# Preliminary Exam - Spring 1985

Problem 1 Let $f ( x ) , 0 \leqslant x < \infty$ , be continuous, differentiable, with $f ( 0 ) =$ 0, and that $f ^ { \prime } ( x )$ is an increasing function of x for $x \geqslant 0$ . Prove that

$$
g ( x ) = { \left\{ \begin{array} { l l } { f ( x ) / x , } & { x > 0 } \\ { f ^ { \prime } ( 0 ) , } & { x = 0 } \end{array} \right. }
$$

is an increasing function of x.

Problem 2 In a commutative group G, let the element a have order r, let b have order $s \left( r , s < \infty \right)$ , and assume that the greatest common divisor of r and s is 1. Show that ab has order rs.

Problem 3 Show that a necessary and sufficient condition for three points a, b, and c in the complex plane to form an equilateral triangle is that

$$
a ^ { 2 } + b ^ { 2 } + c ^ { 2 } = b c + c a + a b .
$$

Problem 4 Let $R > 1$ and let f be analytic on $| z | < R$ except at $z = 1$ where f has a simple pole. If

$$
f ( z ) = \sum _ { n = 0 } ^ { \infty } a _ { n } z ^ { n } \qquad ( | z | < 1 )
$$

is the Maclaurin series for f , show that lim $a _ { n }$ exists. n→∞

Problem 5 Factor $x ^ { 4 } + x ^ { 3 } + x + 3$ completely in $\mathbb { Z } _ { 5 } [ x ]$

Problem 6 Let A and B be two n×n self-adjoint $( i . e .$ , Hermitian) matrices over C such that all eigenvalues of A lie in $[ a , a ^ { \prime } ]$ and all eigenvalues of B lie in $[ b , b ^ { \prime } ]$ . Show that all eigenvalues of $A + B$ lie in $[ a + b , a ^ { \prime } + b ^ { \prime } ]$

Problem 7 Prove that

$$
\int _ { 0 } ^ { \infty } e ^ { - x ^ { 2 } } \cos ( 2 b x ) d x = { \frac { 1 } { 2 } } { \sqrt { \pi } } e ^ { - b ^ { 2 } } .
$$

What restrictions, if any, need be placed on b?

Problem 8 Let $h > 0$ be given. Consider the linear difference equation

$$
\frac { y \left( ( n + 2 ) h \right) - 2 y \left( ( n + 1 ) h \right) + y ( n h ) } { h ^ { 2 } } = - y ( n h ) , \quad n = 0 , 1 , 2 , \ldots .
$$

(Note the analogy with the differential equation $y ^ { \prime \prime } = - y . )$

1. Find the general solution of the equation by trying suitable exponential substitutions.

2. Find the solution with $y ( 0 ) = 0 \ a n d y ( h ) = h$ . Denote it by $S _ { h } ( n h ) , n =$ 1, 2, . . ..

3. Let x be fixed and $h = { \frac { x } { n } }$ . Show that

$$
\operatorname* { l i m } _ { n \to \infty } S _ { \frac { x } { n } } \left( { \frac { n x } { n } } \right) = \sin x .
$$

Problem 9 Define the function ζ by

$$
\zeta ( x ) = \sum _ { n = 1 } ^ { \infty } { \frac { 1 } { n ^ { x } } } \cdot
$$

Prove that $\zeta ( x )$ is defined and has continuous derivatives of all orders in the interval $1 < x < \infty$

Problem 10 For arbitrary elements a, b, and c in a field F, compute the minimal polynomial of the matrix

$$
\left( \begin{array} { c c c } { { 0 } } & { { 0 } } & { { a } } \\ { { 1 } } & { { 0 } } & { { b } } \\ { { 0 } } & { { 1 } } & { { c } } \end{array} \right) .
$$

Problem 11 Let $\mathbf { F } = \{ a + b { \sqrt [ 3 ] { 2 } } + c { \sqrt [ 3 ] { 4 } } \mid a , b , c \in \mathbb { Q } \}$ . Prove that F is a field and each element in F has a unique representation as $a + b { \sqrt [ 3 ] { 2 } } + c { \sqrt [ 3 ] { 4 } }$ with $a , b , c \in \mathbb { Q }$ . Find $ ( 1 - \sqrt [ 3 ] { 2 } ) ^ { - 1 }$ in F.

Problem 12 Prove that

$$
\int _ { 0 } ^ { \infty } { \frac { x ^ { \alpha - 1 } } { 1 + x } } d x = { \frac { \pi } { \sin \pi \alpha } } .
$$

What restrictions must be placed on α $\boldsymbol { \ell }$

Problem 13 Prove that for any $a \in \mathbb { C }$ and any integer $n \geqslant 2$ , the equation $1 + z + a z ^ { n } = 0$ has at least one root in the disc $| z | \leqslant ~ 2$

Problem 14 Show that

$$
I = \int _ { 0 } ^ { \pi } \log ( \sin x ) d x
$$

converges as an improper Riemann integral. Evaluate I.

Problem 15 Let $\zeta = e ^ { \frac { 2 \pi i } { 7 } }$ be a primitive $7 ^ { t h }$ root of unity. Find a cubic polynomial with integer coefficients having $\alpha = \zeta + \zeta ^ { - 1 }$ as a root.

Problem 16 Let f be continuous on $\mathbb { R }$ , and let

$$
f _ { n } ( x ) = { \frac { 1 } { n } } \sum _ { k = 0 } ^ { n - 1 } f \left( x + { \frac { k } { n } } \right) .
$$

Prove that $f _ { n } ( x )$ converges uniformly to a limit on every finite interval $[ a , b ]$

Problem 17 Let $v _ { 1 }$ and $v _ { 2 }$ be two real valued continuous functions on R such that $v _ { 1 } ( x ) < v _ { 2 } ( x )$ for all $x \in \mathbb { R }$ . Let $\varphi _ { 1 } ( t )$ and $\varphi _ { 2 } ( t )$ be, respectively, solutions of the differential equations

$$
\frac { d x } { d t } = v _ { 1 } ( x ) \quad a n d \quad \frac { d x } { d t } = v _ { 2 } ( x )
$$

for $a < t < b$ $I f \varphi _ { 1 } ( t _ { 0 } ) = \varphi _ { 2 } ( t _ { 0 } )$ for some $t _ { 0 } \in ( a , b )$ , show that $\varphi _ { 1 } ( t ) \leqslant \varphi _ { 2 } ( t )$ for all $t \in ( t _ { 0 } , b )$

Problem 18 Let A and B be two $n \times n$ self-adjoint $( i . e .$ , Hermitian) matrices over C and assume A is positive definite. Prove that all eigenvalues of AB are real.

Problem 19 Let F be a finite field. Give a complete proof of the fact that the number of elements of F is of the form $p ^ { r }$ , where $p \geqslant 2$ is a prime number and r is an integer $\geqslant 1$

Problem 20 Let $f ( z )$ be an analytic function that maps the open disc $| z | < 1$ into itself. Show that $| f ^ { \prime } ( z ) | \leqslant 1 / ( 1 - | z | ^ { 2 } )$ .