# Preliminary Exam - Fall 1998

Problem 1 Let M be a metric space with metric d. Let C be a compact subset of M , and let $( U _ { \alpha } ) _ { \alpha \in I }$ be an open cover of C. Show that there exists $\epsilon > 0$ such that, for every $p \in C$ , the open ball $B ( p , \epsilon )$ is contained in at least one of the sets $U _ { \alpha }$

Problem 2 Find a function $y ( x )$ such that $y ^ { ( 4 ) } + y = 0$ for $x \geqslant 0 , y ( 0 ) = 0$ $y ^ { \prime } ( 0 ) = 1 \ a n d \operatorname* { l i m } _ { x \to \infty } y ( x ) = \operatorname* { l i m } _ { x \to \infty } \dot { y } ^ { \prime } ( x ) = 0$

Problem 3 Prove that for any real α > 1, $\int _ { 0 } ^ { \infty } { \frac { d x } { 1 + x ^ { \alpha } } } = { \frac { \pi / \alpha } { \sin ( \pi / \alpha ) } }$

Problem 4 Let f be analytic in the closed unit disc, $w i t h f ( - \log 2 ) = 0$ and $| f ( z ) | \leqslant | e ^ { z } |$ for $a l l \ z \ w i t h \ | z | = 1$ . How large can $| f ( \log 2 ) |$ be?

Problem 5 Let $f : \mathbb { R } ^ { n } \longrightarrow \mathbb { R }$ be a function such that

1. the function g defined by $g ( x , y ) = f ( x + y ) - f ( x ) - f ( y )$ is bilinear,

2. for all $x \in \mathbb { R } ^ { n }$ and $t \in \mathbb { R } , f ( t x ) = t ^ { 2 } f ( x )$

Show that there is a linear transformation $A : \mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ such that $f ( x ) =$ $\langle x , A x \rangle$ where $\langle \cdot , \cdot \rangle$ is the usual inner product on $\mathbb { R } ^ { n }$ (in other words, f is a quadratic form).

Problem 6 Let A and B be linear transformations on a finite dimensional vector space V . Prove that dim ker $( A B ) \leqslant$ dim ker A + dim ker B.

Problem 7 A real symmetric $n \times n$ matrix A is called positive semi-definite $i f x ^ { t } A x \geqslant 0$ for all $x \in \mathbb { R } ^ { n }$ . Prove that A is positive semi-definite if and only if tr $A B \geqslant 0$ for every real symmetric positive semi-definite n×n matrix B.

Problem 8 Let R be a finite ring with identity. Let a be an element of R which is not a zero divisor. Show that a is invertible.

Problem 9 Suppose that G is a finite group such that every Sylow subgroup is normal and abelian. Show that G is abelian.

Problem 10 Let f be a real function on $[ a , b ]$ . Assume that f is differentiable and that f 0 is Riemann integrable. Prove that

$$
\int _ { a } ^ { b } f ^ { \prime } ( x ) d x = f ( b ) - f ( a ) .
$$

Problem 11 Find the minimal value of the areas of hexagons circumscribing the unit circle in $\mathbb { R } ^ { 2 }$

Note: See also Problem ??.

Problem 12 Let $\varphi ( x , y )$ be a function with continuous second order partial derivatives such that

1. $\varphi _ { x x } + \varphi _ { y y } + \varphi _ { x } = 0$ in the punctured plane $\mathbb { R } ^ { 2 } \backslash \{ 0 \}$

$$
{ \mathrm { ~ 2 . ~ } } r \varphi _ { x } \to { \frac { x } { 2 \pi r } } \ a n d \ r \varphi _ { y } \to { \frac { y } { 2 \pi r } } \ a s \ r = { \sqrt { x ^ { 2 } + y ^ { 2 } } } \to 0 .
$$

Let $C _ { R }$ be the circle $x ^ { 2 } + y ^ { 2 } = R ^ { 2 }$ . Show that the line integral

$$
\int _ { C _ { R } } e ^ { x } ( - \varphi _ { y } d x + \varphi _ { x } d y )
$$

is independent of R, and evaluate it.

Problem 13 Let f be an entire function. Define $\Omega \ = \ \mathbb { C } \ \backslash \ ( - \infty , 0 ]$ , the complex plane with the ray $( - \infty , 0 ]$ removed. Suppose that for all $z \in \Omega$ , $| f ( z ) | \leqslant | \log z |$ , where log z is the principal branch of the logarithm. What can one conclude about the function $f ?$

Problem 14 Let $z _ { 1 } , \ldots , z _ { n }$ be distinct complex numbers, and leta $\iota _ { 1 } , \ldots , a _ { n }$ be nonzero complex numbers such that $\begin{array} { r } { \begin{array} { r } { S _ { p } = \sum _ { j = 1 } ^ { n } a _ { j } z _ { j } ^ { p } = 0 f o r p = 0 , 1 , \ldots , m - } \end{array} } \end{array}$ 1 but $S _ { m } \neq 0$ . Here $1 \leqslant$ m $\leqslant n - 1$ How many zeros does the rational function $\begin{array} { r } { f ( z ) = \sum _ { j = 1 } ^ { n } \frac { a _ { j } } { z - z _ { j } } } \end{array}$ have in C? Why is m $\geqslant n$ impossible.

Problem 15 Let A and B be n×n matrices. Show that the eigenvalues of AB are the same as the eigenvalues of BA.

Problem 16 Let B be a 3×3 matrix whose null space is 2-dimensional, and let $\chi ( \lambda )$ be the characteristic polynomial of B. For each assertion below, provide either a proof or a counterexample.

1. $\lambda ^ { 2 }$ is a factor of $\chi ( \lambda )$

2. The trace of B is an eigenvalue of B.

3. B is diagonalizable.

Problem 17 Let F be a finite field with q elements. Denote by $G L _ { n } ( \mathbf { F } )$ the group of invertible n×n matrices with entries if F. What is the order of this group?

Problem 18 Show that the field $\mathbb { Q } ( t _ { 1 } , \ldots , t _ { n } )$ of rational functions in n variables over the rational numbers is isomorphic to a subfield of R.