# Preliminary Exam - Fall 1993

Problem 1 Let X be a metric space and $( x _ { n } )$ a convergent sequence in X with limit $x _ { 0 }$ . Prove that the set $C = \{ x _ { 0 } , x _ { 1 } , x _ { 2 } , \ldots \}$ is compact.

Problem 2 Let A be the additive group of rational numbers, and let M be the multiplicative group of positive rational numbers. Determine all homomorphisms of A into M .

Problem 3 Describe the region in the complex plane where the infinite series

$$
\sum _ { n = 1 } ^ { \infty } { \frac { 1 } { n ^ { 2 } } } \exp \left( { \frac { n z } { z - 2 } } \right)
$$

converges. Draw a sketch of the region.

Problem 4 Let F be a field. For m and n positive integers, let $M _ { m \times n }$ be the vector space of m × n matrices over F. Fix m and n, and fix matrices A and B in $M _ { m \times n }$ . Define the linear transformation T from $M _ { n \times m }$ to $M _ { m \times n } \ b _ { \mathcal { Y } }$ J

$$
T ( X ) = A X B .
$$

Prove that if m $\neq n$ , then T is not invertible.

Problem 5 Let the function $x ( t ) \ ( - \infty < t < \infty )$ be a solution of the $d i f f e r .$ ential equation

$$
\frac { d ^ { 2 } x } { d t ^ { 2 } } - 2 b \frac { d x } { d t } + c x = 0
$$

such that $x ( 0 ) ~ = ~ x ( 1 ) ~ = ~ 0$ (Here, b and c are real constants.) Prove $t h a t x ( n ) = 0$ for every integer n.

Problem 6 Let A, B, and C be finite abelian groups. Prove that if $A \times B$ is isomorphic to $A \times C$ , then B is isomorphic to C.

Problem 7 Let $M _ { n \times n } \ ( n \geqslant 2 )$ be the space of real n×n matrices, identified in the usual way with the Euclidean space $\mathbb { R } ^ { n ^ { 2 } }$ . Let F be the determinant map of $M _ { n \times n }$ into R: $F ( X ) = \operatorname* { d e t } ( X )$ . Find all of the critical points of $F _ { s }$ ; that is, all matrices X such that $D F ( X ) = 0$

Problem 8 Prove that if A is an $n \times n$ matrix over C, and if $A ^ { k } = I$ for some positive integer k, then A is diagonalizable.

Problem 9 Evaluate

$$
\int _ { - \infty } ^ { \infty } { \frac { e ^ { - i x } } { x ^ { 2 } - 2 x + 4 } } d x .
$$

Problem 10 Let f be a continuous real valued function on $[ 0 , \infty )$ . Let A be the set of real numbers a that can be expressed as $a = \operatorname* { l i m } _ { n \to \infty } f ( x _ { n } )$ for some sequence $( x _ { n } )$ in $[ 0 , \infty )$ such that $\scriptstyle \operatorname* { l i m } _ { n \to \infty } x _ { n } = \infty$ . Prove that if A contains the two numbers a and b, then contains the entire interval with endpoints a and b.

Problem 11 Let R be a commutative ring with identity. Let G be a finite subgroup of $R ^ { * }$ , the group of units of R. Prove that if R is an integral domain, then G is cyclic.

Problem 12 Evaluate the integral $\textstyle { \frac { 1 } { 2 \pi i } } \int _ { \gamma } f ( z )$ dz for the function $f ( z ) ~ =$ $z ^ { - 2 } ( 1 - z ^ { 2 } ) ^ { - 1 } e ^ { z }$ and the curve γ depicted $b y$

$$
\cdot \cdot \cdot \operatorname { \mathrm { ~ ‰ ~ } }
$$

Problem 13 Show that there are at least two nonisomorphic nonabelian groups of order 40.

Problem 14 Let n be an integer larger than 1. Is there a differentiable function on [0, ∞) whose derivative equals its $n ^ { t h }$ power and whose value at the origin is positive?

Problem 15 Prove that the matrix

$$
\left( \begin{array} { l l l l } { 0 } & { 5 } & { 1 } & { 0 } \\ { 5 } & { 0 } & { 5 } & { 1 } \\ { 1 } & { 5 } & { 0 } & { 5 } \\ { 0 } & { 1 } & { 5 } & { 0 } \end{array} \right)
$$

has two positive and two negative eigenvalues (taking into account multiplicities).

Problem 16 Let K be a continuous real valued function defined on $[ 0 , 1 ] \times$ [0, 1]. Let F be the family of functions f on [0, 1] of the form

$$
f ( x ) = \int _ { 0 } ^ { 1 } g ( y ) K ( x , y ) d y
$$

with g a real valued continuous function on [0, 1] satisfying $| g | \leqslant 1$ everywhere. Prove that the family F is equicontinuous.

Problem 17 Let w be a positive continuous function on [0, 1], n a positive integer, and $P _ { n }$ the vector space of real polynomials whose degrees are at most n, equipped with the inner product

$$
\langle p , q \rangle = \int _ { 0 } ^ { 1 } p ( t ) q ( t ) w ( t ) \ d t .
$$

1. Prove that $P _ { n }$ has an orthonormal basis $p _ { 0 } , p _ { 1 } , . . . , p _ { n } \ ( i . e . , \langle p _ { j } , p _ { k } \rangle = 1$ for $j = k$ and 0 for $j \neq k )$ such that deg $p _ { k } = k$ for each k.

2. Prove that $\langle p _ { k } , p _ { k } ^ { \prime } \rangle = 0$ for each k.

Problem 18 Let f be a continuous real valued function on [0, 1], and let the function h in the complex plane be defined by

$$
h ( z ) = \int _ { 0 } ^ { 1 } \ f ( t ) \cos { z t } d t .
$$

1. Prove that h is analytic in the entire plane.

2. Prove that h is the zero function only if f is the zero function.