# Preliminary Exam - Fall 1979

Problem 1 Prove that the polynomial

$$
p ( z ) = z ^ { 4 7 } - z ^ { 2 3 } + 2 z ^ { 1 1 } - z ^ { 5 } + 4 z ^ { 2 } + 1
$$

has at least one root in the disc $| z | < 1$

Problem 2 Suppose that f is analytic on the open upper half-plane and satisfies $| f ( z ) | \leqslant 1$ for all $z , f ( i ) = 0$ . How large can $| f ( 2 i ) |$ be under these conditions?

Problem 3 Prove that every finite group of order $> 2$ has a nontrivial automorphism.

Problem 4 Consider the polynomial ring $\mathbb { Z } [ x ]$ and the ideal I generated by 7 and $x - 3$

1. Show that for each $r \in \mathbb { Z } [ x ]$ , there is an integer α satisfying $0 \leqslant \alpha \leqslant 6$ such that $r - \alpha \in \Im$

2. Find α in the special case $r = x ^ { 2 5 0 } + 1 5 x ^ { 1 4 } + x ^ { 2 } + 5$

Problem 5 Let A be a real skew-symmetric matrix $\left( A _ { i j } = - A _ { j i } \right)$ . Prove that A has even rank.

Problem 6 Let N be a linear operator on an n-dimensional vector space, $n > 1$ , such that $N ^ { n } = 0 , N ^ { n - 1 } \neq 0$ . Prove there is no operator X with $X ^ { 2 } = N$

Problem 7 Let V be the vector space of sequences $\left( a _ { n } \right)$ of complex numbers. The shift operator $S : V \to V$ is defined by

$$
S \left( ( a _ { 1 } , a _ { 2 } , a _ { 3 } , . . . ) \right) = ( a _ { 2 } , a _ { 3 } , a _ { 4 } , . . . ) .
$$

1. Find the eigenvectors of S.

2. Show that the subspace W consisting of the sequences $( x _ { n } )$ w $i t h x _ { n + 2 } =$ $x _ { n + 1 } + x _ { n }$ is a two-dimensional, S-invariant subspace of V and exhibit an explicit basis for W .

3. Find an explicit formula for the $n ^ { t h }$ Fibonacci number $f _ { n }$ , where $f _ { 2 } =$ $f _ { 1 } = 1 , f _ { n + 2 } = f _ { n + 1 } + f _ { n } f o r n \geqslant 1$

Note: See also Problem ??.

Problem 8 Prove that

$$
\operatorname* { l i m } _ { n \to \infty } \left( { \frac { 1 } { n + 1 } } + { \frac { 1 } { n + 2 } } + \cdots + { \frac { 1 } { 2 n } } \right) = \log 2 .
$$

Problem 9 Given that

$$
\int _ { - \infty } ^ { \infty } e ^ { - x ^ { 2 } } d x = { \sqrt { \pi } } ,
$$

find f 0(t) explicitly, where

$$
f ( t ) = \int _ { - \infty } ^ { \infty } e ^ { - t x ^ { 2 } } d x , \quad t > 0 .
$$

Problem 10 Solve the differential equations

$$
{ \frac { d x } { d t } } = - 3 x + 1 0 y
$$

$$
{ \frac { d y } { d t } } = - 3 x + 8 y .
$$

Problem 11 An accurate map of California is spread out flat on a table in Evans Hall, in Berkeley. Prove that there is exactly one point on the map lying directly over the point it represents.

Problem 12 Consider the following properties of a functionf : $\mathbb { R } ^ { n } \to \mathbb { R }$

1. f is continuous.

2. The graph of f is connected in $\mathbb { R } ^ { n } \times \mathbb { R }$

Prove or disprove the implications $1 \Rightarrow \ 2 , \ 2 \Rightarrow \ 1$

Problem 13 Let $\{ P _ { n } \}$ be a sequence of real polynomials of degree $\leqslant D$ , a fixed integer. Suppose that $P _ { n } ( x ) \to 0$ pointwise for $0 \leqslant x \leqslant 1$ . Prove that $P _ { n } \to 0$ uniformly on [0, 1].

Problem 14 Let $y = y ( x )$ be a solution of the differential equation $y ^ { \prime \prime } = - | y |$ $w i t h - \infty < x < \infty , y ( 0 ) = 1$ 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.

Problem 15 Suppose f and g are entire functions with $| f ( z ) | \leqslant | g ( z ) |$ for all z. Prove that $f ( z ) = c g ( z )$ for some constant c.

Problem 16 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 17 Let A be an $n \times n$ complex matrix. Prove there is a unitary matrix U such that $B = U A U ^ { - 1 }$ is upper-triangular: $B _ { j k } = 0 ~ f o r ~ j > k$

Problem 18 Let B denote the matrix

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

where $a , b ,$ and c are real and |a|, |b|, and |c| are distinct. Show that there are exactly four symmetric matrices of the form BQ, where Q is a real orthogonal matrix of determinant 1.

Problem 19 Let $M _ { n \times n } ( \mathbf { F } )$ be the ring of n×n matrices over a field F. Prove that it has no 2-sided ideals except $M _ { n \times n } ( \mathbf { F } )$ and {0}.

Problem 20 Let G be the abelian group defined by generators x, y, and $z ,$ and relations

$$
\begin{array} { r } { 1 5 x + 3 y = 0 } \\ { 3 x + 7 y + 4 z = 0 } \\ { 1 8 x + 1 4 y + 8 z = 0 . } \end{array}
$$

1. Express G as a direct product of two cyclic groups.

2. Express G as a direct product of cyclic groups of prime power order.

3. How many elements of G have order 2?