Preliminary Exam - Summer 1985

Problem 1  

1. Show that a real 2$\times$2 matrix $A$ satisfies $A^2 = -I$ if and only if
   
   $$A = \left( \begin{array}{cc} \pm\sqrt{pq-1} & -p \\ q & \mp\sqrt{pq-1}\end{array} \right)$$
   
   
   where $p$ and $q$ are real numbers such that $pq \geq 1$ and both upper or both lower signs should be chosen in the double signs.

2. Show that there is no real 2$\times$2 matrix $A$ such that
   
   $$A^2 = \left( \begin{array}{cc} -1 & 0 \\ 0 & -1-\varepsilon\end{array} \right)$$
   
   
   with $\varepsilon > 0$.

Problem 2  

1. For $0 \leq \theta \leq \frac{\pi}{2}$, show that
   
   $$\sin \theta \geq \frac{2}{\pi}\theta\,.$$
   
   

2. By using Part 1, or by any other method, show that if $\lambda<1$, then
   
   $$\lim_{R\to\infty}R^{\lambda}\int_0^{\frac{\pi}{2}} e^{-R\sin \theta}\,d\theta = 0.$$
   
   

Problem 3   Let ${A}$ be a nonsingular real $n\times n$ matrix. Prove that there exists a unique orthogonal matrix $Q$ and a unique positive definite symmetric matrix $B$ such that $A = QB$.

Problem 4   Let $G$ be a group of order $120$, let $H$ be a subgroup of order $24$, and assume that there is at least one left coset of $H$ (other than $H$ itself) which is equal to some right coset of $H$. Prove that $H$ is a normal subgroup of $G$.

Problem 5   By the Fundamental Theorem of Algebra, the polynomial $x^3+2x^2+7x+1$ has three complex roots, $\alpha_1$, $\alpha_2$, and $\alpha_3$. Compute $\alpha_1^3+\alpha_2^3+\alpha_3^3$. _Fundamental Theorem>of Algebra

Problem 6   Evaluate the integral

$$\int_0^{\infty}\frac{x^{a-1}}{x+1}\,dx$$

where $a$ is a complex number. What restrictions must be put on $a$?

Problem 7   Let

$$f(x) = e^{x^2/2}\int_x^{\infty}e^{-t^2/2}\,dt$$

for $x>0$.

1. Show that $0 < f(x) < 1/x$.

2. Show that $f(x)$ is strictly decreasing for $x>0$.

Problem 8   Let f be a real valued continuous function on a compact interval $[a,b]$. Given $\varepsilon > 0$, show that there is a polynomial p such that
$p(a)= f(a)$, $p'(a) = 0$, and $\vert p(x)-f(x)\vert < \varepsilon$ for $x\in [a,b]$.

Problem 9   Let $u(x),\, 0 \leq x\leq 1$, be a real valued $C^2$ function which satisfies the differential equation

$$u''(x) = e^xu(x).$$

1. Show that if $0 < x_0 < 1$, then $u$ cannot have a positive local maximum at $x_0$. Similarly, show that $u$ cannot have a negative local minimum at $x_0$.
2. Now suppose that $u(0)=u(1)=0$. Prove that $u(x)\equiv 0$, $0\leq x\leq 1$.

Problem 10   Prove that for each $\lambda>1$ the equation $z= \lambda - e^{-z}$ in the half-plane $\Re z \geq 0$ has exactly one root, and that this root is real.

Problem 11  

1. Let $G$ be a cyclic group, and let $a, b \in G$ be elements which are not squares. Prove that $ab$ is a square.

2. Give an example to show that this result is false if the group is not cyclic.

Problem 12   Let ${A}$ be an $n\times n$ real matrix and ${A}^t$ its transpose. Show that ${A}^t{A}$ and ${A}^t$ have the same range.

Problem 13   Let $P(z)$ be a polynomial of degree $< k$ with complex coefficients. Let $\omega_1,\ldots,\omega_k$ be the $k^{th}$ roots of unity in $\mathbb{C}\,^{}$. Prove that

$$\frac{1}{k}\sum_{i=1}^kP(\omega_i) = P(0).$$

Problem 14   Let $M_n(\mbox{\bf {F}})$ denote the ring of $n\times n$ matrices over a field $\mbox{\bf {F}}$. For $n\geq 1$, does there exist a ring homomorphism from $M_{n+1}(\mbox{\bf {F}})$ onto $M_n(\mbox{\bf {F}})$?

Problem 15   For each $k>0$, let $X_k$ be the set of analytic functions $f(z)$ on the open unit disc $\mathbb{D}$ such that

$$% latex2html id marker 992 \sup_{z\in \mbox{$\mathbb{D}^{}$}}\left\{(1-\vert z\vert)^k\left\vert f(z)\right\vert\right\}$$

is finite. Show that $f \in X_k$ if and only if $f' \in X_{k+1}$.

Problem 16   A function $f:[0,1]\to\mbox{$\mathbb{R}^{}$}$ is said to be upper semicontinuous if given $x \in [0,1]$ and $\varepsilon > 0$, there exists a $\delta > 0$ such that if $\vert y-x\vert< \delta$, then $f(y)< f(x) + \varepsilon$. Prove that an upper semicontinuous function f on $[0,1]$ is bounded above and attains its maximum value at some point $p \in [0,1]$. _function,>semicontinuous

Problem 17   Let

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

where all the $a_n$ are nonnegative reals, and the series has radius of convergence $1$. Prove that $f(z)$ cannot be analytically continued to a function analytic in a neighborhood of $z=1$.

Problem 18   Solve the differential equations

$$\frac{dy_1}{dx} = -3y_1 + 10y_2,$$

$$\frac{dy_2}{dx} = -3y_1 + 8y_2.$$

Problem 19   Let $A_1 \geq A_2 \geq \cdots \geq A_k \geq 0$. Evaluate

$$\lim_{n\to\infty}\left(A_1^n+A_2^n+\cdots+A_k^n\right)^{1/n}.$$

Note: See also Problem .

Problem 20   Let F be a field of characteristic $p>0$, $p\neq 3$. If $\alpha$ is a zero of the polynomial $f(x) = x^p-x+3$ in an extension field of $\mbox{\bf {F}}$, show that $f(x)$ has $p$ distinct zeros in the field $\mbox{\bf {F}}(\alpha)$.