Preliminary Exam - Spring 1985

Problem 1   Let $f(x)$, $0\leq x<\infty$, be continuous, differentiable, with $f(0) = 0$, and that $f'(x)$ is an increasing function of $x$ for $x \geq 0$. Prove that

$$g(x) = \left\{ \begin{array}{ll} f(x)/x, & x > 0 \\ f'(0), & x = 0 \end{array} \right.$$

is an increasing function of $x$.

Problem 2   In a commutative group $G$, let the element $a$ have order $r$, let $b$ have order $s\; (r, s < \infty )$, and assume that the greatest common divisor of $r$ and $s$ is $1$. Show that $ab$ has order $rs$.

Problem 3   Show that a necessary and sufficient condition for three points $a$, $b$, and $c$ in the complex plane to form an equilateral triangle is that

$$a^2+b^2+c^2 = bc+ca+ab.$$

Problem 4   Let $R>1$ and let $f$ be analytic on $\vert z\vert<R$ except at $z=1$, where $f$ has a simple pole. If

$$f(z) = \sum_{n=0}^{\infty}a_nz^n \qquad (\vert z\vert < 1)$$

is the Maclaurin series for $f$, show that $\lim_{n\to\infty}a_n$ exists.

Problem 5   Factor $x^4+x^3+x+3$ completely in $\mbox{$\mathbb{Z}^{}$}_5[x]$.

Problem 6   Let $A$ and $B$ be two $n\times n$ self-adjoint (i.e., Hermitian) matrices over $\mathbb{C}\,^{}$ such that all eigenvalues of $A$ lie in $[a,a']$ and all eigenvalues of $B$ lie in $[b,b']$. Show that all eigenvalues of $A + B$ lie in $[a+b,a'+b']$.

Problem 7   Prove that

$$\int_0^{\infty}e^{-x^2}\cos (2bx)\,dx = \frac{1}{2}\sqrt{\pi}e^{-b^2}.$$

What restrictions, if any, need be placed on $b$?

Problem 8   Let $h>0$ be given. Consider the linear difference equation

$$(*) \hspace{.2in} \frac{y \left( (n+2)h \right)-2y\left((n+1)... ...ght)+y(nh)}{h^2}= -y(nh), \hspace{.15in} n = 0, 1, 2, \ldots .$$

(Note the analogy with the differential equation $y''= -y$.)

1. Find the general solution of $(*)$ by trying suitable exponential substitutions.

2. Find the solution with $y(0)=0$ and $y(h)=h$. Denote it by $S_h(nh)$,
   $n = 1, 2, \ldots$.

3. Let $x$ be fixed and $h=x/n$. Show that
   
   $$\lim_{n\to\infty}S_{x/n}(nx/n)=\sin x \, .$$
   
   

Problem 9   Define the function $\zeta$ by

$$\zeta(x) = \sum_{n=1}^{\infty}\frac{1}{n^x}\cdot$$

Prove that $\zeta(x)$ is defined and has continuous derivatives of all orders in the interval $1<x<\infty$.

Problem 10   For arbitrary elements $a$, $b$, and $c$ in a field $\mbox{\bf {F}}$, compute the minimal polynomial of the matrix

$$\left( \begin{array}{ccc} 0 & 0 & a \\ 1 & 0 & b \\ 0 & 1 & c\end{array} \right).$$

Problem 11   Let $\mbox{\bf {F}} = \{a+b\sqrt[3]{2}+c\sqrt[3]{4} \;\vert\; a, b, c \in \mbox{$\mathbb{Q}\,^{}$}\}$. Prove that $\mbox{\bf {F}}$ is a field and each element in $\mbox{\bf {F}}$ has a unique representation as $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ with $a, b, c \in \mbox{$\mathbb{Q}\,^{}$}$. Find $\left( 1-\sqrt[3]{2} \, \right)^{-1}$ in $\mbox{\bf {F}}$.

Problem 12   Prove that

$$\int_0^{\infty}\frac{x^{\alpha -1}}{1+x}\,dx = \frac{\pi}{\sin \pi\alpha } \cdot$$

What restrictions must be placed on $\alpha$?

Problem 13   Prove that for any $a\in\mbox{$\mathbb{C}\,^{}$}$ and any integer $n\geq 2$, the equation $1+z+az^n=0$ has at least one root in the disc $\vert z\vert\leq~2$.

Problem 14   Show that

$$I = \int_0^{\pi}\log (\sin x)\,dx$$

converges as an improper Riemann integral. Evaluate $I$.

Problem 15   Let $${\zeta = e^{\frac{2\pi i}{7}}}$$ be a primitive $7^{th}$ root of unity. Find a cubic polynomial with integer coefficients having $\alpha = \zeta + \zeta^{-1}$ as a root.

Problem 16   Let f be continuous on $\mathbb{R}^{}$, and let

$$f_n(x) = \frac{1}{n}\sum_{k=0}^{n-1}f\left(x+\frac{k}{n}\right).$$

Prove that $f_n(x)$ converges uniformly to a limit on every finite interval $[a,b]$.

Problem 17   Let $v_1$ and $v_2$ be two real valued continuous functions on $\mathbb{R}^{}$ such that $v_1(x)<v_2(x)$ for all $x\in\mbox{$\mathbb{R}^{}$}$. Let $\varphi_1(t)$ and $\varphi_2(t)$ be, respectively, solutions of the differential equations

$$\frac{dx}{dt} =v_1(x)\quad and \quad \frac{dx}{dt} =v_2(x)$$

for $a<t<b$. If $\varphi_1(t_0)=\varphi_2(t_0)$ for some $t_0\in (a,b)$, show that $\varphi_1(t)\leq\varphi_2(t)$ for all $t\in (t_0,b)$.

Problem 18   Let $A$ and $B$ be two $n\times n$ self-adjoint (i.e., Hermitian) matrices over $\mathbb{C}\,^{}$ and assume $A$ is positive definite. Prove that all eigenvalues of AB are real.

Problem 19   Let $\mbox{\bf {F}}$ be a finite field. Give a complete proof of the fact that the number of elements of $\mbox{\bf {F}}$ is of the form $p^r$, where $p\geq 2$ is a prime number and $r$ is an integer $\geq 1$.

Problem 20   Let $f(z)$ be an analytic function that maps the open disc $\vert z\vert<1$ into itself. Show that $\vert f'(z)\vert\leq 1/(1-\vert z\vert^2)$.