Preliminary Exam - Fall 1979

Problem 1   Prove that the polynomial

$$p(z)=z^{47}-z^{23}+2z^{11}-z^5+4z^2+1$$

has at least one root in the disc $\vert z\vert<1$.

Problem 2   Suppose that $f$ is analytic on the open upper half-plane and satisfies $\vert f(z)\vert\leq 1$ for all $z$, $f(i)=0$. How large can $\vert f(2i)\vert$ be under these conditions?

Problem 3   Prove that every finite group of order $> 2$ has a nontrivial automorphism.

Problem 4   Consider the polynomial ring $\mbox{$\mathbb{Z}^{}$}[x]$ and the ideal $\mathfrak{I}$ generated by $7$ and $x-3$.

1. Show that for each $r \in \mbox{$\mathbb{Z}^{}$}[x]$, there is an integer $\alpha$ satisfying $0 \leq \alpha \leq 6$ such that $r-\alpha \in \mathfrak{I}$.

2. Find $\alpha$ in the special case $r = x^{250}+15x^{14}+x^2+5$.

Problem 5   Let $A$ be a real skew-symmetric matrix $(A_{ij} = -A_{ji})$. Prove that $A$ has even rank. _matrix,>skew-symmetric

Problem 6   Let $N$ be a linear operator on an $n$-dimensional vector space, $n>1$, such that $N^n = 0, \; N^{n-1} \; \neq 0$. Prove there is no operator X with $X^2=N$.

Problem 7   Let $V$ be the vector space of sequences $(a_n)$ of _Fibonacci numbers complex numbers. The shift operator $S : V \to V$ is defined by

$$S \left( (a_1, a_2, a_3, \ldots )\right)= (a_2, a_3, a_4, \ldots ).$$

1. Find the eigenvectors of $S$.
2. Show that the subspace $W$ consisting of the sequences $(x_n)$ with $x_{n+2} = x_{n+1} + x_n$ is a two-dimensional, $S$-invariant subspace of $V$ and exhibit an explicit basis for $W$.
3. Find an explicit formula for the $n^{th}$ Fibonacci number $f_n$, where $f_2 = f_1 = 1$, $f_{n+2} = f_{n+1} + f_n$ for $n \geq 1$.

Note: See also Problem .

Problem 8   Prove that

$$\lim_{n \to \infty}\left(\frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n}\right) = \log 2 .$$

Problem 9   Given that

$$\int_{-\infty}^{\infty}e^{-x^2}\,dx = \sqrt{\pi} \; ,$$

find $f'(t)$ explicitly, where

$$f(t) = \int_{-\infty}^{\infty}e^{-tx^2}\,dx, \hspace{.2in} t > 0.$$

Problem 10   Solve the differential equations

$$\begin{eqnarray*} \frac{dx}{dt} & = & -3x + 10y \\ \frac{dy}{dt} & = & -3x + 8y\,. \end{eqnarray*}$$

Problem 11   An accurate map of California is spread out flat on a table in Evans Hall, in Berkeley. Prove that there is exactly one point on the map lying directly over the point it represents.

Problem 12   Consider the following properties of a map $f:\mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{}$}$:

1. $f$ is continuous.
2. The graph of $f$ is connected in $\mbox{$\mathbb{R}^{n}$}\times\mbox{$\mathbb{R}^{}$}$.

Prove or disprove the implications 1 $\Rightarrow$ 2, 2 $\Rightarrow$ 1.

Problem 13   Let $\{P_n\}$ be a sequence of real polynomials of degree $\leq D$, a fixed integer. Suppose that $P_n(x) \to 0$ pointwise for $0\leq x \leq 1$. Prove that $P_n \to 0$ uniformly on $[0,1]$.

Problem 14   Let $y=y(x)$ be a solution of the differential equation $y''=-\vert y\vert$ with $-\infty<x<\infty$, $y(0)=1$ and $y'(0)=0$.

1. Show that $y$ is an even function.

2. Show that $y$ has exactly one zero on the positive real axis.

Problem 15   Suppose $f$ and $g$ are entire functions with $\vert f(z)\vert\leq \vert g(z)\vert$ for all $z$. Prove that $f(z) = cg(z)$ for some constant $c$.

Problem 16   Prove that

$$\int_0^{\infty}\frac{x^{\alpha -1}}{1+x}\,dx = \frac{\pi}{\sin \pi\alpha } \cdot$$

What restrictions must be placed on $\alpha$?

Problem 17   Let $A$ be an $n\times n$ complex matrix. Prove there is a unitary matrix $U$ such that $B =UAU^{-1}$ is upper-triangular: $B_{jk}=0$ for $j>k$.

Problem 18   Let $B$ denote the matrix

$$\left( \begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array} \right)$$

where $a$, $b$, and $c$ are real and $\vert a\vert$, $\vert b\vert$, and $\vert c\vert$ are distinct. Show that there are exactly four symmetric matrices of the form $BQ$, where $Q$ is a real orthogonal matrix of determinant $1$.

Problem 19   Let $M_{n\times n}(\mbox{\bf {F}})$ be the ring of $n\times n$ matrices over a field $\mbox{\bf {F}}$. Prove that it has no 2-sided ideals except $M_{n\times n}(\mbox{\bf {F}})$ and $\{0\}$.

Problem 20   Let $G$ be the abelian group defined by generators $x$, $y$, and $z$, and relations

$$\begin{eqnarray*} 15x + 3y & = & 0 \\ 3x + 7y + 4z & = & 0 \\ 18x + 14y + 8z & = & 0. \end{eqnarray*}$$

1. Express $G$ as a direct product of two cyclic groups.
2. Express $G$ as a direct product of cyclic groups of prime power order.
3. How many elements of $G$ have order $2$?