# Preliminary Exam - Summer 1985

Problem 1 1. Show that a real ${ \mathcal { Q } } \times { \mathcal { Q } }$ matrix A satisfies $A ^ { 2 } = - I$ if and only if

$$
A = \left( \begin{array} { c c } { { \pm \sqrt { p q - 1 } } } & { { - p } } \\ { { q } } & { { \mp \sqrt { p q - 1 } } } \end{array} \right)
$$

where p and q are real numbers such that pq $\geqslant 1$ and both upper or both lower signs should be chosen in the double signs.

2. Show that there is no real ${ \it 2 } \times { \it 2 }$ matrix A such that

$$
A ^ { 2 } = { \binom { - 1 } { 0 } } \ { \begin{array} { l } { 0 } \\ { - 1 - \varepsilon } \end{array} } \quad
$$

with $\varepsilon > 0$

Problem 2 1. For $\begin{array} { r } { 0 \leqslant \theta \leqslant \frac { \pi } { 2 } } \end{array}$ , show that

$$
\sin \theta \geqslant { \frac { 2 } { \pi } } \theta .
$$

2. By using Part 1, or by any other method, show that if $\lambda < 1$ , then

$$
\operatorname* { l i m } _ { R \to \infty } R ^ { \lambda } \int _ { 0 } ^ { \frac { \pi } { 2 } } e ^ { - R \sin \theta } d \theta = 0 .
$$

Problem 3 Let A be a nonsingular real $n \times n$ matrix. Prove that there exists a unique orthogonal matrix Q and a unique positive definite symmetric matrix B such that $A = Q B$

Problem 4 Let G be a group of order 120, let H be a subgroup of order 24, and assume that there is at least one left coset of H (other than H itself ) which is equal to some right coset of H. Prove that H is a normal subgroup of G.

Problem 5 By the Fundamental Theorem of Algebra, the polynomial $x ^ { 3 } +$ $2 x ^ { 2 } + 7 x + 1$ has three complex roots, $\alpha _ { 1 } , \alpha _ { 2 }$ , and α3. Compute $\alpha _ { 1 } ^ { 3 } + \alpha _ { 2 } ^ { 3 } + \alpha _ { 3 } ^ { 3 }$

Problem 6 Evaluate the integral

$$
\int _ { 0 } ^ { \infty } { \frac { x ^ { a - 1 } } { x + 1 } } d x
$$

where a is a complex number. What restrictions must be put on a $\ell$

Problem 7 Let

$$
f ( x ) = e ^ { x ^ { 2 } / 2 } \int _ { x } ^ { \infty } e ^ { - t ^ { 2 } / 2 } d t
$$

for $x > 0$

1. Show that $0 < f ( x ) < { \frac { 1 } { x } } .$

2. Show that f (x) is strictly decreasing for $x > 0$

Problem 8 Let f be a real valued continuous function on a compact interval $[ a , b ]$ Given $\varepsilon > 0$ , show that there is a polynomial p such $t h a t p ( a ) = f ( a )$ , $p ^ { \prime } ( a ) = 0$ , and $| p ( x ) - f ( x ) | < \varepsilon$ for $x \in [ a , b ]$

Problem 9 Let $u ( x ) , 0 \leqslant x \leqslant 1$ , be a real valued $C ^ { 2 }$ function which satisfies the differential equation

$$
u ^ { \prime \prime } ( x ) = e ^ { x } u ( x ) .
$$

1. Show that if $0 < x _ { 0 } < 1$ , then u cannot have a positive local maximum at $x _ { 0 }$ . Similarly, show that u cannot have a negative local minimum at $x _ { 0 }$

2. Now suppose that $u ( 0 ) = u ( 1 ) = 0$ . Prove that $u ( x ) \equiv 0 , 0 \leqslant x \leqslant 1$

Problem 10 Prove that for each $\lambda > 1$ the equation $z = \lambda - e ^ { - z }$ in the half-plane $\Re z \geqslant 0$ has exactly one root, and that this root is real.

Problem 11 1. Let G be a cyclic group, and let $a , b \in G$ be elements which are not squares. Prove that ab is a square.

2. Give an example to show that this result is false if the group is not cyclic.

Problem 12 Let A be an $n \times n$ real matrix and $A ^ { t }$ its transpose. Show that $A ^ { t } A$ and At have the same range.

Problem 13 Let $P ( z )$ be a polynomial of degree $< k$ with complex coefficients. Let $\omega _ { 1 } , \ldots , \omega _ { k }$ be the $k ^ { t h }$ roots of unity in C . Prove that

$$
{ \frac { 1 } { k } } \sum _ { i = 1 } ^ { k } P ( \omega _ { i } ) = P ( 0 ) .
$$

Problem 14 Let $M _ { n } ( \mathbf { F } )$ denote the ring of n×n matrices over a field F. For $n \geqslant 1$ , does there exist a ring homomorphism from $M _ { n + 1 } ( \mathbf { F } )$ onto $M _ { n } ( \mathbf { F } ) \mathbf { \mathcal { Q } }$

Problem 15 For each $k > 0$ , let $X _ { k }$ be the set of analytic functions $f ( z )$ on the open unit disc D such that

$$
\operatorname* { s u p } _ { z \in \mathbb { D } } \left\{ ( 1 - | z | ) ^ { k } \left| f ( z ) \right| \right\}
$$

is finite. Show that $f \in X _ { k }$ if and only if $f ^ { \prime } \in X _ { k + 1 }$

Problem 16 A function $f : [ 0 , 1 ] \to \mathbb { R }$ is said to be upper semicontinuous $i f$ given $x \in [ 0 , 1 ]$ and $\varepsilon > 0$ , there exists $a \delta > 0$ such that if $| y - x | < \delta ,$ then $f ( y ) < f ( x ) + \varepsilon$ . Prove that an upper semicontinuous function f on [0, 1] is bounded above and attains its maximum value at some point $p \in [ 0 , 1 ]$

Problem 17 Let

$$
f ( z ) = \sum _ { n = 0 } ^ { \infty } a _ { n } z ^ { n }
$$

where all the $a _ { n }$ are nonnegative reals, and the series has radius of convergence 1. Prove that $f ( z )$ cannot be analytically continued to a function analytic in a neighborhood $o f z = 1$

Problem 18 Solve the differential equations

$$
\begin{array} { c } { { \displaystyle \frac { d y _ { 1 } } { d x } = - 3 y _ { 1 } + 1 0 y _ { 2 } , \hfill } } \\ { { \displaystyle \frac { d y _ { 2 } } { d x } = - 3 y _ { 1 } + 8 y _ { 2 } . \hfill } } \end{array}
$$

Problem 19 Let $A _ { 1 } \geqslant A _ { 2 } \geqslant \dots \geqslant A _ { k } \geqslant 0$ . Evaluate

$$
\operatorname* { l i m } _ { n \to \infty } \left( A _ { 1 } ^ { n } + A _ { 2 } ^ { n } + \cdots + A _ { k } ^ { n } \right) ^ { 1 / n } .
$$

Note: See also Problem ??.

Problem 20 Let F be a field of characteristic $p > 0 , p \neq 3$ . If α is a zero of the polynomial $f ( x ) = x ^ { p } - x + 3$ in an extension field of F, show that $f ( x )$ has p distinct zeros in the field F(α).