# Preliminary Exam - Summer 1981

Problem 1 Let

$$
y ( h ) = 1 - 2 \sin ^ { 2 } ( 2 \pi h ) , ~ f ( y ) = \frac { 2 } { 1 + \sqrt { 1 - y } } \cdotp
$$

Justify the statement

$$
f \left( y ( h ) \right) = 2 - 4 \sqrt { 2 } \pi \left| h \right| + O ( h ^ { 2 } )
$$

where

$$
\operatorname* { l i m } _ { h \to 0 } { \frac { O ( h ^ { 2 } ) } { h ^ { 2 } } } < \infty .
$$

Problem 2 Let G be a finite group, and let ϕ be an automorphism of G which leaves fixed only the identity element of G.

1. Show that every element of G may be written in the form $g ^ { - 1 } \varphi ( g )$

2. If ϕ has order 2 $( i . e . , \varphi \cdot \varphi = \mathrm { i d } )$ show that ϕ is given by the formula $g \mapsto g ^ { - 1 }$ and that G is an abelian group whose order is odd.

Problem 3 Prove or disprove: The set Q of rational numbers is the intersection of a countable family of open subsets of R.

Problem 4 Let $f : \mathbb { R } \to \mathbb { R }$ be continuous, with

$$
\int _ { - \infty } ^ { \infty } | f ( x ) | d x < \infty .
$$

Show that there is a sequence $( x _ { n } )$ such that $x _ { n } \to \infty , x _ { n } f ( x _ { n } ) \to 0 _ { ! }$ and xnf(−xn) → 0 as $n \to \infty$

Problem 5 Let S denote the vector space of real $n \times n$ skew-symmetric matrices. For a nonsingular matrix A, compute the determinant of the linear map $T _ { A } : S  S , T _ { A } ( X ) = A X A ^ { t }$

Problem 6 Let $S \mathbb { O } ( 3 )$ denote the group of orthogonal transformations $o f \mathbb { R } ^ { 3 }$ of determinant 1. Let $Q \subset S \mathbb { O } ( 3 )$ be the subset of symmetric transformations $\neq I$ . Let $P ^ { 2 }$ denote the space of lines through the origin in $\mathbb { R } ^ { 3 }$

1. Show that $P ^ { 2 }$ and $S \mathbb { O } ( 3 )$ are compact metric spaces (in their usual topologies).

2. Show that $P ^ { 2 }$ and $Q$ are homeomorphic.

Problem 7 Compute

$$
{ \frac { 1 } { 2 \pi i } } \int _ { C } { \frac { d z } { \sin { \frac { 1 } { z } } } } ,
$$

where C is the circle $| z | = { \frac { 1 } { 5 } } .$ , positively oriented.

Problem 8 Show that $x ^ { 1 0 } + x ^ { 9 } + x ^ { 8 } + \cdot \cdot \cdot + x + 1$ is irreducible over $\mathbb { Q }$ . How about $x ^ { 1 1 } + x ^ { 1 0 } + \cdot \cdot \cdot + x + 1 ?$

Problem 9 Let $f : \mathbb { R } \to \mathbb { R }$ be the function of period 2π such that $f ( x ) = x ^ { 3 }$ $f o r - \pi \leqslant x < \pi$

1. Prove that the Fourier series for f has the form $\sum _ { 1 } ^ { \infty } b _ { n }$ sin nx and write an integral formula for $b _ { n }$ (do not evaluate it).

2. Prove that the Fourier series converges for all x.

3. Prove

$$
\sum _ { n = 1 } ^ { \infty } b _ { n } ^ { 2 } = { \frac { 2 \pi ^ { 6 } } { 7 } } \cdotp
$$

Problem 10 $L e t \ S$ be a vector space of complex sequences $\{ a _ { n } \} _ { n = 1 } ^ { \infty }$ . Define the map $T : S  S$ by $T \{ a _ { 1 } , a _ { 2 } , a _ { 3 } , . . . \} = \{ a _ { 2 } , a _ { 3 } , . . . \}$

1. Describe the eigenvectors of T .

2. Consider the difference equation $x _ { n + 2 } \ = \ x _ { n + 1 } + x _ { n }$ Show that the solutions of this equation define a two-dimensional subspace $E \subset S$ and that $T ( E ) \subset E$ . Find an explicit basis for E.

3. The Fibonacci numbers are defined recursively by $f _ { 1 } = 1 , f _ { 2 } = 1$ , and $f _ { n + 2 } = f _ { n + 1 } + f _ { n }$ for $n \geqslant 1$ . Find and explicit formula for $f _ { n }$

Note: See also Problem $? ?$

Problem 11 Show that the equation

$$
x \left( 1 + \log \left( \frac { 1 } { \varepsilon \sqrt { x } } \right) \right) = 1 , \quad x > 0 , \quad \varepsilon > 0 \ ,
$$

has, for each sufficiently small $\varepsilon > 0$ , exactly two solutions. Let $x ( \varepsilon )$ be the smaller one. Show that

$$
1 . ~ x ( \varepsilon ) \to 0 ~ a s \varepsilon \to 0 + ;
$$

yet for any $s > 0$

$$
\mathcal { Q } . ~ \varepsilon ^ { - s } x ( \varepsilon )  \infty ~ a s ~ \varepsilon  0 + .
$$

Problem 12 Show that no commutative ring with identity has additive group isomorphic to $\mathbb { Q } / \mathbb { Z }$

Problem 13 Let G be an additive group, and $u , v : G \to G$ homomorphisms. Show that the map $f : G  G , f ( x ) = x - v ( u ( x ) )$ is surjective if the map $h : G \to G , h ( x ) = x - u \left( v ( x ) \right)$ is surjective.

Problem 14 Let $I \subset \mathbb { R }$ be the open interval from 0 to 1.Let $f : I  \mathbb { C }$ be $C ^ { 1 } \ { \mathrm { ~ } } ( i . e .$ , the real and imaginary parts are continuously differentiable). Suppose that $f ( t ) \to 0 , f ^ { \prime } ( t ) \to C \neq 0$ as $t  0 +$ . Show that the function $g ( t ) = | f ( t ) |$ is C1 for sufficiently small $t > 0$ and that lim $g ^ { \prime } ( t )$ exists, and t→0+ evaluate the limit.

Problem 15 Let V be a finite-dimensional vector space over the rationals Q and let M be an automorphism of V such that M fixes no nonzero vector in V . Suppose that M p is the identity map on V , where p is a prime number. Show that the dimension of V is divisible by $p - 1$

Problem 16 Let $\left\{ f _ { n } \right\}$ be a sequence of continuous functions defined from $[ 0 , 1 ] \to \mathbb { R }$ such that

$$
\int _ { 0 } ^ { 1 } \left( f _ { n } ( x ) - f _ { m } ( x ) \right) ^ { 2 } d x \to 0 a s n , m \to \infty .
$$

Let $K : [ 0 , 1 ] \times [ 0 , 1 ]  \mathbb { R }$ be continuous. Define $g _ { n } : [ 0 , 1 ] \to \mathbb { R }$ by

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

Prove that the sequence $\left\{ g _ { n } \right\}$ converges uniformly.

Problem 17 Suppose that $f ( z )$ and $g ( z )$ are entire functions such that $| f ( z ) | \leqslant$ $| g ( z ) |$ for all z. Show that $f ( z ) = c g ( z )$ for some constant $c \in \mathbb { C }$ •

Problem 18 Let A and B be square matrices of rational numbers such that $C A C ^ { - 1 } = B$ for some real matrix C. Prove that such a C can be chosen to have rational entries.

Problem 19 Prove that the number of roots of the equationz $^ { 2 n } + \alpha ^ { 2 } z ^ { 2 n - 1 } +$ $\beta ^ { 2 } = 0$ (n a natural number, α and β real, nonzero) that have positive real part is

1. n if n is even, and

2. n − 1 if n is odd.

Problem 20 Let $y = y ( x )$ be a solution of the differential equation $y ^ { \prime \prime } = - | y |$ with $- \infty < x < \infty , y ( 0 ) = 1$ , and $y ^ { \prime } ( 0 ) = 0$

1. Show that y is an even function.

2. Show that y has exactly one zero on the positive real axis.