ^ { 3 } , \qquad b ^ { - 1 } a ^ { 2 } b = a ^ { 3 } .
$$

Is G trivial?

Problem 5 Let $0 < a _ { 0 } \leqslant a _ { 1 } \leqslant \cdots \leqslant a _ { n }$ . Prove that the equation

$$
a _ { 0 } z ^ { n } + a _ { 1 } z ^ { n - 1 } + \cdot \cdot \cdot + a _ { n } = 0
$$

has no roots in the disc $| z | < 1$

Problem 6 Suppose f is a differentiable real valued function such that $f ^ { \prime } ( x ) >$ f (x) for all $x \in \mathbb { R }$ and $f ( 0 ) = 0$ . Prove that $f ( x ) > 0$ for all positive x.

Problem 7 Let V be the vector space of all real $\mathcal { B } { \times } \mathcal { B }$ matrices and let A be the diagonal matrix

$$
\left( { \begin{array} { l l l } { 1 } & { 0 } & { 0 } \\ { 0 } & { 2 } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} } \right) ~ .
$$

Calculate the determinant of the linear transformation T on V defined $b y T ( X ) =$ ${ \scriptstyle { \frac { 1 } { 2 } } } ( A X + X A )$

Problem 8 Let n be a positive integer.

1. Show that the binomial coefficient

$$
c _ { n } = { \binom { 2 n } { n } }
$$

is even.

2. Prove that $c _ { n }$ is divisible by 4 if and only if n is not a power of 2.

Problem 9 Determine the complex numbers z for which the power series

$$
\sum _ { n = 1 } ^ { \infty } { \frac { z ^ { n } } { n ^ { \log n } } }
$$

and its term by term derivatives of all orders converge absolutely.

Problem 10 For complex numbers $\alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { k }$ , prove

$$
\operatorname* { l i m } _ { n } \operatorname* { s u p } \left| \sum _ { j = 1 } ^ { k } \alpha _ { j } ^ { n } \right| ^ { 1 / n } = \ \operatorname* { s u p } _ { j } | \alpha _ { j } | .
$$

Note: See also Problem ??.

Problem 11 Let $s ( y )$ and $t ( y )$ be real differentiable functions of $y , - \infty <$ $y < \infty$ , such that the complex function

$$
f ( x + i y ) = e ^ { x } \left( s ( y ) + i t ( y ) \right)
$$

is complex analytic with $s ( 0 ) = 1$ and $t ( 0 ) = 0$ . Determine $s ( y )$ and $t ( y )$

Problem 12 Determine (with proofs) which of the following polynomials are irreducible over the field Q of rationals.

1. $x ^ { 2 } + 3$

2. $x ^ { 2 } - 1 6 9$

3. $x ^ { 3 } + x ^ { 2 } + x + 1$

$$
\begin{array} { r } { 4 \cdot \ x ^ { 3 } + 2 x ^ { 2 } + 3 x + 4 . } \end{array}
$$

Problem 13 Let $f : [ 0 , \pi ] \to$ R be continuous and such that

$$
\int _ { 0 } ^ { \pi } f ( x ) \sin n x d x = 0
$$

for all integers $n \geqslant 1$ . Is f identically 0?

Problem 14 Let A be a real $n \times n$ matrix such that $\langle A x , x \rangle \geqslant 0$ for every real n-vector x. Show that $A u = 0$ if and only if $A ^ { t } u = 0$

Problem 15 Let $f ( z )$ be analytic on the open unit disc $\mathbb { D } { = } \{ z | | z | { < } 1 \}$ . Prove that there is a sequence $\left( z _ { n } \right)$ in D such that $| z _ { n } | \to 1$ and $\left( f ( z _ { n } ) \right)$ is bounded.

Problem 16 A square matrix A is nilpotent if $A ^ { k } = 0$ for some positive integer k.

1. If A and B are nilpotent, is $A + B$ nilpotent? Proof or counterexample.

2. Prove: If A is nilpotent, then $I - A$ is invertible.

Problem 17 Let $f : \mathbb { R } ^ { 3 } \longrightarrow \mathbb { R } ^ { 2 }$ and assume that 0 is a regular value of f $( i . e .$ , the differential of f has rank 2 at each point of $f ^ { - 1 } ( 0 ) )$ . Prove that $\mathbb { R } ^ { 3 } \setminus f ^ { - 1 } ( 0 )$ is arcwise connected.

Problem 18 Let E be the set of all continuous real valued functions u : $[ 0 , 1 ] \to \mathbb { R }$ satisfying

$$
| u ( x ) - u ( y ) | \leqslant | x - y | , \quad 0 \leqslant x , y \leqslant 1 , \quad u ( 0 ) = 0 .
$$

Let $\varphi : E  \mathbb { R }$ be defined by

$$
\varphi ( u ) = \int _ { 0 } ^ { 1 } \left( u ( x ) ^ { 2 } - u ( x ) \right) d x .
$$

Show that ϕ achieves its maximum value at some element of E.

Problem 19 Let V be a finite-dimensional vector space over the rationals $\mathbb { Q }$ and let M be an automorphism of V such that M fixes no nonzero vector in V . Suppose that M p is the identity map on $V ,$ where p is a prime number. Show that the dimension of V is divisible by $p - 1$

Problem 20 Let $M _ { 2 \times 2 }$ be the four-dimensional vector space of all ${ \mathcal { Q } } \times { \mathcal { Q } }$ real matrices and define $f : M _ { 2 \times 2 } \longrightarrow M _ { 2 \times 2 }$ by $f ( X ) = X ^ { 2 }$

1. Show that f has a local inverse near the point

$$
X = { \binom { 1 } { 0 } } \ 1 ) .
$$

2. Show that f does not have a local inverse near the point

$$
X = \left( \begin{array} { c c } { { 1 } } & { { 0 } } \\ { { 0 } } & { { - 1 } } \end{array} \right) .
$$