# Preliminary Exam - Spring 1987

Problem 1 A standard theorem states that a continuous real valued function on a compact set is bounded. Prove the converse: If K is a subset of $\mathbb { R } ^ { n }$ and if every continuous real valued function on K is bounded, then K is compact.

Problem 2 Let the transformation T from the subse $\mathcal { W } = \{ ( u , v ) \mid u > v \}$ $o f \mathbb { R } ^ { 2 }$ into $\mathbb { R } ^ { 2 }$ be defined by $T ( u , v ) = ( u + v , u ^ { 2 } + v ^ { 2 } )$

1. Prove that T is locally one-to-one.

2. Determine the range of T , and show that T is globally one-to-one.

Problem 3 Let f be a complex valued function in the open unit disc, D, of the complex plane such that the functions $g = f ^ { 2 }$ and $h = f ^ { 3 }$ are both analytic. Prove that f is analytic in D.

Problem 4 Let F be a finite field with q elements and let x be an indeterminate. For f a polynomial in ${ \bf F } [ x ]$ , let $\varphi _ { f }$ denote the corresponding function of F into F, defined by $\varphi _ { f } ( a ) = f ( a ) , ( a \in \mathbf { F } )$ . Prove that if ϕ is any function of F into F, then there is an f in ${ \bf F } [ x ]$ such that $\varphi = \varphi _ { f }$ . Prove that f is uniquely determined by $\varphi$ to within addition of a multiple of $x ^ { q } - x$

Problem 5 Let f be a continuous real valued function on R satisfying

$$
| f ( x ) | \leqslant { \frac { C } { 1 + x ^ { 2 } } } ,
$$

where C is a positive constant. Define the function F on R by

$$
F ( x ) = \sum _ { n = - \infty } ^ { \infty } f ( x + n ) .
$$

1. Prove that F is continuous and periodic with period 1.

2. Prove that if G is continuous and periodic with period 1, then

$$
\int _ { 0 } ^ { 1 } F ( x ) G ( x ) d x = \int _ { - \infty } ^ { \infty } f ( x ) G ( x ) d x .
$$

Problem 6 Let f be an analytic function in the open unit disc of the complex plane such that $| f ( z ) | \leqslant C / ( 1 - | z | )$ for all z in the disc, where C is a positive constant. Prove that $| f ^ { \prime } ( z ) | \leqslant 4 C / ( 1 - | z | ) ^ { 2 }$

Problem 7 Let p, q and r be continuous real valued functions on R, with $p > 0$ . Prove that the differential equation

$$
p ( t ) x ^ { \prime \prime } ( t ) + q ( t ) x ^ { \prime } ( t ) + r ( t ) x ( t ) = 0
$$

is equivalent to $( i . e .$ , has exactly the same solutions $\boldsymbol { a s } )$ a differential equation of the form

$$
( a ( t ) x ^ { \prime } ( t ) ) ^ { \prime } + b ( t ) x ( t ) = 0 ,
$$

where a is continuously differentiable and b is continuous.

Problem 8 Prove that if the nonconstant polynomial $p ( z )$ , with complex coefficients, has all of its roots in the half-plane $\Re z ~ > ~ 0$ , then all of the roots of its derivative are in the same half-plane.

Problem 9 Let A be an $m \times n$ matrix with rational entries and b an mdimensional column vector with rational entries. Prove or disprove: If the equation Ax = b has a solution x in $\mathbb { C } ^ { n }$ , then it has a solution with x in $\mathbb { Q } ^ { n }$

Problem 10 Prove that any finite group of order n is isomorphic to a subgroup $o f \mathbb { O } ( n )$ , the group of $n \times n$ orthogonal real matrices.

Problem 11 Show that the equation $a e ^ { x } = 1 + x + x ^ { 2 } / 2$ , where a is a positive constant, has exactly one real root.

Problem 12 Evaluate the integral

$$
I = \int _ { 0 } ^ { 1 / 2 } { \frac { \sin { x } } { x } } d x
$$

to an accuracy of two decimal places; that is, find a number $I ^ { * }$ such that|I − $I ^ { * } | < 0 . 0 0 5$

Problem 13 Let f be a real valued $C ^ { 1 }$ function defined in the punctured plane $\mathbb { R } ^ { 2 } \setminus \{ ( 0 , 0 ) \}$ . Assume that the partial derivatives $\partial f / \partial x$ and $\partial f / \partial y$ are uniformly bounded; that is, there exists a positive constant M such that $| \partial f / \partial x | \leqslant M$ and $| \partial f / \partial y | \leqslant M$ for all points $( x , y ) \neq ( 0 , 0 )$ . Prove that

$$
\operatorname* { l i m } _ { ( x , y ) \longrightarrow ( 0 , 0 ) } f ( x , y )
$$

exists.

Problem 14 1. Show that, to within isomorphism, there is just one noncyclic group G of order 4.

2. Show that the group of automorphisms of G is isomorphic to the permutation group $S _ { 3 }$

Problem 15 Prove or disprove: If the function f is analytic in the entire complex plane, and if f maps every unbounded sequence to an unbounded sequence, then f is a polynomial.

Problem 16 Let F be a uniformly bounded, equicontinuous family of real valued functions on the metric space $( X , d )$ . Prove that the function

$$
g ( x ) = \operatorname* { s u p } \{ f ( x ) \mid f \in { \mathcal { F } } \}
$$

is continuous.

Problem 17 Let V be a finite-dimensional linear subspace of $C ^ { \infty } ( \mathbb { R } )$ (the space of complex valued, infinitely differentiable functions). Assume that V is closed under D, the operator of differentiation (i.e., $f \in V \Rightarrow D f \in V )$ Prove that there is a constant coefficient differential operator

$$
L = \sum _ { k = 0 } ^ { n } a _ { k } D ^ { k }
$$

such that V consists of all solutions of the differential equation $L f = 0$

Problem 18 Let A and B be two diagonalizable $n \times n$ complex matrices such that $\mathrm { A B } = \mathrm { B A }$ . Prove that there is a basis for $\mathbb { C } ^ { n }$ that simultaneously diagonalizes A and B.

Problem 19 Let F be a field. Prove that every finite subgroup of the multiplicative group of nonzero elements of F is cyclic.

Problem 20 Evaluate

$$
I = \int _ { 0 } ^ { \pi } \frac { \cos 4 \theta } { 1 + \cos ^ { 2 } \theta } d \theta .
$$