# Preliminary Exam - Fall 1983

Problem 1 Evaluate

$$
\int _ { 0 } ^ { \infty } ( \operatorname { s e c h } x ) ^ { 2 } \cos \lambda x d x
$$

where λ is a real constant and

$$
\operatorname { s e c h } x = { \frac { 2 } { e ^ { x } + e ^ { - x } } } \cdotp
$$

Problem 2 Let $M _ { n \times n } ( \mathbf { F } )$ be the ring of n×n matrices over a field F. For $n \geqslant 1$ does there exist a ring homomorphism from $M _ { ( n + 1 ) \times ( n + 1 ) } ( \mathbf { F } )$ onto $M _ { n \times n } ( \mathbf { F } ) \boldsymbol { \ ? }$

Problem 3 Let $f : \mathbb { R } ^ { n } \backslash \{ 0 \}  \mathbb { R }$ be a function which is continuously differentiable and whose partial derivatives are uniformly bounded:

$$
\left| { \frac { \partial f } { \partial x _ { i } } } ( x _ { 1 } , \ldots , x _ { n } ) \right| \leqslant M
$$

for all $( x _ { 1 } , \ldots , x _ { n } ) \neq ( 0 , \ldots , 0 )$ . Show that if $n \geqslant 2$ , then f can be extended to a continuous function defined on all of Rn. Show that this is false if $n = 1$ by giving a counterexample.

Problem 4 Prove or disprove (by giving a counterexample), the following assertion: Every infinite sequence $x _ { 1 } , x _ { 2 } , . . .$ . of real numbers has either a nondecreasing subsequence or a nonincreasing subsequence.

Problem 5 Let A be the $n \times n$ matrix which has zeros on the main diagonal and ones everywhere else. Find the eigenvalues and eigenspaces of A and compute det(A).

Problem 6 Consider the polynomial

$$
p ( z ) = z ^ { 5 } + z ^ { 3 } + 5 z ^ { 2 } + 2 .
$$

How many zeros (counting multiplicities) does p have in the annular region1 < $| z | < 2 \ell$

Problem 7 Let G be a finite group and suppose that $G \times G$ has exactly four normal subgroups. Show that G is simple and nonabelian.

Problem 8 Let A be a linear transformation on $\mathbb { R } ^ { 3 }$ whose matrix (relative to the usual basis for $\mathbb { R } ^ { 3 } )$ is both symmetric and orthogonal. Prove that A is either plus or minus the identity, or a rotation by 180◦ about some axis in $\mathbb { R } ^ { 3 }$ , or a reflection about some two-dimensional subspace of $\mathbb { R } ^ { 3 }$

Problem 9 For which real values of p does the differential equation

$$
y ^ { \prime \prime } + 2 p y ^ { \prime } + y = 3
$$

admit solutions $y = f ( x )$ with infinitely many critical points?

Problem 10 Let $f : [ 0 , \infty ) \to \mathbb { R }$ be a uniformly continuous function with the property that

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

exists (as a finite limit). Show that

$$
\operatorname* { l i m } _ { x \to \infty } f ( x ) = 0 .
$$

Problem 11 Prove or supply a counterexample: If f and g are $C ^ { 1 }$ real valued functions on $( 0 , 1 )$ , if

$$
\operatorname* { l i m } _ { x \to 0 } f ( x ) = \operatorname* { l i m } _ { x \to 0 } g ( x ) = 0 ,
$$

if g and g0 never vanish, and if

$$
\operatorname* { l i m } _ { x \to 0 } { \frac { f ( x ) } { g ( x ) } } = c ,
$$

then

$$
\operatorname* { l i m } _ { x \to 0 } { \frac { f ^ { \prime } ( x ) } { g ^ { \prime } ( x ) } } = c .
$$

Problem 12 Let $r _ { 1 } , r _ { 2 } , \ldots , r _ { n }$ be distinct complex numbers. Show that a rational function of the form

$$
f ( z ) = { \frac { b _ { 0 } + b _ { 1 } z + \cdot \cdot \cdot + b _ { n - 2 } z ^ { n - 2 } + b _ { n - 1 } z ^ { n - 1 } } { ( z - r _ { 1 } ) ( z - r _ { 2 } ) \cdot \cdot \cdot ( z - r _ { n } ) } }
$$

can be written as a sum

$$
f ( z ) = { \frac { A _ { 1 } } { z - r _ { 1 } } } + { \frac { A _ { 2 } } { z - r _ { 2 } } } + \cdots + { \frac { A _ { n } } { z - r _ { n } } }
$$

for suitable constants $A _ { 1 } , \ldots , A _ { n }$

Problem 13 1. Let $u ( t )$ be a real valued differentiable function of a real variable t which satisfies an inequality of the form

$$
u ^ { \prime } ( t ) \leqslant a u ( t ) , \quad t \geqslant 0 , \quad u ( 0 ) \leqslant b ,
$$

where a and b are positive constants. Starting from first principles, derive an upper bound for u(t) for $t > 0$

2. Let $x ( t ) = ( x _ { 1 } ( t ) , x _ { 2 } ( t ) , \ldots , x _ { n } ( t ) )$ be a differentiable function from R to $\mathbb { R } ^ { n }$ which satisfies a differential equation of the form

$$
x ^ { \prime } ( t ) = f ( x ( t ) ) ,
$$

where $f : \mathbb { R } ^ { n }  \mathbb { R } ^ { n }$ is a continuous function. Assuming that f satisfies the condition

$$
\langle f ( y ) , y \rangle \leqslant \| y \| ^ { 2 } , \quad y \in \mathbb { R } ^ { n }
$$

derive an inequality showing that the norm $\| x ( t ) \|$ grows, at most, exponentially.

Problem 14 Let V be a finite-dimensional complex vector space and let A and B be linear operators on V such that $A B = B A$ . Prove that if A and B can each be diagonalized, then there is a basis for V which simultaneously diagonalizes A and B.

Problem 15 1. Let f be a complex function which is analytic on an open set containing the disc $| z | \leqslant 1$ , and which is real valued on the unit circle. Prove that f is constant.

2. Find a nonconstant function which is analytic at every point of the complex plane except for a single point on the unit circle $| z | = 1$ , and which is real valued at every other point of the unit circle.

Problem 16 Let $F ( t ) = ( f _ { i j } ( t ) )$ be an $n \times n$ matrix of continuously differentiable functions $f _ { i j } : \mathbb { R } \to \mathbb { R }$ , and let

$$
u ( t ) = \mathrm { t r } \left( F ( t ) ^ { 3 } \right) .
$$

Show that u is differentiable and

$$
u ^ { \prime } ( t ) = 3 \operatorname { t r } \left( F ( t ) ^ { 2 } F ^ { \prime } ( t ) \right) .
$$

Problem 17 Prove that every finite integral domain is a field.

Problem 18 Let $f , g : \mathbb { R } \to \mathbb { R }$ be smooth functions with $f ( 0 ) = 0$ and $f ^ { \prime } ( 0 ) \neq 0$ . Consider the equation $f ( x ) = t g ( x ) , t \in \mathbb { R }$

1. Show that in a suitably small interval $| t | < \delta$ , there is a unique continuous function x(t) which solves the equation and satisfies $x ( 0 ) = 0$

2. Derive the first order Taylor expansion of $x ( t )$ about $t = 0$

Problem 19 Prove that if p is a prime number, then the polynomial

$$
f ( x ) = x ^ { p - 1 } + x ^ { p - 2 } + \cdot \cdot \cdot + 1
$$

is irreducible in $\mathbb { Q } [ x ]$

Problem 20 Let m and n be positive integers, with $m < n$ . Let $M _ { m \times n }$ be the space of linear transformations of $\mathbb { R } ^ { m }$ into $\mathbb { R } ^ { n }$ (considered as n×m matrices) and let L be the set of transformations in $M _ { m \times n }$ which have rank m.

1. Show that L is an open subset of $M _ { m \times n }$

2. Show that there is a continuous function $T : L \to M _ { m \times n }$ such thatT ${ \bf \nabla } ^ { \prime } ( A ) A =$ $I _ { m }$ for all A, where $I _ { m }$ is the identity on $\mathbb { R } ^ { m }$