# Preliminary Exam - Spring 1984

Problem 1 Evaluate

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

for $a > 0$

Problem 2 For a p-group of order $p ^ { 4 }$ , assume the center of G has order $p ^ { 2 }$ Determine the number of conjugacy classes of G.

Problem 3 Let $f : [ 0 , 1 ] \to \mathbb { R }$ be continuous function, wi $t h f ( 0 ) = f ( 1 ) = 0$ Assume that $f ^ { \prime \prime }$ exists on $0 \textless x \textless 1$ 1, with $f ^ { \prime \prime } + 2 f ^ { \prime } + f \geqslant 0$ . Show that $f ( x ) \leqslant 0$ for all $0 \leqslant x \leqslant 1$

Problem 4 Which number is larger, $\pi ^ { 3 }$ or $3 ^ { \pi } \ell$

Problem 5 Let A and B be complex $n \times n$ matrices such thatA $\scriptstyle 1 B = B A ^ { 2 }$ and assume A has no eigenvalues of absolute value 1. Prove that A and B have a common (nonzero) eigenvector.

Problem 6 Let a be a positive real number. Define a sequence $( x _ { n } ) \ b y$

$$
x _ { 0 } = 0 , \quad x _ { n + 1 } = a + x _ { n } ^ { 2 } , \quad n \geqslant 0 .
$$

Find a necessary and sufficient condition on a in order that a finite limit lim $x _ { n }$ should exist.   
n→∞

Problem 7 Find the number of roots of

$$
z ^ { 7 } - 4 z ^ { 3 } - 1 1 = 0
$$

which lie between the two circles $| z | = 1$ and $| z | = 2$

Problem 8 Show that the system of differential equations

$$
{ \frac { d } { d t } } \left( { \begin{array} { l } { x } \\ { y } \\ { z } \end{array} } \right) = \left( { \begin{array} { l l l } { 0 } & { 1 } & { 0 } \\ { 2 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 3 } \end{array} } \right) \left( { \begin{array} { l } { x } \\ { y } \\ { z } \end{array} } \right)
$$

has a solution which tends to ∞ as $t  - \infty$ and tends to the origin as $t \to + \infty$

Problem 9 Let A be a real $m \times n$ matrix with rational entries and let b be an m-tuple of rational numbers. Assume that the system of equations $A x = b$ has a solution x in complex n-space C n. Show that the equation has a solution vector with rational components, or give a counterexample.

Problem 10 Let R be a principal ideal domain and let I and J be nonzero ideals in R. Show that $\Im \Im = \Im \cap \Im$ if and only if $\Im + \Im = R$

Problem 11 Prove the following statement or supply a counterexample: If A and B are real $n \times n$ matrices which are similar over $\mathbb { C }$ , then A and B are similar over R.

Problem 12 Consider the equation

$$
{ \frac { d y } { d x } } = y - \sin y .
$$

Show that there is an $\varepsilon > 0$ such that $i f \left| y _ { 0 } \right| < \varepsilon$ , then the solution $y = f ( x )$ with $f ( 0 ) = y _ { 0 }$ satisfies

$$
\operatorname* { l i m } _ { x \to - \infty } f ( x ) = 0 .
$$

Problem 13 Let I be an open interval in R containing zero. Assume that f0 exists on a neighborhood of zero and $f ^ { \prime \prime } ( 0 )$ exists. Show that

$$
f ( x ) = f ( 0 ) + f ^ { \prime } ( 0 ) \sin x + { \frac { 1 } { 2 } } f ^ { \prime \prime } ( 0 ) \sin ^ { 2 } x + o ( x ^ { 2 } )
$$

$( o ( x ^ { 2 } )$ denotes a quantity such that ${ \frac { o ( x ^ { 2 } ) } { x ^ { 2 } } }  0 \ a s \ x  0 )$

Problem 14 Let F be a field and let X be a finite set. Let $R ( X , \mathbf { F } )$ be the ring of all functions from X to F, endowed with the pointwise operations. What are the maximal ideals of $R ( X , \mathbf { F } ) \ ?$

Problem 15 Let F be a continuous complex valued function on the interval [0, 1]. Let

$$
f ( z ) = \int _ { 0 } ^ { 1 } { \frac { F ( t ) } { t - z } } d t ,
$$

$f o r \ z \ a$ complex number not in [0, 1].

1. Prove that f is an analytic function.

2. Express the coefficients of the Laurent series of f about ∞ in terms of F . Use the result to show that F is uniquely determined by f .

Problem 16 Prove, or supply a counterexample: If A is an invertible $n \times n$ complex matrix and some power of A is diagonal, then A can be diagonalized.

Problem 17 Prove that the Taylor coefficients at the origin of the function

$$
f ( z ) = { \frac { z } { e ^ { z } - 1 } }
$$

are rational numbers.

Problem 18 Prove or supply a counterexample: If the function f from R to R has both a left limit and a right limit at each point of R, then the set of discontinuities of f is, at most, countable.

Problem 19 Let $f ( x ) = x \log { ( 1 + x ^ { - 1 } ) } , 0 < x < \infty .$

1. Show that f is strictly monotonically increasing.

2. Compute lim f (x) as $x \to 0$ and $x \to \infty$

Problem 20 Determine all finitely generated abelian groups G which have only finitely many automorphisms.