Preliminary Exam - Spring 1981

Problem 1   Let $\vec{\imath}$, $\vec{\jmath}$, and $\vec{k}$ be the usual unit vectors in $\mathbb{R}^{3}$. Let $\vec{F}$ denote the vector field

$$(x^2+y-4)\vec{\imath} + 3xy\vec{\jmath} + (2xz+z^2)\vec{k}.$$

1. Compute $\nabla \times \vec{F}$ (the curl of $\vec{F}$).

2. Compute the integral of $\nabla \times \vec{F}$ over the surface $x^2+y^2+z^2=16$, $z\geq 0$.

Problem 2   Let $T$ be a linear transformation of a vector space $V$ into itself. Suppose $x\in V$ is such that $T^m x = 0$, $T^{m-1}x\neq0$ for some positive integer $m$. Show that $x,\, Tx,\ldots , T^{m-1}x$ are linearly independent.

Problem 3   Let $D$ be an ordered integral domain and $a \in D$. Prove that

$$a^2-a+1>0.$$

Problem 4   Consider the system of differential equations

$$\begin{eqnarray*} \frac{dx}{dt} & = & y + x(1-x^2-y^2) \\ \frac{dy}{dt} & = & -x+y(1-x^2-y^2). \end{eqnarray*}$$

1. Show that for any $x_0$ and $y_0$, there is a unique solution $(x(t),y(t))$ defined for all $t\in \mbox{$\mathbb{R}^{}$}$ such that $x(0)=x_0$, $y(0)=y_0$.
2. Show that if $x_0 \neq 0$ and $y_0 \neq 0$, the solution referred to in Part 1 approaches the circle $x^2+y^2=1$ as $t\to\infty$.

Problem 5   Decompose $x^4-4$ and $x^3-2$ into irreducibles over $\mathbb{R}^{}$, over $\mathbb{Z}^{}$, and over $\mbox{$\mathbb{Z}^{}$}_3$ (the integers modulo $3$).

Problem 6   Suppose the complex polynomial

$$\sum_{k=0}^na_kz^k$$

has $n$ distinct roots $r_1,\ldots,r_n \in \mbox{$\mathbb{C}\,^{}$}$. Prove that if $\vert b_k-a_k\vert$ is sufficiently small then

$$\sum_{k=0}^nb_kz^k$$

has $n$ roots which are smooth functions of $b_0,\ldots,b_n$.

Problem 7   Evaluate

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

Problem 8   Let $f:[0,1] \to \mbox{$\mathbb{R}^{}$}$ be continuous. Prove that there is a real polynomial $P(x)$ of degree $\leq 10$ which minimizes (for all such polynomials)

$$\sup_{0\leq x \leq 1}\vert f(x)-P(x)\vert.$$

Problem 9   Show that the following three conditions are all equivalent for a real 3$\times$3 symmetric matrix $A$, whose eigenvalues are $\lambda_1$, $\lambda_2$, and $\lambda_3$:

1. $\mathrm{tr}\, A$ is not an eigenvalue of $A$.

2. $(a+b)(b+c)(a+c) \neq 0$.

3. The map $L:S \to S$ is an isomorphism, where $S$ is the space of 3$\times$3 real skew-symmetric matrices and $L(W) = AW + WA$.

Problem 10  

1. Give an example of a sequence of $C^1$ functions
   
   $$% latex2html id marker 915 f_k:[0,\infty) \to \mbox{$\mathbb{R}^{}$}, \quad k=0,1,2, \ldots$$
   
   
   such that $f_k(0)=0$ for all k, and $f_k'(x) \to f_0'(x)$ for all $x$ as $k \to \infty$, but $f_k(x)$ does not converge to $f_0(x)$ for all $x$ as $k \to \infty$.

2. State an extra condition which would imply that $f_k(x) \to f_0(x)$ for all $x$ as $k \to \infty$.

Problem 11   Evaluate

$$\int_C\frac{e^z-1}{z^2(z-1)}\,dz$$

where $C$ is the closed curve shown below:

file=../Fig/Pr/Sp81-11,width=4.5in

Problem 12   For $x \in \mbox{$\mathbb{R}^{}$}$, let

$$A_x = \left( \begin{array}{cccc} x & 1 & 1 & 1 \\ 1 & x & 1 & 1 \\ 1 & 1 & x & 1 \\ 1 & 1 & 1 & x \end{array} \right).$$

1. Prove that $\det (A_x) = (x-1)^3(x+3)$.
2. Prove that if $x\neq 1,-3$, then $A_x^{-1}= -(x-1)^{-1}(x+3)^{-1}A_{-x-2}$.

Problem 13   Which of the following series converges?

1. 
   
   $$\sum_{n=1}^{\infty}\frac{(2n)!(3n)!}{n!(4n)!} \,\cdot$$
   
   

2. 
   
   $$\sum_{n=1}^{\infty}\frac{1}{n^{1+1/n}} \,\cdot$$
   
   

Problem 14   The set of real 3$\times$3 symmetric matrices is a real, finite-dimensional vector space isomorphic to $\mathbb{R}^{6}$. Show that the subset of such matrices of signature $(2,1)$ is an open connected subspace in the usual topology on $\mathbb{R}^{6}$.

Problem 15   Let $\mbox{\bf {M}}$ be one of the following fields: $\mathbb{R}^{}$, $\mathbb{C}\,^{}$, $\mathbb{Q}\,^{}$, and $\mbox{\bf {F}}_{9}$ (the field with nine elements). Let $\mathfrak{I} \subset \mbox{\bf {M}}[x]$ be the ideal generated by $x^4+2x-2$. For which choices of $\mbox{\bf {M}}$ is the ring $\mbox{\bf {M}}[x]/\mathfrak{I}$ a field?

Problem 16   Let $f(x)$ be a real valued function defined for all $x\geq 1$, satisfying $f(1)=1$ and

$$f'(x) = \frac{1}{x^2+f(x)^2} \cdot$$

Prove that

$$\lim_{x\to\infty}f(x)$$

exists and is less than $1+\frac{\pi}{4} \cdot$

Problem 17   Let b be a real nonzero $n\times 1$ matrix (a column vector). Set $M=bb^t$ (an $n\times n$ matrix) where $b^t$ denotes the transpose of b.

1. Prove that there is an orthogonal matrix $Q$ such that $QMQ^{-1}=D$ is diagonal, and find $D$.

2. Describe geometrically the linear transformation $M:\mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{n}$}$.

Problem 18   Describe the two regions in $(a,b)$-space for which the function

$$f_{a,b}(x,y) = ay^2 + bx$$

restricted to the circle $x^2+y^2=1$, has exactly two, and exactly four critical points, respectively.

Problem 19   Let $G$ be a finite group. A conjugacy class is a set of the form

$$C(a) = \{bab^{-1} \;\vert\; b \in G \}$$

for some $a \in G$.

1. Prove that the number of elements in a conjugacy class divides the order of $G$.

2. Do all conjugacy classes have the same number of elements?

3. If $G$ has only two conjugacy classes, prove $G$ has order $2$.

Problem 20   Let $f:[0,1] \to \mbox{$\mathbb{R}^{}$}$ be continuous with $f(0)=0$. Show there is a continuous concave function $g:[0,1] \to \mbox{$\mathbb{R}^{}$}$ such that $g(0)=0$ and $g(x)\geq f(x)$ for all $x \in [0,1]$.

Note: A function $g:I \to \mbox{$\mathbb{R}^{}$}$ is concave if

$$g \left( tx + (1-t)y \right) \geq tg(x) + (1-t)g(y)$$

for all $x$ and $y$ in $I$ and $0\leq t \leq 1$. _function,>concave