# Preliminary Exam - Fall 1978

Problem 1 Let $f : \mathbb { R } \to \mathbb { R }$ satisfy $f ( x ) \leqslant f ( y )$ for $x \leqslant y$ . Prove that the set where f is not continuous is finite or countably infinite.

Problem 2 Let $\left\{ g _ { n } \right\}$ be a sequence of Riemann integrable functions from [0, 1] into R such that $| g _ { n } ( x ) | \leqslant 1$ for all n, x. Define

$$
G _ { n } ( x ) = \int _ { 0 } ^ { x } g _ { n } ( t ) d t .
$$

Prove that a subsequence of $\{ G _ { n } \}$ converges uniformly.

Problem 3 Let $M _ { n \times n }$ denote the vector space of $n \times n$ real matrices. Prove that there are neighborhoods U and V in $M _ { n \times n }$ of the identity matrix such that for every A in U , there is a unique X in V such that $X ^ { 4 } = A$

Problem 4 Evaluate

$$
\int _ { 0 } ^ { 2 \pi } { \frac { d \theta } { 1 - 2 r \cos \theta + r ^ { 2 } } }
$$

where $r ^ { 2 } \neq 1$

Problem 5 Let $f ( z ) = a _ { 0 } + a _ { 1 } z + \cdot \cdot \cdot + a _ { n } z ^ { n }$ be a complex polynomial of degree $n > 0$ . Prove

$$
{ \frac { 1 } { 2 \pi i } } \int _ { | z | = R } z ^ { n - 1 } | f ( z ) | ^ { 2 } d z = a _ { 0 } { \overline { { a } } } _ { n } R ^ { 2 n } .
$$

Problem 6 Solve the differential equation

$$
{ \frac { d y } { d x } } = x ^ { 2 } y - 3 x ^ { 2 } , \quad y ( 0 ) = 1 .
$$

Problem 7 Let H be a subgroup of a finite group G.

1. Show that H has the same number of left cosets as right cosets.

2. Let G be the group of symmetries of the square. Find a subgroup H such that xH 6= Hx for some x.

Problem 8 Let M be the $n \times n$ matrix over a field F, all of whose entries are equal to 1.

1. Find the characteristic polynomial of M .

2. Is M diagonalizable?

3. Find the Jordan Canonical Form of M and discuss the extent to which the Jordan form depends on the characteristic of the field F.

Problem 9 For x, $y \in \mathbb { C } ^ { n }$ , let $\langle x , y \rangle$ be the Hermitian inner product $\sum _ { j } x _ { j } { \overline { { y } } } _ { j }$ Let T be a linear operator on $\mathbb { C } ^ { n }$ such that $\langle T x , T y \rangle \ : = \ : 0 \ : \ : i f \ : \langle x , y \rangle \ ' = \ : 0$ Prove that $T = k S$ for some scalar k and some operator S which is unitary: $\langle S x , S y \rangle = \langle x , y \rangle$ for all x and y.

Problem 10 How many homomorphisms are there from the group $\mathbb { Z } _ { 2 } \times \mathbb { Z } _ { 2 }$ to the symmetric group on three objects?

Problem 11 Let $W \subset \mathbb { R } ^ { n }$ be an open connected set and f a real valued function on W such that all partial derivatives of f are 0. Prove that f is constant.

Problem 12 Let $f : \mathbb { R } ^ { n } \to \mathbb { R }$ have the following properties: f is differentiable on $\mathbb { R } ^ { n } \setminus \{ 0 \} , f$ is continuous at 0, and

$$
\operatorname* { l i m } _ { p \to 0 } { \frac { \partial f } { \partial x _ { i } } } ( p ) = 0
$$

for $i = 1 , \ldots , n$ . Prove that f is differentiable at 0.

Problem 13 1. Show that $i f u , v : \mathbb { R } ^ { 2 } \to$ R are continuously differentiable and ${ \frac { \partial u } { \partial y } } = { \frac { \partial v } { \partial x } }$ , then $u = { \frac { \partial f } { \partial x } } , v = { \frac { \partial f } { \partial y } }$ for some $f : \mathbb { R } ^ { 2 } \to \mathbb { R }$

2. Prove there is no $f : \mathbb { R } ^ { 2 } \setminus \{ 0 \} \to \mathbb { R }$ such that

$$
{ \frac { \partial f } { \partial x } } = { \frac { - y } { x ^ { 2 } + y ^ { 2 } } } \quad a n d \quad { \frac { \partial f } { \partial y } } = { \frac { x } { x ^ { 2 } + y ^ { 2 } } } \cdotp
$$

Problem 14 Give examples of conformal maps as follows:

1. from $\{ z \mid | z | < 1 \}$ onto $\{ z \mid \Re z < 0 \}$

2. from $\{ z \mid | z | < 1 \}$ onto itself, with $f ( 0 ) = 0$ and $f ( 1 / 2 ) = i / 2$

3. from $\{ z \mid z \neq 0 , 0 < \arg z < { \frac { 3 \pi } { 2 } } \}$ onto $\{ z \mid z \neq 0 , 0 < \arg z < { \frac { \pi } { 2 } } \}$

Problem 15 Suppose $h ( z )$ is analytic in the whole plane, $h ( 0 ) = 3 + 4 i$ , and $| h ( z ) | \leqslant 5 ~ i f | z | < 1$ . What is $h ^ { \prime } ( 0 ) \ d s ^ { 2 }$

Problem 16 Which of the following matrices are similar as matrices over R?

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

$$
( d ) \left( { \begin{array} { c c c } { 1 } & { 0 } & { 0 } \\ { 1 } & { 1 } & { 0 } \\ { 0 } & { 1 } & { 1 } \end{array} } \right) , \left( e \right) \left( { \begin{array} { c c c } { 1 } & { 1 } & { 0 } \\ { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} } \right) , \left( f \right) \left( { \begin{array} { c c c } { 0 } & { 1 } & { 1 } \\ { 0 } & { 1 } & { 0 } \\ { 1 } & { 0 } & { 0 } \end{array} } \right) .
$$

Problem 17 Find all automorphisms of the additive group of rational numbers.

Problem 18 Prove that every finite multiplicative group of complex numbers is cyclic.

Problem 19 Let

$$
A = \left( \begin{array} { c c } { { 1 } } & { { 2 } } \\ { { 1 } } & { { - 1 } } \end{array} \right) .
$$

Express $A ^ { - 1 }$ as a polynomial in A with real coefficients.

Problem 20 Let $M _ { n \times n }$ be the vector space of real $n \times n$ matrices, identified with $\mathbb { R } ^ { n ^ { 2 } }$ . Let $X \subset M _ { n \times n }$ be a compact set. Let $S \subset \mathbb { C }$ be the set of all numbers that are eigenvalues of at least one element of X. Prove that S is compact.