Preliminary Exam - Spring 1977

Problem 1   Suppose $f$ is a differentiable function from the reals into the reals. Suppose $f'(x)>f(x)$ for all $x\in\mbox{$\mathbb{R}^{}$}$, and $f(x_0)=0$. Prove that $f(x)>0$ for all $x>x_0$.

Problem 2   Suppose that $f$ is a real valued function of one real variable such that

$$\lim_{x\to c}f(x)$$

exists for all $c \in [a,b]$. Show that $f$ is Riemann integrable on $[a,b]$.

Problem 3  

1. Evaluate $P_{n-1}(1)$, where $P_{n-1}(x)$ is the polynomial
   
   $$P_{n-1}(x) = \frac{x^n-1}{x-1} \cdot$$
   
   

2. Consider a circle of radius $1$, and let $Q_1,Q_2,\ldots,Q_n$ be the vertices of a regular $n$-gon inscribed in the circle. Join $Q_1$ to $Q_2,Q_3,\ldots,Q_n$ by segments of a straight line. You obtain $(n-1)$ segments of lengths $\lambda_2,\lambda_3,\ldots,\lambda_n$. Show that
   
   $$\prod_{i=2}^n\lambda_i = n.$$
   
   

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Problem 4   Prove the Fundamental Theorem of Algebra: Every nonconstant polynomial with complex coefficients has a complex root. _Fundamental Theorem>of Algebra

Problem 5   Let $\mbox{\bf {F}} \subset \mbox{\bf {K}}$ be fields, and $a$ and $b$ elements of $\mbox{\bf {K}}$ which are algebraic over $\mbox{\bf {F}}$. Show that $a+b$ is algebraic over $\mbox{\bf {F}}$.

Problem 6   Let $G$ be the collection of 2$\times$2 real matrices with nonzero determinant. Define the product of two elements in $G$ as the usual matrix product. _group,>center _matrix,>orthogonal

1. Show that $G$ is a group.
2. Find the center $Z$ of $G$; that is, the set of all elements $z$ of $G$ such that $az=za$ for all $a\in G$.
3. Show that the set $O$ of real orthogonal matrices is a subgroup of $G$ (a matrix is orthogonal if $AA^t=I$, where $A^t$ denotes the transpose of $A$). Show by example that $O$ is not a normal subgroup.
4. Find a nontrivial homomorphism from $G$ onto an abelian group.

Problem 7   A matrix of the form

$$\left( \begin{array}{ccccc} 1 & a_0 & a_0^2 & \ldots & a_0^n ... ...\vdots \\ 1 & a_n & a_n^2 & \ldots & a_n^n \end{array} \right)$$

where the $a_i$ are complex numbers, is called a Vandermonde matrix.

1. Prove that the Vandermonde matrix is invertible if $a_0,a_1,\ldots,a_n$ are all different.

2. If $a_0,a_1,\ldots,a_n$ are all different, and $b_0,b_1,\ldots,b_n$ are complex numbers, prove that there is a unique polynomial $f$ of degree $n$ with complex coefficients such that $f(a_0)=b_0$, $f(a_1)=b_1$,..., $f(a_n)=b_n$.

Problem 8   Find a list of real matrices, as long as possible, such that

• the characteristic polynomial of each matrix is $(x-1)^{5}(x+1)$,
• the minimal polynomial of each matrix is $(x-1)^{2}(x+1)$,
• no two matrices in the list are similar to each other.

Problem 9   Find the solution of the differential equation

$$y''-2y'+y = 0,$$

subject to the conditions

$$y(0)=1, \hspace{.2in} y'(0)=1.$$

Problem 10   In $\mathbb{R}^{2}$, consider the region ${\cal A}$ defined by $x^2+y^2>1$. Find differentiable real valued functions $f$ and $g$ on ${\cal A}$ such that $\frac{\partial f}{\partial x} = \frac{\partial g}{\partial y}$ but there is no real valued function $h$ on ${\cal A}$ such that $f = \frac{\partial h}{\partial y}$ and $g = \frac{\partial h}{\partial x}\cdot$

Problem 11   Let the sequence $a_0,a_1,\ldots$ be defined by the equation

$$1 - x^2 + x^4 - x^6 + \cdots = \sum_{n=0}^{\infty}a_n(x-3)^n \quad (0<x<1).$$

Find

$$\limsup_{n\to \infty}\left(\vert a_n\vert^{\frac{1}{n}}\right).$$

Problem 12   Let $p$ be an odd prime. Let $Q(p)$ be the set of integers $a$, $0 \leq a \leq p-1$, for which the congruence

$$x^2 \equiv a \pmod{p}$$

has a solution. Show that $Q(p)$ has cardinality $(p+1)/2$.

Problem 13   Consider the family of square matrices $A(\theta)$ defined by the solution of the matrix differential equation _matrix,>orthogonal

$$\frac{dA(\theta)}{d\theta}=BA(\theta)$$

with the initial condition $A(0) = I$, where $B$ is a constant square matrix.

1. Find a property of $B$ which is necessary and sufficient for $A(\theta)$ to be orthogonal for all $\theta$; that is, $A(\theta)^t= A(\theta)^{-1}$, where $A(\theta)^t$ denotes the transpose of $A(\theta)$.

2. Find the matrices $A(\theta)$ corresponding to
   
   $$B=\left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right)$$
   
   
   and give a geometric interpretation.

Problem 14   A square matrix $A$ is nilpotent if $A^k = 0$ for some positive integer $k$.

1. If $A$ and $B$ are nilpotent, is $A + B$ nilpotent?

2. Prove: If $A$ and $B$ are nilpotent matrices and $AB = BA$, then $A + B$ is nilpotent.

3. Prove: If $A$ is nilpotent then $I + A$ and $I-A$ are invertible.

Problem 15   Let $f(z)$ be a nonconstant meromorphic function. A complex number $w$ is called a period of $f$ if $f(z+w)=f(z)$ for all $z$.

1. Show that if $w_1$ and $w_2$ are periods, so are $n_1w_1+n_2w_2$ for all integers $n_1$ and $n_2$.

2. Show that there are, at most, a finite number of periods of $f$ in any bounded region of the complex plane.

Problem 16   Evaluate

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

Problem 17   Let $\mbox{$\mathbb{Q}\,^{}$}_{+}$ be the multiplicative group of positive rational numbers.

1. Is $\mbox{$\mathbb{Q}\,^{}$}_{+}$ torsion free?
2. Is $\mbox{$\mathbb{Q}\,^{}$}_{+}$ free?

Problem 18  

1. In $\mbox{$\mathbb{R}^{}$}[x]$, consider the set of polynomials $f(x)$ for which $f(2)= f'(2)=f''(2)=0$. Prove that this set forms an ideal and find its monic generator.
2. Do the polynomials such that $f(2)=0$ and $f'(3)=0$ form an ideal?

Problem 19   Suppose that $u(x,t)$ is a continuous function of the real variables x and t with continuous second partial derivatives. Suppose that u and its first partial derivatives are periodic in $x$ with period $1$, and that

$$\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \cdot$$

Prove that

$$E(t) = \frac{1}{2}\int_0^1\left(\left(\frac{\partial u}{\part... ...ght)^2+\left(\frac{\partial u}{\partial x}\right)^2\right)\,dx$$

is a constant independent of $t$.

Problem 20   Let $h:[0,1)\to\mbox{$\mathbb{R}^{}$}$ be a function defined on the half-open interval $[0,1)$. Prove that if $h$ is uniformly continuous, there exists a unique continuous function $g:[0,1]\to\mbox{$\mathbb{R}^{}$}$ such that $g(x)=h(x)$ for all $x\in [0,1)$.