# Preliminary Exam - Fall 1992

Problem 1 Are the matrices $A = \left( \begin{array} { l l l } { 1 } & { 0 } & { 0 } \\ { - 1 } & { 1 } & { 1 } \\ { - 1 } & { 0 } & { 2 } \end{array} \right)$ and $\mathbf { B } = { \left( \begin{array} { l l l } { 1 } & { 1 } & { 0 } \\ { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 2 } \end{array} \right) }$ similar?

Problem 2 Let I denote the ideal in $\mathbb { Z } [ x ]$ , the ring of polynomials with coefficients in Z, generated by $x ^ { 3 } + x + 1$ and 5. Is I a prime ideal?

Problem 3 1. How many roots does the polynomial defined by $p ( z ) =$ $2 z ^ { 5 } + 4 z ^ { 2 } + 1$ have in the disc $| z | < 1 \ell$

2. How many roots does the same polynomial have on the real axis?

Problem 4 Let the real valued function $y ( t ) \ ( 0 \leqslant t < \infty )$ solve the initial value problem

$$
y ^ { \prime \prime } = - | y | , y ( 0 ) = 1 , y ^ { \prime } ( 0 ) = 0 .
$$

Prove that there is exactly one $t > 0$ such that $y ( t ) = 0$

Problem 5 Let G be a group and H and K subgroups such that H has a finite index in G. Prove that K ∩ H has a finite index in $K$

Problem 6 Let $( X _ { 1 } , d _ { 1 } )$ and $( X _ { 2 } , d _ { 2 } )$ be metric spaces andf $: X _ { 1 }  X _ { 2 }$ a continuous surjective map such that $d _ { 1 } ( p , q ) \in d _ { 2 } ( f ( p ) , f ( q ) )$ for every pair of points $p , q$ in $X _ { 1 }$

1. If $X _ { 1 }$ is complete, must $X _ { 2 }$ be complete? Give a proof or a counterexample.

2. If $X _ { 2 }$ is complete, must $X _ { 1 }$ be complete? Give a proof or a counterexample.

Problem 7 Evaluate

$$
\int _ { C } { \frac { e ^ { z } } { z ( 2 z + 1 ) ^ { 2 } } } \ d z ,
$$

where C is the unit circle with counterclockwise orientation.

Problem 8 Let F be a field, V a finite-dimensional vector space over $\mathbf { F } ,$ and T a linear transformation of V into V whose minimum polynomial, $\mu ,$ is irreducible over F.

1. Let v be a nonzero vector in V and let $V _ { 1 }$ be the subspace spanned by v and its images under the positive powers of T . Prove that dim $V _ { 1 } =$ deg µ.

2. Prove that deg µ divides dim V .

Problem 9 Let the function f be analytic in the region $| z | > 1$ of the complex plane. Prove that if f is real valued on the interval (1, ∞) of the real axis, then f is also real valued on the interval $( - \infty , - 1 )$ .

Problem 10 How many Sylow 2-subgroups does the dihedral group $D _ { n }$ of order 2n have, when n is odd?

Problem 11 Let V be a finite-dimensional vector space over a field F, and let $B : V \times V \to \mathbf { F }$ be a bilinear map (not necessarily symmetric). Define the subspaces $V _ { 1 }$ and $V _ { 2 }$ by

$$
V _ { 1 } = \{ x \in V : B ( x , y ) = 0 \quad f o r \ a l l \quad y \quad { \dot { \imath } } n \quad V \}
$$

$$
V _ { 2 } = \{ y \in V : B ( x , y ) = 0 \quad f o r \ a l l \quad x \quad i n \quad V \} .
$$

Prove that dim $V _ { 1 } = \dim V _ { 2 }$

Problem 12 Let $\left\{ f _ { n } \right\}$ be a sequence of real valued $C ^ { 1 }$ functions on [0, 1] such that, for all $n _ { \ i }$

$$
| f _ { n } ^ { \prime } ( x ) | \leqslant { \frac { 1 } { \sqrt { x } } } \quad ( 0 < x \leqslant 1 ) ,
$$

$$
\int _ { 0 } ^ { 1 } f _ { n } ( x ) \ d x = 0 .
$$

Prove that the sequence has a subsequence that converges uniformly on [0, 1].

Problem 13 Evaluate

$$
\int _ { - \infty } ^ { \infty } { \frac { \sin x } { x ( 1 + x ^ { 2 } ) } } d x .
$$

Problem 14 Let f and g be positive continuous functions on R, with $g \leqslant f$ everywhere. Assume the initial value problem

$$
\frac { d x } { d t } = f ( x ) , \quad x ( 0 ) = 0 ,
$$

has a solution defined on all of R. Prove that the initial value problem

$$
\frac { d x } { d t } = g ( x ) , \quad x ( 0 ) = 0 ,
$$

also has a solution defined on all of R.

Problem 15 Let G be the group of all real ${ \mathcal { Q } } \times { \mathcal { Q } }$ matrices of the form $\left( \begin{array} { c c } { { a } } & { { b } } \\ { { 0 } } & { { a ^ { - 1 } } } \end{array} \right)$ with $a > 0$ . Let N be the subgroup of those matrices in G having $a = 1$

(a) Prove that N is a normal subgroup of G and that $G / N$ is isomorphic to R.

(b) Find a proper normal subgroup of G that contains N properly.

Problem 16 Let f be a C1 function of Rn into $\mathbb { R } ^ { n }$ such that Df has rank n everywhere. Assume f is proper (which, by definition, means $f ^ { - 1 } ( K )$ is compact whenever K is compact). Prove that $f ( \mathbb { R } ^ { n } ) = \mathbb { R } ^ { n }$

Problem 17 Let s be a real number, and let the function u be defined in $\mathbb { C } \backslash ( - \infty , 0 ]$ by

$$
u ( r e ^ { i \theta } ) = r ^ { s } \cos s \theta \qquad ( r > 0 , \quad - \pi < \theta < \pi ) .
$$

Prove that u is a harmonic function.

Problem 18 Let k be a positive integer. Determine those real numbers c for which every sequence $( x _ { n } )$ of real numbers satisfying the recurrence relation

$$
{ \frac { 1 } { 2 } } \left( x _ { n + 1 } + x _ { n - 1 } \right) = c x _ { n }
$$

has period k (i.e., xn+k = xn for all n).