Preliminary Exam - Fall 1982

Problem 1   Let $f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ be a continuous nowhere vanishing function, and consider the differential equation

$$(*) \qquad \frac{dy}{dx} = f(y).$$

1. For each real number $c$, show that $(*)$ has a unique, continuously differentiable solution $y=y(x)$ on a neighborhood of $0$ which satisfies the initial condition $y(0)=c$.

2. Deduce the conditions on $f$ under which the solution $y$ exists for all $x \in \mbox{$\mathbb{R}^{}$}$, for every initial value $c$.

Problem 2   Consider the polynomial ring $R = \mbox{$\mathbb{Z}^{}$}[x]$ and the ideal $\mathfrak{I}$ of $R$ generated by $7$ and $x-3$.

1. Show that for each $r \in R$, 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 3   Let

$$\cot (\pi z) = \sum_{n=-\infty}^{\infty}a_nz^n$$

be the Laurent expansion for $\cot(\pi z)$ on the annulus $1<\vert z\vert<2$. Compute the $a_n$ for $n<0$.

Problem 4   Let $M$ be an $n\times n$ matrix of real numbers. Prove or disprove: The dimension of the subspace of $\mathbb{R}^{n}$ generated by the rows of $M$ is equal to the dimension of the subspace of $\mathbb{R}^{n}$ generated by the columns of $M$.

Problem 5   Let $\varphi_1, \varphi_2, \ldots,\varphi_n,\ldots$ be nonnegative continuous functions on $[0,1]$ such that the limit

$$\lim_{n \to \infty}\int_0^1x^k\varphi_n(x)\,dx$$

exists for every $k = 0,1,\ldots$ . Show that the limit

$$\lim_{n \to \infty}\int_0^1f(x)\varphi_n(x)\,dx$$

exists for every continuous function f on $[0,1]$.

Problem 6   Let $T$ be a linear transformation on a finite-dimensional $\mathbb{C}\,^{}$-vector space V, and let $f$ be a polynomial with coefficients in $\mathbb{C}\,^{}$. If $\lambda$ is an eigenvalue of $T$, show that $f(\lambda)$ is an eigenvalue of $f(T)$. Is every eigenvalue of $f(T)$ necessarily obtained in this way?

Problem 7   Evaluate

$$\int_{-\infty}^{\infty}\frac{\cos \pi x}{4x^2-1}\,dx\, .$$

Problem 8   Let

$$% latex2html id marker 890 G=\left\{\left( \begin{array}{cc... ...right)\;\vert\; a,b \in \mbox{$\mathbb{R}^{}$},\; a>0 \right\}$$

$$% latex2html id marker 891 N=\left\{\left( \begin{array}{cc... ...{array} \right)\;\vert\; b\in \mbox{$\mathbb{R}^{}$} \right\}.$$

1. Show that $N$ is a normal subgroup of $G$ and prove that $G/N$ is isomorphic to $\mathbb{R}^{}$.

2. Find a normal subgroup $N'$ of $G$ satisfying $N\subset N' \subset G$ (where the inclusions are proper), or prove that there is no such subgroup.

Problem 9   Let f be a real valued continuous nonnegative function on $[0,1]$ such that

$$f(t)^2 \leq 1 + 2\int_0^tf(s)\,ds$$

for $t \in [0,1]$. Show that $f(t) \leq 1 + t$ for $t \in [0,1]$.

Problem 10   Let $a$ and $b$ be complex numbers whose real parts are negative or $0$. Prove the inequality $\vert e^a-e^b\vert \leq \vert a-b\vert$.

Problem 11  

1. Prove that there is no continuous map from the closed interval $[0,1]$ onto the open interval $(0,1)$.

2. Find a continuous surjective map from the open interval $(0,1)$ onto the closed interval $[0,1]$.

3. Prove that no map in Part 2 can be bijective.

Problem 12   Let $A$ and $B$ be complex $n\times n$ matrices having the same rank. Suppose that $A^2 = A$ and $B^2 = B$. Prove that $A$ and $B$ are similar.

Problem 13   Let $f_1,f_2,\ldots$ be continuous functions on $[0,1]$ satisfying $f_1\geq f_2 \geq \cdots$ and such that $\lim_{n \to \infty}f_n(x) = 0$ for each x. Must the sequence $\{f_n\}$ converge to $0$ uniformly on $[0,1]$?

Problem 14   Let $A$ be an $n\times n$ complex matrix, and let $B$ be the Hermitian transpose of $A$ (i.e., $b_{ij}=\overline{a}_{ji}$). Suppose that $A$ and $B$ commute with each other. Consider the linear transformations $\alpha$ and $\beta$ on $\mathbb{C}\,^{n}$ defined by $A$ and $B$. Prove that $\alpha$ and $\beta$ have the same image and the same kernel.

Problem 15   Let $A$ be a subgroup of an abelian group $B$. Assume that $A$ is a direct summand of $B$, i.e., there exists a subgroup $X$ of $B$ such that $A\cap X =0$ and such that $B = X + A$. Suppose that $C$ is a subgroup of $B$ satisfying $A \subset C \subset B$. Is $A$ necessarily a direct summand of $C$?

Problem 16   Find all pairs of $C^{\infty}$ functions $x(t)$ and $y(t)$ on $\mathbb{R}^{}$ satisfying

$$x'(t)=2x(t)-y(t), \qquad y'(t)=x(t).$$

Problem 17   Evaluate

$$I = \int_{-\infty}^{\infty}\frac{\sin^3x}{x^3}\,dx\, .$$

Problem 18   Let $K$ be a continuous function on the unit square
$0\leq x,y \leq1$ satisfying $\vert K(x,y)\vert<1$ for all $x$ and $y$. Show that there is a continuous function $f(x)$ on $[0,1]$ such that we have

$$f(x) + \int_0^1K(x,y)f(y)\,dy = e^{x^2}\, .$$

Can there be more than one such function $f$?

Problem 19   Let $a$ and $b$ be nonzero complex numbers and
$f(z)=az+bz^{-1}$. Determine the image under $f$ of the unit circle $\{z \; \vert\; \vert z\vert = 1\}$.

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

$$\begin{array}{l} 2x+4y+6z = 0, \\ 8x+4z = 4y, \\ 6x = 8y + 2z. \end{array}$$

Write $G$ as a product of cyclic groups. How many elements of $G$ have order $2$?