# Preliminary Exam - Fall 1995

Problem 1 Let G be a group generated by n elements. Find an upper bound $N ( n , k )$ for the number of subgroups H of G with the index $[ G : H ] = k$

Problem 2 Let A be a finite subset of the unit disc in the plane, and let $N ( A , r )$ be the set of points at distance $\leqslant r$ from A, where $0 < r < 1$ . Show that the length of the boundary $N ( A , r )$ is, at most, C/r for some constant C independent of A.

Problem 3 Find the radius of convergence R of the Taylor series about $z = 1$ of the function $f ( z ) = 1 / ( 1 + z ^ { 2 } + z ^ { 4 } + z ^ { 6 } + z ^ { 8 } + z ^ { 1 0 } )$ . Express the answer in terms of real numbers and square roots only.

Problem 4 Suppose A and B are real $n \times n$ matrices and C is a complex $n \times n$ matrix such that

$$
{ \cal C } A { \cal C } ^ { - 1 } = B \ .
$$

Find a real $n \times n$ matrix D such that $D A D ^ { - 1 } = B$

Problem 5 Prove that $\mathbb { Q } \left[ x , y \right] / \langle x ^ { 2 } + y ^ { 2 } - 1 \rangle$ is an integral domain and that its field of fractions is isomorphic to the field of rational functions $\mathbb { Q } \left( t \right)$

Problem 6 Determine all real numbers $L > 1$ so that the boundary value problem

$$
\begin{array} { c } { { x ^ { 2 } y ^ { \prime \prime } ( x ) + y ( x ) = 0 , \qquad 1 \leqslant x \leqslant L } } \\ { { \qquad y ( 1 ) = y ( L ) = 0 } } \end{array}
$$

has a nonzero solution.

Problem 7 Let $\textstyle g ( z ) = \sum _ { n = 0 } ^ { \infty } g _ { n } z ^ { n }$ and $\textstyle h ( z ) = \sum _ { n = 0 } ^ { \infty } h _ { n } z ^ { n }$ be entire functions. Find a formula for the coefficients $f _ { n }$ in the Taylor expansion about $z = 0 \ o f$

$$
f ( z ) = \frac { 1 } { 2 \pi i } \int _ { | w | = 1 } g ( z / w ) h ( w ) \frac { d w } { w } \cdotp
$$

Problem 8 Show that an $n \times n$ matrix of complex numbers A satisfying

$$
| a _ { i i } | > \sum _ { j \neq i } | a _ { i j } |
$$

for $1 \leqslant i \leqslant n$ must be invertible.

Problem 9 Let $x _ { 1 }$ be a real number, $0 < x _ { 1 } < 1$ , and define a sequence by   
$x _ { n + 1 } = x _ { n } - x _ { n } ^ { n + 1 }$ . Show that lim inf $x _ { n } > 0$ n →∞

Problem 10 Let F be a field and $\mathbf { F } ^ { * }$ be the multiplicative group of nonzero elements. Let G be a subgroup of $\mathbf { F } ^ { * }$ of finite order n. Show that G is cyclic.

Problem 11 Let $f ( z ) = u ( z ) + i v ( z )$ be holomorphic on $| z | < 1$ , u and v real. Show that

$$
\int _ { 0 } ^ { 2 \pi } u ( r e ^ { i \theta } ) ^ { 2 } d \theta = \int _ { 0 } ^ { 2 \pi } v ( r e ^ { i \theta } ) ^ { 2 } d \theta
$$

for 0 < r < 1 if $u ( 0 ) ^ { 2 } = v ( 0 ) ^ { 2 }$

Problem 12 Let f and f0 be continuous on $[ 0 , \infty )$ and $f ( x ) = 0$ for $x \geqslant$ $1 0 ^ { 1 0 }$ . Show that

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

Problem 13 Show that

$$
( 1 + z + z ^ { 2 } + \cdot \cdot \cdot + z ^ { 9 } ) ( 1 + z ^ { 1 0 } + z ^ { 2 0 } + \cdot \cdot \cdot + z ^ { 9 0 } ) ( 1 + z ^ { 1 0 0 } + z ^ { 2 0 0 } + \cdot \cdot \cdot + z ^ { 9 0 0 } ) \cdot \cdot \cdot = \frac { 1 } { 1 - z }
$$

for |z| < 1.

Problem 14 Let $f ( x ) \ \in \ \mathbb { Q } \left[ x \right]$ be a polynomial with rationalcoefficients. Show that there is a $g ( x ) \in \mathbb { Q } \left[ x \right] , g \neq 0$ , such that $f ( x ) g ( x ) = a _ { 2 } x ^ { 2 } + a _ { 3 } x ^ { 3 } +$ $a _ { 5 } x ^ { 5 } + \cdots + a _ { p } x ^ { p }$ is a polynomial in which only prime exponents appear.

Problem 15 Let $f : \mathbb { R }  \mathbb { R }$ be a $C ^ { \infty }$ function. Assume that $f ( x )$ has a local minimum at $x = 0$ . Prove there is a disc centered on the y axis which lies above the graph of f and touches the graph at $( 0 , f ( 0 ) )$ .

Problem 16 Let A and B be nonsimilar $n \times n$ complex matrices with the same minimal and the same characteristic polynomial. Show that $n \geqslant 4$ and the minimal polynomial is not equal to the characteristic polynomial.

Problem 17 Let $f _ { 1 } , f _ { 2 } , \ldots , f _ { n }$ be continuous real valued functions on $[ a , b ]$ Show that the set $\{ f _ { 1 } , \ldots , f _ { n } \}$ is linearly dependent on [a, b] if and only if

$$
\operatorname* { d e t } \left( \int _ { a } ^ { b } f _ { i } ( x ) f _ { j } ( x ) d x \right) = 0 .
$$

Problem 18 Let $f : \mathbb { R } \to \mathbb { R }$ be a nonzero $C ^ { \infty }$ function such that $f ( x ) f ( y ) =$ $f \left( { \sqrt { x ^ { 2 } + y ^ { 2 } } } \right)$ for all x and y and that $f ( x )  0 \ a s \ | x |  \infty$

1. Prove that f is an even function and that f (0) is 1.

2. Prove that f satisfies the differential equation $f ^ { \prime } ( x ) = f ^ { \prime \prime } ( 0 ) x f ( x )$ , and find the most general function satisfying the given conditions.