Preliminary Exam - Fall 1981

Problem 1   Evaluate the integral

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

Problem 2   Consider an autonomous system of differential equations

$$\frac{dx_i}{dt} = F_i(x_1,\ldots,x_n),$$

where $F=(F_1,\ldots,F_n):\mbox{$\mathbb{R}^{n}$}\to\mbox{$\mathbb{R}^{n}$}$ is a $C^1$ vector field.

1. Let $U$ and $V$ be two solutions on $a<t<b$. Assuming that
   
   $$\langle DF(x)z, z \rangle \leq 0$$
   
   
   for all x, z in $\mathbb{R}^{n}$, show that $\vert U(t)-V(t)\vert^2$ is a decreasing function of $t$.

2. Let $W(t)$ be a solution defined for $t>0$. Assuming that
   
   $$\langle DF(x)z, z \rangle \leq -\vert z\vert^2,$$
   
   
   show that there exists $C\in\mbox{$\mathbb{R}^{n}$}$ such that
   
   $$\lim_{t\to\infty}W(t) = C.$$
   
   

Problem 3   Let $\mathrm{S}_{n}$ be the group of all permutations of $n$ objects and let $G$ be a subgroup of $\mathrm{S}_{n}$ of order $p^k$, where $p$ is a prime not dividing $n$. Show that $G$ has a fixed point; that is, one of the objects is left fixed by every element of $G$. _group,>fixed point

Problem 4   Prove the following three statements about real $n\times n$ matrices.

1. If $A$ is an orthogonal matrix whose eigenvalues are all different from $-1$, then $I_n + A$ is nonsingular and $S=(I_n-A)(I_n + A)^{-1}$ is skew-symmetric.

2. If $S$ is a skew-symmetric matrix, then $A=(I_n - S)(I_n + S)^{-1}$ is an orthogonal matrix with no eigenvalue equal to $-1$.

3. The correspondence $A \leftrightarrow S$ from Parts 1 and 2 is one-to-one.

Problem 5   The Fibonacci numbers $f_1,f_2,\ldots$ are defined recursively by $f_1=1$, _Fibonacci numbers $f_2=2$, and $f_{n+1}=f_n+f_{n-1}$ for $n\geq 2$. Show that

$$\lim_{n\to\infty}\frac{f_{n+1}}{f_n}$$

exists, and evaluate the limit.

Note: See also Problem .

Problem 6   Let $f$ and $g$ be continuous functions on $\mathbb{R}^{}$ such that
$f(x+1)=f(x)$, $g(x+1)=g(x)$, for all $x \in \mbox{$\mathbb{R}^{}$}$. Prove that

$$\lim_{n\to \infty}\int_{0}^{1}f(x)g(nx)\,dx = \int_{0}^{1}f(x)\,dx\,\int_{0}^{1}g(x)\,dx\, .$$

Problem 7   Find a specific polynomial with rational coefficients having $\sqrt{2}+\sqrt[3]{3}$ as a root.

Problem 8  

1. How many zeros does the function $f(z)=3z^{100}-e^z$ have inside the unit circle (counting multiplicities)?

2. Are the zeros distinct?

Problem 9   Let $M_{2\times 2}$ be the vector space of all real 2$\times$2 matrices. Let

$$A = \left( \begin{array}{cc} 1 & 2 \\ -1 & 3 \end{array} \ri... ... = \left( \begin{array}{cc} 2 & 1 \\ 0 & 4\end{array} \right)$$

and define a linear transformation $L:M_{2\times 2} \to M_{2\times 2}$ by L(X) = AXB. Compute the trace and the determinant of $L$.

Problem 10   Let $A=\left( a_{ij} \right)$ be an $n\times n$ matrix whose entries $a_{ij}$ are real valued differentiable functions defined on $\mathbb{R}^{}$. Assume that the determinant $\det(A)$ of $A$ is everywhere positive. Let $B=\left( b_{ij} \right)$ be the inverse matrix of $A$. Prove the formula

$$\frac{d}{dt}\log \left( \det (A) \right) = \sum_{i,j=1}^n\frac{da_{ij}}{dt}b_{ji}.$$

Problem 11   Consider the complex 3$\times$3 matrix

$$A=\left( \begin{array}{ccc} a_0 & a_1 & a_2 \\ a_2 & a_0 & a_1 \\ a_1 & a_2 & a_0 \end{array} \right),$$

where $a_0, a_1, a_2 \in \mbox{$\mathbb{C}\,^{}$}$.

1. Show that $A=a_0I_3 + a_1E + a_2E^2$, where
   
   $$E=\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right).$$
   
   

2. Use Part 1 to find the complex eigenvalues of $A$.

3. Generalize Parts 1 and 2 to n$\times$n matrices.

Problem 12   Let a, b be real constants and let

$$u(x,y) = \frac{a^2+b^2+x^2-y^2}{(a-x)^2+(b-y)^2} \cdot$$

Show that u is harmonic and find an entire function f(z) whose real part is u.

Correction: $u$ cannot be the real part of an entire function. Why? Change $u$ slightly and do the problem.

Problem 13   Let $f$ be a real valued function on $\mathbb{R}^{n}$ of class $C^2$. A point $x \in \mbox{$\mathbb{R}^{n}$}$ is a critical point of f if all the partial derivatives of f vanish at x; a critical point is nondegenerate if the $n\times n$ matrix

$$\left( \frac{\partial^2f}{\partial x_i \partial x_j}(x) \right)$$

is nonsingular.

Let x be a nondegenerate critical point of f. Prove that there is an open neighborhood of x which contains no other critical points (i.e., the nondegenerate critical points are isolated).

Problem 14   Let $V:\mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{}$}$ be a $C^1$ function and consider the system of second order differential equations

$$x_{i}''(t) =f_i\left(x(t)\right), \hspace{.2in} 1 \leq i \leq n,$$

where

$$f_i = -\frac{\partial V}{\partial x_i}\cdot$$

Let $x(t)= \left( x_1(t),\ldots,x_n(t) \right)$ be a solution of this system on a finite interval $a<t<b$.

1. Show that the function
   
   $$H(t) = \frac{1}{2}\langle x'(t), x'(t) \rangle + V(x(t))$$
   
   
   is constant for $a<t<b$.

2. Assuming that $V(x)\geq M > -\infty$ for all $x \in \mbox{$\mathbb{R}^{n}$}$, show that $x(t)$, $x'(t)$, and $x''(t)$ are bounded on $a<t<b$, and then prove all three limits
   
   $$\lim_{t\to b}x(t), \quad \lim_{t\to b}x'(t), \quad \lim_{t\to b}x''(t)$$
   
   
   exist.

Problem 15   Let $f$ be a holomorphic map of the unit disc $\mathbb{D}=\{z \;\vert\; \vert z\vert < 1\}$ into itself, which is not the identity map $f(z)=z$. Show that $f$ can have, at most, one fixed point.

Problem 16   Let $G$ be a group with three normal subgroups $N_1, N_2, N_3$. Suppose $N_i \cap N_j = \{e\}$ and $N_iN_j = G$ for all $i,j$ with $i\neq j$. Show that $G$ is abelian and $N_i$ is isomorphic to $N_j$ for all $i,j$.

Problem 17   Let $f$ be a continuous function on $[0,1]$. Evaluate the following limits.

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

2. 
   
   $$\lim_{n\to\infty}n\int_0^1x^nf(x)\,dx\, .$$
   
   

Problem 18   Let $A$ and $B$ be two real $n\times n$ matrices. Suppose there is a complex invertible $n\times n$ matrix $U$ such that $A =UBU^{-1}$. Show that there is a real invertible $n\times n$ matrix $V$ such that $A =VBV^{-1}$. (In other words, if two real matrices are similar over $\mathbb{C}\,^{}$, then they are similar over $\mathbb{R}^{}$.)

Problem 19   Either prove or disprove (by a counterexample) each of the following statements:

1. Let $f:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$, $g:\mbox{$\mathbb{R}^{}$}\to \mbox{$\mathbb{R}^{}$}$ be such that
   
   $$\lim_{ t \to a} g(t) = b \;\;and\;\; \lim_{ t \to b} f(t) = c.$$
   
   
   Then
   
   $$\lim_{ t \to a} f\left( g(t) \right) = c.$$
   
   

2. If $f:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$ is continuous and $U$ is an open set in $\mathbb{R}^{}$, then $f(U)$ is an open set in $\mathbb{R}^{}$.

3. Let f be of class $C^\infty$ on the interval $-1 < x < 1$. Suppose that $\vert f^{(n)}(x)\vert\leq 1$ for all $n \geq 1$ and all x in the interval. Then f is real analytic; that is, it has a convergent power series expansion in a neighborhood of each point of the interval. _function,>real analytic

Problem 20   Let $G$ be a group of order $10$ which has a normal subgroup of order $2$. Prove that $G$ is abelian.