# Preliminary Exam - Fall 1981

Problem 1 Evaluate the integral

$$
\int _ { - \infty } ^ { \infty } { \frac { \cos x } { 1 + x ^ { 4 } } } d x .
$$

Problem 2 Consider an autonomous system of differential equations

$$
\frac { d x _ { i } } { d t } = F _ { i } ( x _ { 1 } , \ldots , x _ { n } ) ,
$$

where $F = ( F _ { 1 } , \dots , F _ { n } ) : \mathbb { R } ^ { n } \longrightarrow \mathbb { R } ^ { n }$ is $a C ^ { 1 }$ vector field.

1. Let U and V be two solutions on $a < t < b$ . Assuming that

$$
\langle D F ( x ) z , z \rangle \leqslant 0
$$

for all $x , z \ i n \mathbb { R } ^ { n }$ , show that $\| U ( t ) - V ( t ) \| ^ { 2 }$ is a decreasing function of t.

2. Let $W ( t )$ be a solution defined for $t > 0$ . Assuming that

$$
\langle D F ( x ) z , z \rangle \leqslant - \| z \| ^ { 2 } ,
$$

show that there exists $C \in \mathbb { R } ^ { n }$ such that

$$
\operatorname* { l i m } _ { t  \infty } W ( t ) = C .
$$

Problem 3 Let $\mathrm { S } _ { n }$ be the group of all permutations of n objects and let G be a subgroup of $\mathrm { S } _ { n }$ of order $p ^ { k }$ , where p is a prime not dividing n. Show that G has a fixed point; that is, one of the objects is left fixed by every element of G.

Problem 4 Prove the following three statements about real $n \times n$ matrices.

1. If A is an orthogonal matrix whose eigenvalues are all different from −1, then $I + A$ is nonsingular and $S = ( I - A ) ( I + A ) ^ { - 1 }$ is skewsymmetric.

2. If S is a skew-symmetric matrix, then $A = ( I - S ) ( I + S ) ^ { - 1 }$ is an orthogonal matrix with no eigenvalue equal to −1.

3. The correspondence $A  S$ from Parts 1 and 2 is one-to-one.

Problem 5 The Fibonacci numbers $f _ { 1 } , f _ { 2 } , \ldots$ . are defined recursively by $f _ { 1 } =$ $1 , f _ { 2 } = 2$ , and $f _ { n + 1 } = f _ { n } + f _ { n - 1 }$ for $n \geqslant 2$ . Show that

$$
\operatorname* { l i m } _ { n \to \infty } { \frac { f _ { n + 1 } } { f _ { n } } }
$$

exists, and evaluate the limit.

Note: See also Problem ??.

Problem 6 Let f and g be continuous functions on R such tha $t f ( x + 1 ) =$ $f ( x ) , g ( x + 1 ) = g ( x )$ , for all $x \in \mathbb { R }$ . Prove that

$$
\operatorname* { l i m } _ { n \to \infty } \int _ { 0 } ^ { 1 } f ( x ) g ( n x ) d x = \int _ { 0 } ^ { 1 } f ( x ) d x \int _ { 0 } ^ { 1 } g ( x ) d x .
$$

Problem 7 Find a specific polynomial with rational coefficients having √ ${ \sqrt { 2 } } +$ $\sqrt [ 3 ] { 3 }$ as a root.

Problem 8 1. How many zeros does the function $f ( z ) = 3 z ^ { 1 0 0 } - e ^ { z }$ have inside the unit circle (counting multiplicities)?

2. Are the zeros distinct?

Problem 9 Let $M _ { 2 \times 2 }$ be the vector space of all real ${ \it 2 } \times { \it 2 }$ matrices. Let

$$
A = { \left( \begin{array} { l l } { 1 } & { 2 } \\ { - 1 } & { 3 } \end{array} \right) }
$$

$$
B = { \binom { 2 } { 0 } } \ { \frac { 1 } { 4 } } )
$$

and define a linear transformation $L : \ : \ : M _ { 2 \times 2 } \ :  \ : \ : M _ { 2 \times 2 }$ by $L ( X ) \ = \ A X B$ Compute the trace and the determinant of L.

Problem 10 Let $A = \left( a _ { i j } \right)$ be an n $\times \ n$ matrix whose entries $a _ { i j }$ are real valued differentiable functions defined on R. Assume that the determinant det(A) of A is everywhere positive. Let $B = ( b _ { i j } )$ be the inverse matrix of A. Prove the formula

$$
{ \frac { d } { d t } } \log \left( \operatorname* { d e t } ( A ) \right) = \sum _ { i , j = 1 } ^ { n } { \frac { d a _ { i j } } { d t } } b _ { j i } .
$$

Problem 11 Consider the complex $\mathcal { B } \times \mathcal { B }$ matrix

$$
A = \left( \begin{array} { l l l } { { a _ { 0 } } } & { { a _ { 1 } } } & { { a _ { 2 } } } \\ { { a _ { 2 } } } & { { a _ { 0 } } } & { { a _ { 1 } } } \\ { { a _ { 1 } } } & { { a _ { 2 } } } & { { a _ { 0 } } } \end{array} \right) ,
$$

where $a _ { 0 } , a _ { 1 } , a _ { 2 } \in \mathbb { C }$

1. Show that $A = a _ { 0 } I _ { 3 } + a _ { 1 } E + a _ { 2 } E ^ { 2 }$ , where

$$
E = \left( \begin{array} { l l l } { { 0 } } & { { 1 } } & { { 0 } } \\ { { 0 } } & { { 0 } } & { { 1 } } \\ { { 1 } } & { { 0 } } & { { 0 } } \end{array} \right) .
$$

2. Use Part 1 to find the complex eigenvalues of A.

3. Generalize Parts 1 and 2 to n×n matrices.

Problem 12 Let a, b be real constants and let

$$
u ( x , y ) = \frac { a ^ { 2 } + b ^ { 2 } + x ^ { 2 } - y ^ { 2 } } { ( a - x ) ^ { 2 } + ( b - y ) ^ { 2 } } \cdot
$$

Show that u is harmonic and find an entire function $f ( z )$ whose real part is u.

Correction: u cannot be the real part of an entire function. Why? Change u slightly and do the problem.

Problem 13 Let f be a real valued function on $\mathbb { R } ^ { n }$ of class $C ^ { 2 }$ . A point $x \in \mathbb { R } ^ { n }$ is a critical point of f if all the partial derivatives of f vanish at x; a critical point is nondegenerate if the $n \times n$ matrix

$$
\left( \frac { \partial ^ { 2 } f } { \partial x _ { i } \partial x _ { j } } ( x ) \right)
$$

is nonsingular.

Let x be a nondegenerate critical point of f. Prove that there is an open neighborhood of x which contains no other critical points (i.e., the nondegenerate critical points are isolated).

Problem 14 Let $V : \mathbb { R } ^ { n } \longrightarrow \mathbb { R }$ be a $C ^ { 1 }$ function and consider the system of second order differential equations

$$
x _ { i } ^ { \prime \prime } ( t ) = f _ { i } \left( x ( t ) \right) , \ \quad 1 \leqslant i \leqslant n ,
$$

where

$$
f _ { i } = - { \frac { \partial V } { \partial x _ { i } } } \cdot
$$

Let $x ( t ) = ( x _ { 1 } ( t ) , \ldots , x _ { n } ( t ) )$ be a solution of this system on a finite interval $a < t < b$

## 1. Show that the function

$$
H ( t ) = \frac { 1 } { 2 } \langle x ^ { \prime } ( t ) , x ^ { \prime } ( t ) \rangle + V ( x ( t ) )
$$

is constant for $a < t < b$

2. Assuming that $V ( x ) \geqslant M > - \infty$ for all $x \in \mathbb { R } ^ { n }$ , show that $x ( t ) , x ^ { \prime } ( t )$ and $x ^ { \prime \prime } ( t )$ are bounded on $a < t < b$ , and then prove all three limits

$$
\operatorname* { l i m } _ { t \to b } x ( t ) , \quad \operatorname* { l i m } _ { t \to b } x ^ { \prime } ( t ) , \quad \operatorname* { l i m } _ { t \to b } x ^ { \prime \prime } ( t )
$$

exist.

Problem 15 Let f be a holomorphic map of the unit disc $\mathbb { D } = \{ z \mid | z | < 1 \}$ into itself, which is not the identity map $f ( z ) = z$ . Show that f can have, at most, one fixed point.

Problem 16 Let G be a group with three normal subgroups $N _ { 1 } , N _ { 2 } , N _ { 3 }$ . Suppose $N _ { i } \cap N _ { j } = \{ e \}$ and $N _ { i } N _ { j } = G$ for all $i , j$ with $i \neq j$ . Show that G is abelian and $N _ { i }$ is isomorphic to $N _ { j }$ for all $i , j$

Problem 17 Let f be a continuous function on [0, 1]. Evaluate the following limits.

1.

$$
\operatorname* { l i m } _ { n \to \infty } \int _ { 0 } ^ { 1 } x ^ { n } f ( x ) d x .
$$

2.

$$
\operatorname* { l i m } _ { n \to \infty } n \int _ { 0 } ^ { 1 } x ^ { n } f ( x ) d x .
$$

Problem 18 Let A and B be two real $n \times n$ matrices. Suppose there is a complex invertible $n \times n$ matrix U such that $A = U B U ^ { - 1 }$ . Show that there is a real invertible $n \times n$ matrix V such that $A = V B V ^ { - 1 }$ . (In other words, if two real matrices are similar over C , then they are similar over R.)

Problem 19 Either prove or disprove (by a counterexample) each of the following statements:

1. Let $f : \mathbb { R } \to \mathbb { R } , g : \mathbb { R } \to \mathbb { R }$ be such that

$$
\operatorname* { l i m } _ { t \to a } g ( t ) = b a n d \operatorname* { l i m } _ { t \to b } f ( t ) = c .
$$

Then

$$
\operatorname* { l i m } _ { t \to a } f \left( g ( t ) \right) = c .
$$

2. If $f : \mathbb { R }  \mathbb { R }$ is continuous and U is an open set in R, then $f ( U )$ is an open set in R.

3. Let f be of class $C ^ { \infty }$ on the interval (−1, 1). Suppose that $| f ^ { ( n ) } ( x ) | \leqslant 1$ for all $n \geqslant 1$ and all x in the interval. Then f is real analytic; that is, it has a convergent power series expansion in a neighborhood of each point of the interval.

Problem 20 Let G be a group of order 10 which has a normal subgroup of order 2. Prove that G is abelian.