Preliminary Exam - Fall 1985

Problem 1   Evaluate the integral

$$\int_0^{\infty}\frac{1-\cos ax}{x^2}\,dx$$

for $a \in \mathbb{R}$.

Problem 2   Prove that for every $\lambda >1$, the equation $ze^{\lambda-z}=1$ has exactly one root in the disc $\vert z\vert<1$ and that this root is real.

Problem 3  

1. How many different monic irreducible polynomials of degree $2$ are there over the field $\mbox{$\mathbb{Z}^{}$}_5$?

2. How many different monic irreducible polynomials of degree $3$ are there over the field $\mbox{$\mathbb{Z}^{}$}_5$?

Problem 4   Let $G$ be a finite subgroup of the group $\mathbb{C}\,^{*}$ of nonzero complex numbers under multiplication. Prove that $G$ is cyclic.

Problem 5   How many roots does the polynomial $z^4+3z^2+z+1$ have in the right half $z$-plane?

Problem 6   Let $k$ be real, $n$ an integer $\geq 2$, and let ${A}=(a_{ij})$ be the $n\times n$ matrix such that all diagonal entries $a_{ii}=k$, all entries $a_{i\,i\pm1}$ immediately above or below the diagonal equal $1$, and all other entries equal $0$. For example, if $n=5$,

$${A} = \left( \begin{array}{ccccc} k & 1 & 0 & 0 & 0 \\ 1 & k... ... 0 & 0 & 1 & k & 1 \\ 0 & 0 & 0 & 1 & k\end{array} \right)\,.$$

Let $\lambda_{min}$ and $\lambda_{max}$ denote the smallest and largest eigenvalues of ${A}$, respectively. Show that $\lambda_{min}\leq k-1$ and $\lambda_{max}\geq k+1$.

Problem 7   Let $y(t)$ be a real valued solution, defined for
$0<t<\infty$, of the differential equation

$$\frac{dy}{dt} = e^{-y}-e^{-3y}+e^{-5y}.$$

Show that $y(t) \to +\infty$ as $t \to +\infty$.

Problem 8   Let $f(x)$, $0\leq x \leq 1$, be a real valued continuous function. Show that

$$\lim_{n\to\infty}(n+1)\int_0^1x^nf(x)\,dx = f(1).$$

Problem 9   Let $A$ be the symmetric matrix

$$\frac{1}{6}\left( \begin{array}{rrr} 13 & -5 & -2 \\ -5 & 13 & -2 \\ -2 & -2 & 10 \end{array} \right).$$

We denote by $x$ the column vector

$$\left( \begin{array}{c} x_1 \\ x_2 \\ x_3\end{array} \right)$$

$x_i\in\mbox{$\mathbb{R}^{}$}$, and by $x^t$ its transpose $(x_1,x_2,x_3)$. Let $\vert x\vert$ denote the length of the vector $x$. As $x$ ranges over the set of vectors for which $x^t A x = 1$, show that $\vert x\vert$ is bounded, and determine its least upper bound.

Problem 10   Let $f$ and $f_n$, $n=1,2,\ldots$, be functions from $\mathbb{R}^{}$ to $\mathbb{R}^{}$. Assume that $f_n(x_n)\to f(x)$ as $n\to \infty$ whenever $x_n \to x$. Show that $f$ is continuous. Note: The functions $f_n$ are not assumed to be continuous.

Problem 11   Let $G$ be a subgroup of the symmetric group on six letters, $\mathrm{S}_{6}$. Assume that $G$ has an element of order $6$. Prove that $G$ has a normal subgroup $H$ of index $2$.

Problem 12   Evaluate

$$\int_0^{2\pi}e^{e^{i\theta}}\,d\theta\,.$$

Problem 13   Let $f(z)$ be analytic on the right half-plane $H = \{ z\;\vert\; \Re z > 0 \}$ and suppose $\vert f(z)\vert \leq 1$ for $z \in H$. Suppose also that $f(1) = 0$. What is the largest possible value of $\vert f'(1)\vert$?

Problem 14   Suppose that and are endomorphisms of a finite-dimensional vector space over a field . Prove or disprove the following statements:

1. Every eigenvector of is also an eigenvector of .
2. Every eigenvalue of is also an eigenvalue of .

Problem 15   Let $0 \leq a \leq 1$ be given. Determine all nonnegative continuous functions $f$ on $[0,1]$ which satisfy the following three conditions:

$$\int_{0}^{1}f(x)\,dx = 1,$$

$$\int_{0}^{1}xf(x)\,dx = a,$$

$$\int_{0}^{1}x^2f(x)\,dx = a^2.$$

Problem 16   Let $f(x)=x^5-8x^3+9x-3$ and $g(x)=x^4-5x^2-6x+3$. Prove that there is an integer $d$ such that the polynomials $f(x)$ and $g(x)$ have a common root in the field $\mbox{$\mathbb{Q}\,^{}$}(\sqrt{d}\,)$. What is $d$?

Problem 17   Let $(M,d)$ be a nonempty complete metric space. Let $S$ map $M$ into $M$, and write $S^2$ for $S\circ S$; that is, $S^2(x) = S \left( S(x) \right)$. Suppose that $S^2$ is a strict contraction; that is, there is a constant $\lambda < 1$ such that for all points $x, y \in M, d \left( S^2(x),S^2(y) \right) \leq \lambda d(x,y)$. Show that $S$ has a unique fixed point in $M$. _map>strict contraction

Problem 18   Let $G$ be a group. For any subset $X$ of $G$, define its centralizer $C(X)$ to be $\{y \in G \;\vert\; xy = yx, \forall x \in X\}$. Prove the following:

1. If $X \subset Y$, then $C(Y) \subset C(X)$.

2. $X \subset C \left( C(X) \right)$.

3. $C(X) = C \left( C \left( C(X) \right) \right)$.

Problem 19   An n$\times$n real matrix $T$ is positive definite if _matrix,>positive definite $T$ is symmetric and $\langle Tx,x \rangle >0$ for all nonzero vectors $x \in \mbox{$\mathbb{R}^{n}$}$, where $\langle u, v \rangle$ is the standard inner product. Suppose that $A$ and $B$ are two positive definite real matrices.

1. Show that there is a basis $\{v_1,v_2,\ldots,v_n\}$ of $\mathbb{R}^{n}$ and real numbers
   $\lambda_1,\lambda_2,\ldots,\lambda_n$ such that, for $1 \leq i,j \leq n$:
   
   $$\langle Av_i,v_j \rangle=\left\{ \begin{array}{cc} 1 & i=j \\ 0 & i \neq j \end{array} \right.$$
   
   
   and
   
   $$\langle Bv_i,v_j \rangle =\left\{ \begin{array}{cc} \lambda_i & i=j \\ 0 & i \neq j \end{array} \right.$$
   
   

2. Deduce from Part 1 that there is an invertible real matrix $U$ such that $U^tAU$ is the identity matrix and $U^tBU$ is diagonal.

Problem 20   Let $f$ be a differentiable function on $[0,1]$ and let

$$\sup_{0<x<1}\,\vert f'(x)\vert = M < \infty.$$

Let $n$ be a positive integer. Prove that

$$\left\vert \sum_{j=0}^{n-1}\frac{f(j/n)}{n} - \int_0^1f(x)\,dx \right\vert \leq \frac{M}{2n}\cdot$$