# Preliminary Exam - Summer 1983

Problem 1 The number 21982145917308330487013369 is the thirteenth power of a positive integer. Which positive integer?

Problem 2 Let $f : \mathbb { C } \to \mathbb { C }$ be an analytic function such that

$$
\left( 1 + | z | ^ { k } \right) ^ { - 1 } \frac { d ^ { m } f } { d z ^ { m } }
$$

is bounded for some k and m. Prove that $d ^ { n } f / d z ^ { n }$ is identically zero for sufficiently large n. How large must n be, in terms of k and m?

Problem 3 Let A be an $n \times n$ complex matrix, and let $\chi$ and $\mu$ be the characteristic and minimal polynomials of A. Suppose that

$$
\chi ( x ) = \mu ( x ) ( x - i ) ,
$$

$$
\mu ( x ) ^ { 2 } = \chi ( x ) ( x ^ { 2 } + 1 ) .
$$

Determine the Jordan Canonical Form of A.

Problem 4 Outline a proof, starting from basic properties of the real numbers, of the following theorem: Let $f : [ a , b ] \to \mathbb { R }$ be a continuous function such that $f ^ { \prime } ( x ) = 0$ for all $x \in ( a , b )$ . Then $f ( b ) = f ( a )$

Problem 5 Let $b _ { 1 } , b _ { 2 } , \ldots$ . be positive real numbers with

$$
\operatorname* { l i m } _ { n \to \infty } b _ { n } = \infty { \mathrm { ~ } } a n d \operatorname* { l i m } _ { n \to \infty } ( b _ { n } / b _ { n + 1 } ) = 1 .
$$

Assume also that $b _ { 1 } < b _ { 2 } < b _ { 3 } < \cdot \cdot \cdot$ Show that the set of quotients $( b _ { m } / b _ { n } ) _ { 1 \leqslant n < m }$ is dense in $( 1 , \infty )$

Problem 6 Let V be a real vector space of dimension n with a positive definite inner product. We say that two bases $\left( { { a } _ { i } } \right)$ and $\left( b _ { i } \right)$ have the same orientation if the matrix of the change of basis from $\left( { { a } _ { i } } \right)$ to $\left( b _ { i } \right)$ has a positive determinant. Suppose now that $( a _ { i } )$ and $\left( b _ { i } \right)$ are orthonormal bases with the same orientation. Show that $( a _ { i } + 2 b _ { i } )$ is again a basis of V with the same orientation as $\left( { { a } _ { i } } \right)$

Problem 7 Compute

$$
\int _ { 0 } ^ { \infty } { \frac { \log x } { x ^ { 2 } + a ^ { 2 } } } d x
$$

where $a > 0$ is a constant.

Problem 8 Let $G _ { 1 } , G _ { 2 }$ , and $G _ { 3 }$ be finite groups, each of which is generated by its commutators (elements of the form $x y x ^ { - 1 } y ^ { - 1 } )$ . Let A be a subgroup of $G _ { 1 } \times G _ { 2 } \times G _ { 3 }$ , which maps surjectively, by the natural projection map, to the partial products $G _ { 1 } \times G _ { 2 } , G _ { 1 } \times G _ { 3 }$ and $G _ { 2 } \times G _ { 3 }$ . Show that A is equal to $G _ { 1 } \times G _ { 2 } \times G _ { 3 }$

Problem 9 Suppose Ω is a bounded domain in C with a boundary consisting of a smooth Jordan curve γ. Let f be holomorphic on a neighborhood of the closure $o f \Omega$ , and suppose that $f ( z ) \neq 0$ for $z \in \gamma$ . Let $z _ { 1 } , \ldots , z _ { k }$ be the zeros of f in Ω, and let $n _ { j }$ be the order of the zero of f at $z _ { j } \ ( f o r \ j = 1 , \dots , k )$

1. Use Cauchy’s integral formula to show that

$$
{ \frac { 1 } { 2 \pi i } } \int _ { \gamma } { \frac { f ^ { \prime } ( z ) } { f ( z ) } } d z = \sum _ { j = 1 } ^ { k } n _ { j } .
$$

2. Suppose that f has only one zero $z _ { 1 }$ in Ω with multiplicity $n _ { 1 } = 1$ . Find a boundary integral involving f whose value is the point $z _ { 1 }$

Problem 10 Let f be a twice differentiable real valued function on $[ 0 , 2 \pi ]$ with $\begin{array} { r } { \int _ { 0 } ^ { 2 \pi } f ( x ) d x = 0 = f ( 2 \pi ) - f ( 0 ) } \end{array}$ . Show that

$$
\int _ { 0 } ^ { 2 \pi } \left( f ( x ) \right) ^ { 2 } d x \leqslant \int _ { 0 } ^ { 2 \pi } \left( f ^ { \prime } ( x ) \right) ^ { 2 } d x .
$$

Problem 11 Find the eigenvalues, eigenvectors, and the Jordan Canonical Form of

$$
A = \left( \begin{array} { l l l } { { 2 } } & { { 1 } } & { { 1 } } \\ { { 1 } } & { { 2 } } & { { 1 } } \\ { { 1 } } & { { 1 } } & { { 2 } } \end{array} \right) ,
$$

considered as a matrix with entries in $\mathbf { F } _ { 3 } = \mathbb { Z } / 3 \mathbb { Z }$

Problem 12 Prove that a continuous function from R to R which maps open sets to open sets must be monotonic.

Problem 13 Let A be an n×n complex matrix, all of whose eigenvalues are equal to 1. Suppose that the set $\{ A ^ { n } \mid n = 1 , 2 , . . . \}$ is bounded. Show that A is the identity matrix.

Problem 14 $L e t G$ be a transitive subgroup of the group $S _ { n }$ of permutations of n objects $\{ 1 , \ldots , n \}$ . Suppose that G is a simple group and that ∼ is an equivalence relation on $\{ 1 , \ldots , n \}$ such that $i \sim j$ implies that $\sigma ( i ) \sim \sigma ( j )$ for all $\sigma \in G$ . What can one conclude about the relation $\sim \ell$

Problem 15 Let f be analytic on and inside the unit circle $C = \{ z \mid | z | = 1 \}$ Let L be the length of the image of C under f. Show that $L \geqslant 2 \pi | f ^ { \prime } ( 0 ) |$

Problem 16 Let Ω be an open subset $o f \mathbb { R } ^ { 2 }$ , and let $f : \Omega \to { \mathbb { R } } ^ { 2 }$ be a smooth map. Assume that f preserves orientation and maps any pair of orthogonal curves to a pair of orthogonal curves. Show that f is holomorphic. Note: Here we identify $\mathbb { R } ^ { 2 }$ with C .

Problem 17 Let A be an $n \times n$ Hermitian matrix satisfying the condition

$$
A ^ { 5 } + A ^ { 3 } + A = 3 I .
$$

Show that $A = I$

Problem 18 Find all real valued $C ^ { 1 }$ solutions $y ( x )$ of the differential equation

$$
x \frac { d y } { d x } + y = x \quad ( - 1 < x < 1 ) .
$$

Problem 19 Compute the area of the image of the unit disc $\{ z | \ | z | < 1 \}$ under the map $f ( z ) = z + z ^ { 2 } / 2$

Problem 20 Let $f : \mathbb { R } \to \mathbb { R }$ be continuously differentiable, periodic of period 1, and nonnegative. Show that

$$
{ \frac { d } { d x } } \left( { \frac { f ( x ) } { 1 + c f ( x ) } } \right) \to 0 \quad ( a s c \to \infty )
$$

uniformly in x.