Preliminary Exam - Spring 1998

Problem 1   Prove that the polynomial $z^4+z^3+1$ has exactly one root in the quadrant $\{z=x+iy\,\vert\, x,y >0\}$.

Problem 2   Let $f$ be analytic in an open set containing the closed unit disc. Suppose that $\vert f(z)\vert > m$ for $\vert z\vert=1$ and $\vert f(0)\vert < m$. Prove that $f(z)$ has at least one zero in the open unit disc $\vert z\vert < 1$.

Problem 3   Let $M$ be a non-empty complete metric space. Let $T:M\to M$ such that $T\circ T=T^2$ is a strict contraction. Prove that $T$ has a unique fixed point in $M$, i.e., there is a unique point $x_0$ with $T(x_0)=x_0$.

Problem 4   Using the properties of the Riemann integral, show that if $f$ is a non-negative continuous function on $[0,1]$, and $\int^1_0 f(x)dx=0$, then $f(x)=0$ for all $x\in [0,1]$.

Problem 5   Let $A$ be the ring of real 2$\times$2 matrices of the form $\bigl({a\atop 0}{b\atop c}\bigr)$. What are the 2-sided ideals in $A$? Justify your answer.

Problem 6   Let $G$ be the group $\mbox{$\mathbb{Q}\,^{}$}/\mbox{$\mathbb{Z}^{}$}$. Show that for every positive integer $t$, $G$ has a unique cyclic subgroup of order $t$.

Problem 7   Suppose that $A$ and $B$ are two commuting $n\times n$ complex matrices. Show that they have a common eigenvector.

Problem 8   Let $m\geq 0$ be an integer. Let $a_1,a_2,\dots ,a_m$ be integers and let

$$f(x)=\sum^m_{i=1} \frac{a_ix^i}{i!}$$

Show that if $d\geq 0$ is an integer then $f(x)^d/d!$ can be expressed in the form

$$\sum^{md}_{i=0} \frac{b_ix_i}{i!}\,\cdot$$

where the $b_i$ are integers.

Problem 9   Let $M_1= \bigl({3\atop 1}{2\atop 4}\bigr)$, $M_2= \bigl({5\atop -3}{7\atop -4}\bigr)$, $M_3= \bigl({5\atop -3}{6.9\atop -4}\bigr)$. For which (if any) $i$, $1\leq i\leq 3$, is the sequence $(M_i^n)$ bounded away from $\infty$? For which $i$ is the sequence bounded away from $0$?

Problem 10   Let $P_i=(a\cos\theta_i,b\sin\theta_i)$, $i=1,2,3$, be a triple of points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. What is the maximal value of the area of the triangle $\Delta P_1P_2P_3$? Determine those triangles with maximal area.

Problem 11   Let $A,B,\dots ,F$ be real coefficients. Show that the
quadratic form

$$Ax^2+2Bxy+Cy^2+2Dxz+2Eyz+Fz^2$$

is positive definite if and only if

$$A > 0, \qquad \left\vert\begin{array}{cc} A &B\\ B & C\end{a... ...gin{array}{ccc} A&B&D\\ B&C&E\\ D&E&F\end{array}\right\vert>0.$$

Problem 12   Given the fact that $${\int^\infty_{-\infty}e^{-x^2}dx=\sqrt{\pi}}$$, evaluate the integral

$$I=\int^\infty_{-\infty}\int^\infty_{-\infty}e^{-(x^2+(y-x)^2+y^2)}dx \ dy \ .$$

Problem 13   Let $a$ be a complex number with $\vert a\vert < 1$. Evaluate the integral

$$\int_{\vert z\vert=1} \frac{\vert dz\vert}{\vert z-a\vert^2}$$

Problem 14   Let $K$ be a real constant. Suppose that $y(t)$ is a positive differentiable function satisfying $y'(t)\leq Ky(t)$ for $t\geq 0$. Prove that $y(t)\leq e^{Kt}y(0)$ for $t\geq 0$.

Problem 15   For continuous real valued functions $f,g$ on the interval $[-1,1]$ define the inner product $\langle f,g\rangle=\int^1_{-1} f(x)g(x)dx$. Find that polynomial of the form $p(x)=a+bx^2-x^4$ which is orthogonal on $[-1,1]$ to all lower order polynomials.

Problem 16   Let $a > 0$. Show that the complex function

$$f(z)=\frac{1+z+az^2}{1-z+az^2}$$

satisfies $\vert f(z)\vert <1$ for all $z$ in the open left half-plane $\Re z < 0$.

Problem 17   Let $A$ be an $n\times n$ complex matrix with $\mathrm{tr}\,A=0$. Show that $A$ is similar to a matrix with all $0$'s along the main diagonal.

Problem 18   Let $N$ be a nilpotent complex matrix. Let $r$ be a positive integer. Show that there is a $n\times n$ complex matrix $A$ with

$$A^r=I+N.$$