Preliminary Exam - Fall 1992

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

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

Problem 3  

1. How many roots does the polynomial $p(z) = 2z^5+4z^2+1$ have in the disc $\vert z\vert < 1$?

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

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

$$y'' = -\vert y\vert, \ \ y(0) = 1, \ y'(0) = 0.$$

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 $K \cap H$ has a finite index in $K$.

Problem 6   Let $(X_1,d_1)$ and $(X_2,d_2)$ be metric spaces and $f: X_1 \to X_2$ a continuous surjective map such that $d_1(p,q) \leq d_2(f(p),f(q))$ 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

$$\int _C \ \frac{e^z}{z(2z+1)^2} \ dz,$$

where $C$ is the unit circle with counterclockwise orientation.

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

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 $\dim V_1 = \deg \mu$.

2. Prove that $\deg \mu$ divides $\dim V$.

Problem 9   Let the function $f$ be analytic in the region $\vert z\vert > 1$ of the complex plane. Prove that if $f$ is real valued on the interval $(1,\infty )$ 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 $\mbox{\bf {F}}$, and let $B: V \times V \to \mbox{\bf {F}}$ be a bilinear map (not necessarily symmetric). Define the subspaces $V_1$ and $V_2$ by

$$% latex2html id marker 872 V_1 = \{x \in V: B(x,y) = 0 \ {\mbox{\ \ for all \ \ $y$\ \ \ in \ \ $V$}}\}$$

$$% latex2html id marker 873 V_2 = \{y \in V: B(x,y) = 0 \ {\mbox{\ \ for all \ \ $x$\ \ \ in \ \ $V$}}\}.$$

Prove that $\dim V_1 = \dim V_2$.

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

$$\vert f'_{n} (x)\vert \leq \ x^{-\frac{1}{2}} \; \; \; (0 < x \leq 1),$$

$$\int^1_0 \ f_n(x) \ dx = 0\,.$$

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

Problem 13   Evaluate

$$\int^\infty_{-\infty} \ \frac{\sin x}{x(1+x^2)} \ dx\, .$$

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

$$\frac{dx}{dt} = f(x), \ \ \ x(0) = 0,$$

has a solution defined on all of $\mathbb{R}$. Prove that the initial value problem

$$\frac{dx}{dt} = g(x), \ \ \ x(0) = 0,$$

also has a solution defined on all of $\mathbb{R}$.

Problem 15   Let $G$ be the group of all real 2$\times$2 matrices of the form $\left( \begin{array}{cc} a & b \\ 0 & a^{-1} \end{array} \right)$ 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 $\mathbb{R}$.

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

Problem 16   Let f be a $C^1$ function of $\mathbb{R}^{n}$ 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 $f(\mbox{$\mathbb{R}^{n}$}) = \mbox{$\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

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

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

$$\frac{1}{2}\left(x_{n+1}+x_{n-1}\right) = cx_n$$

has period k (i.e., $x_{n+k} = x_n$ for all n).