# Preliminary Exam - Fall 1996

Problem 1 Let M be the set of real valued continuous functions f on [0, 1] such that $f ^ { \prime }$ is continuous on $[ 0 , 1 ]$ , with the norm

$$
\| f \| = \operatorname* { s u p } _ { 0 \leqslant x \leqslant 1 } | f ( x ) | + \operatorname* { s u p } _ { 0 \leqslant x \leqslant 1 } | f ^ { \prime } ( x ) | ~ .
$$

Which subsets of M are compact?

Problem 2 A real-valued function f on a closed bounded interval $[ a , b ]$ is said to be upper semicontinuous provided that for every $\varepsilon > 0$ and $p \in [ a , b ]$ there is a $\delta = \delta ( \varepsilon , p ) > 0$ such that if $x \in [ a , b ]$ and $| x - p | < \delta$ then $f ( x ) < f ( p ) + \varepsilon$ . Prove that an upper semicontinuous function is bounded above on $[ a , b ]$

Problem 3 Evaluate the integral

$$
I = \int _ { 0 } ^ { \infty } { \frac { \sqrt { x } } { 1 + x ^ { 2 } } } \ d x .
$$

Problem 4 Does there exist a function $f ,$ , analytic in the punctured plane $\mathbb { C } \ \backslash \ \{ 0 \}$ , such that

$$
| f ( z ) | \geqslant { \frac { 1 } { \sqrt { | z | } } }
$$

for all nonzero $z ?$

Problem 5 Prove that any linear transformation $T : \mathbb { R } ^ { 3 } \longrightarrow \mathbb { R } ^ { 3 }$ has

1. a one-dimensional invariant subspace

2. a two-dimensional invariant subspace.

Problem 6 Let A and B be real ${ \it 2 } \times { \it 2 }$ matrices such that

$$
A ^ { 2 } = B ^ { 2 } = I \ , \qquad A B + B A = 0 \ .
$$

Show that there exists a real ${ \mathcal { Q } } \times { \mathcal { Q } }$ matrix $T$ such that

$$
\begin{array} { r } { T A T ^ { - 1 } = \left( \begin{array} { c c } { 1 } & { 0 } \\ { 0 } & { - 1 } \end{array} \right) \ , T B T ^ { - 1 } = \left( \begin{array} { c c } { 0 } & { 1 } \\ { 1 } & { 0 } \end{array} \right) \ . } \end{array}
$$

Problem 7 Suppose p is a prime. Show that every element of ${ \cal G L } _ { 2 } ( { \bf F } _ { p } )$ has order dividing either $p ^ { 2 } - 1$ or $p ( p - 1 )$ .

Problem 8 Show the denominator $o f \left( { 1 \atop n } \right)$ is a power of 2 for all integers n.

Problem 9 For positive integers a, b and c show that

$$
\operatorname* { g c d } \left\{ a , \operatorname { l c m } \{ b , c \} \right\} = \operatorname { l c m } \left\{ \operatorname* { g c d } \{ a , b \} , \operatorname* { g c d } \{ a , c \} \right\} .
$$

Problem 10 If f is $a C ^ { 2 }$ function on an open interval, prove that

$$
\operatorname * { l i m } _ { h \to 0 } \ { \frac { f ( x + h ) - 2 f ( x ) + f ( x - h ) } { h ^ { 2 } } } = f ^ { \prime \prime } ( x ) \ .
$$

Problem 11 Let f be continuous and nonnegative on [0, 1] and suppose that

$$
f ( t ) ^ { 2 } \leqslant 1 + 2 \int _ { 0 } ^ { t } f ( s ) d s ~ .
$$

Prove that $f ( t ) \leqslant 1 + t$ for $0 \leqslant t \leqslant 1$

Problem 12 Find the number of roots, counted with their multiplicities, of

$$
z ^ { 7 } - 4 z ^ { 3 } - 1 1 = 0
$$

which lie between the circles $\left| z \right| = 1 \ a n d \ \left| z \right| = 2$

Problem 13 Define $F : \mathbb { C } ^ { 3 } \to \mathbb { C } ^ { 3 } \quad b y$

$$
F ( u , v , w ) = ( u + v + w , \ u v + v w + u w , \ u v w ) \ .
$$

Show that F is onto but not one-to-one.

Problem 14 Let f be holomorphic on and inside the unit circle C. Let L be the length of the image of C under f. Show that

$$
L \geqslant 2 \pi | f ^ { \prime } ( 0 ) | \ .
$$

Problem 15 Is there a real ${ \mathcal { Q } } \times { \mathcal { Q } }$ matrix A such that

$$
A ^ { 2 0 } = \left( \begin{array} { c c } { { - 1 } } & { { 0 } } \\ { { 0 } } & { { - 1 - \varepsilon } } \end{array} \right) ?
$$

Exhibit such an A or prove there is none.

Problem 16 Let

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

Show that every real matrix B such that $A B = B A$ has the form

$$
B = a I + b A + c A ^ { 2 }
$$

for some real numbers $a , b ,$ and c.

Problem 17 Let $\mathbb { Z } [ x ]$ be the ring of polynomials in the indeterminate x with coefficients in the ring Z of integers. Let ${ \mathfrak { I } } \subset \mathbb { Z } [ x ]$ be the ideal generated by 13 and $x - 4$ . Find an integer m such that $0 \leqslant m \leqslant 1 2$ and

$$
( x ^ { 2 6 } + x + 1 ) ^ { 7 3 } - m \in \Im .
$$

Problem 18 Prove that any finite group is isomorphic to

1. a subgroup of the group of permutations of n objects

2. a subgroup of the group of permutations of n objects which consists only of even permutations.