Preliminary Exam - Spring 1980

Problem 1   Let $f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ be the unique function such that
$f(x)=x$ if $-\pi \leq x < \pi$ and $f(x+2n\pi )=f(x)$ for all $n\in \mbox{$\mathbb{Z}^{}$}$.

1. Prove that the Fourier series of $f$ is
   
   $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}2\sin nx}{n}\cdot$$
   
   

2. Prove that the series does not converge uniformly.
3. For each $x\in\mbox{$\mathbb{R}^{}$}$, find the sum of the series.

Problem 2   Let $f_n:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$ be differentiable for each n = 1, 2, ... with $\vert f_n'(x)\vert\leq 1$ for all n, x. Assume

$$\lim_{n\to\infty}f_n(x) = g(x)$$

for all x. Prove that $g:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ is continuous.

Problem 3   Let $P_2$ denote the set of real polynomials of degree $\leq 2$. Define the map $J : P_2 \to \mbox{$\mathbb{R}^{}$}$ by

$$J(f) = \int_0^1f(x)^2\,dx\, .$$

Let $Q = \{f \in P_2 \;\vert\; f(1) = 1\}$. Show that $J$ attains a minimum value on $Q$ and determine where the minimum occurs.

Problem 4   Let $a>0$ be a constant $\neq 2$. Let $C_a$ denote the positively oriented circle of radius $a$ centered at the origin. Evaluate

$$\int_{C_a}\frac{z^2+e^z}{z^2(z-2)}\,dz\,.$$

Problem 5   Let

$$f(z) = \sum_{n=0}^{\infty}a_nz^n$$

be an analytic function in the open unit disc $\mathbb{D}^{}$. Assume that

$$\sum_{n=2}^{\infty}n\vert a_n\vert \leq \vert a_1\vert\quad with\quad a_{1} \neq 0.$$

Prove that $f$ is injective.

Problem 6   $G$ is a group of order $n$, $H$ a proper subgroup of order $m$, and $(n/m)!<2n$. Prove $G$ has a proper normal subgroup different from the identity.

Problem 7   Let $n\geq 2$ be an integer such that $2^n+n^2$ is prime. Prove that

$$n \equiv 3 \pmod{6}.$$

Problem 8   Let $A$ and $B$ be $n\times n$ complex matrices. Prove or disprove each of the following statements:

1. If $A$ and $B$ are diagonalizable, so is $A + B$.
2. If $A$ and $B$ are diagonalizable, so is $AB$.
3. If $A^2=A$, then $A$ is diagonalizable.
4. If $A$ is invertible and $A^2$ is diagonalizable, then $A$ is diagonalizable.

Problem 9   Let

$$A = \left( \begin{array}{cc} 1 & 2 \\ 3 & 4\end{array} \right).$$

Show that every real matrix $B$ such that $AB = BA$ has the form $sI+tA$, where $s, t \in \mbox{$\mathbb{R}^{}$}$.

Problem 10   Consider the differential equation

$$x' = \frac{x^3-x}{1+e^x}\cdot$$

1. Find all its constant solutions.
2. Discuss $${\lim_{t\to \infty}x(t)}$$, where $x(t)$ is the solution such that $x(0)=\frac{1}{2} \cdot$

Problem 11   Let ${\cal S} = \{(x,y,z) \in \mbox{$\mathbb{R}^{3}$} \;\vert\; x^2+y^2+z^2=1\}$ denote the unit sphere in $\mathbb{R}^{3}$. Evaluate the surface integral over S:

$$\int_{\cal S} (x^2+y+z)\,dA\, .$$

Problem 12   Let $M_{3\times 3}$ denote the vector space of real 3$\times$3 matrices. For any matrix $A \in M_{3\times 3}$, define the linear operator $L_A:M_{3\times 3} \to M_{3\times 3}$, $L_A(B)=AB$. Suppose that the determinant of $A$ is $32$ and the minimal polynomial is $(t-4)(t-2)$. What is the trace of $L_A$?

Problem 13   Let $G$ be a group of permutations of a set $S$ of $n$ elements. Assume $G$ is transitive; that is, for any $x$ and $y$ in $S$, there is some $\sigma \in G$ with $\sigma(x) = y$. _group of permutations,>transitive

1. Prove that $n$ divides the order of $G$.

2. Suppose $n=4$. For which integers $k\geq 1$ can such a $G$ have order $4k$?

Problem 14   Find a real matrix $B$ such that

$$B^4 = \left( \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & -1 & 1 \end{array} \right).$$

Problem 15   Show that a vector space over an infinite field cannot be the union of a finite number of proper subspaces.

Problem 16   Let $f : \mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{n}$}$ be continuously differentiable. Assume the Jacobian matrix $\left(\partial f_i/\partial x_j\right)$ has rank n everywhere. Suppose $f$ is proper; that is, $f^{-1}(K)$ is compact whenever $K$ is compact. Prove $f(\mbox{$\mathbb{R}^{n}$}) = \mbox{$\mathbb{R}^{n}$}$. _function,>proper

Problem 17   $S_9$ is the group of permutations of the set of integers from $1$ to $9$.

1. Exhibit an element of $S_9$ of order $20$.

2. Prove that no element of $S_9$ has order $18$.

Problem 18   For each $t\in \mbox{$\mathbb{R}^{}$}$, let $P(t)$ be a symmetric real $n\times n$ matrix whose entries are continuous functions of $t$. Suppose for all $t$ that the eigenvalues of $P(t)$ are all $\leq -1$. Let $x(t)=\left( x_1(t),\ldots,x_n(t) \right)$ be a solution of the vector differential equation

$$\frac{dx}{dt} = P(t)x.$$

Prove that

$$\lim_{t \to \infty}x(t)=0.$$

Problem 19   Let

$$f(z) = \sum_{n=0}^{\infty}c_nz^n$$

be analytic in the disc $\mbox{$\mathbb{D}^{}$} = \{z \in \mbox{$\mathbb{C}\,^{}$} \;\vert\; \vert z\vert<1\}$. Assume $f$ maps $\mbox{$\mathbb{D}^{}$}$ one-to-one onto a domain $G$ having area $A$. Prove

$$A = \pi\sum_{n=1}^{\infty}n\vert c_n\vert^2.$$

Problem 20   Does there exist an analytic function mapping the annulus

$$A = \{z \;\vert\; 1 \leq \vert z\vert \leq 4\}$$

onto the annulus

$$B = \{z \;\vert\; 1 \leq \vert z\vert \leq 2\}$$

and taking $C_1 \to C_1, C_4 \to C_2$, where $C_r$ is the circle of radius r?