Preliminary Exam - Spring 1987

Problem 1   A standard theorem states that a continuous real valued function on a compact set is bounded. Prove the converse: If $K$ is a subset of $\mathbb{R}^{n}$ and if every continuous real valued function on $K$ is bounded, then $K$ is compact.

Problem 2   Let the transformation $T$ from the subset
$U=\{(u,v)\;\vert\; u > v \}$ of $\mathbb{R}^{2}$ into $\mathbb{R}^{2}$ be defined by $T(u,v)=(u+v, u^2+v^2)$.

1. Prove that $T$ is locally one-to-one.
2. Determine the range of $T$, and show that $T$ is globally one-to-one.

Problem 3   Let $f$ be a complex valued function in the open unit disc, $\mathbb{D}$, of the complex plane such that the functions $g = f^2$ and $h = f^3$ are both analytic. Prove that is analytic in $\mathbb{D}$.

Problem 4   Let $\mbox{\bf {F}}$ be a finite field with $q$ elements and let $x$ be an indeterminate. For $f$ a polynomial in $\mbox{\bf {F}}[x]$, let $\varphi_f$ denote the corresponding function of $\mbox{\bf {F}}$ into $\mbox{\bf {F}}$, defined by $\varphi_f(a)=f(a),\, (a\in \mbox{\bf {F}})$. Prove that if $\varphi$ is any function of $\mbox{\bf {F}}$ into $\mbox{\bf {F}}$, then there is an $f$ in $\mbox{\bf {F}}[x]$ such that $\varphi=\varphi_f$. Prove that $f$ is uniquely determined by $\varphi$ to within addition of a multiple of $x^q-x$.

Problem 5   Let $f$ be a continuous real valued function on $\mathbb{R}^{}$ satisfying

$$\vert f(x)\vert \leq C/(1 + x^2),$$

where $C$ is a positive constant. Define the function $F$ on $\mathbb{R}^{}$ by

$$F(x) = \sum_{n=-\infty}^{\infty}f(x+n)\, .$$

1. Prove that $F$ is continuous and periodic with period $1$.

2. Prove that if $G$ is continuous and periodic with period $1$, then
   
   $$\int_{0}^{1}F(x)G(x)\,dx = \int_{-\infty}^{\infty}f(x)G(x)\,dx\, .$$
   
   

Problem 6   Let $f$ be an analytic function in the open unit disc of the complex plane such that $\vert f(z)\vert\leq C/(1-\vert z\vert)$ for all $z$ in the disc, where $C$ is a positive constant. Prove that $\vert f'(z)\vert\leq 4C/(1-\vert z\vert)^2$.

Problem 7   Let p, q and r be continuous real valued functions on $\mathbb{R}^{}$, with $p>0$. Prove that the differential equation

$$p(t)x''(t) + q(t)x'(t) + r(t)x(t) = 0$$

is equivalent to (i.e., has exactly the same solutions as) a differential equation of the form

$$\left( a(t)x'(t) \right)' + b(t)x(t) = 0,$$

where a is continuously differentiable and b is continuous.

Problem 8   Prove that if the nonconstant polynomial $p(z)$, with complex coefficients, has all of its roots in the half-plane $\Re z > 0$, then all of the roots of its derivative are in the same half-plane.

Problem 9   Let $A$ be an $m\times n$ matrix with rational entries and $b$ an $m$-dimensional column vector with rational entries. Prove or disprove: If the equation $Ax = b$ has a solution $x$ in $\mathbb{C}\,^{n}$, then it has a solution with $x$ in $\mathbb{Q}\,^{n}$.

Problem 10   Prove that any finite group of order $n$ is isomorphic to a subgroup of $\mathbb{O}(n)$, the group of $n\times n$ orthogonal real matrices.

Problem 11   Show that the equation $ae^x=1+x+x^2/2$, where $a$ is a positive constant, has exactly one real root.

Problem 12   Evaluate the integral

$$I = \int_{0}^{1/2}\frac{\sin x}{x}\,dx$$

to an accuracy of two decimal places; that is, find a number $I^*$ such that $\vert I-I^*\vert<0.005$.

Problem 13   Let $f$ be a real valued $C^1$ function defined in the punctured plane $\mbox{$\mathbb{R}^{2}$}\setminus\{(0,0)\}$. Assume that the partial derivatives $\partial f/\partial x$ and $\partial f/\partial y$ are uniformly bounded; that is, there exists a positive constant $M$ such that $\vert\partial f/\partial x\vert\leq M$ and $\vert\partial f/\partial y\vert\leq M$ for all points $(x,y) \neq (0,0)$. Prove that

$$\lim_{(x,y)\to (0,0)}f(x,y)$$

exists. _function,>uniformly bounded

Problem 14  

1. Show that, to within isomorphism, there is just one noncyclic group $G$ of order $4$.

2. Show that the group of automorphisms of $G$ is isomorphic to the permutation group $S_3$.

Problem 15   Prove or disprove: If the function $f$ is analytic in the entire complex plane, and if $f$ maps every unbounded sequence to an unbounded sequence, then $f$ is a polynomial.

Problem 16   Let $\cal F$ be a uniformly bounded, equicontinuous family of real valued functions on the metric space $(X,d)$. Prove that the function

$$g(x)=\sup\{f(x) \;\vert\; f \in {\cal F} \}$$

is continuous.

Problem 17   Let $V$ be a finite-dimensional linear subspace of $C^{\infty}(\mbox{$\mathbb{R}^{}$})$ (the space of complex valued, infinitely differentiable functions). Assume that $V$ is closed under $D$, the operator of differentiation (i.e., $f\in V\Rightarrow Df\in V$). Prove that there is a constant coefficient differential operator

$$L=\sum_{k=0}^{n}a_kD^k$$

such that $V$ consists of all solutions of the differential equation $Lf=0$.

Problem 18   Let ${A}$ and ${B}$ be two diagonalizable $n\times n$ complex matrices such that $\rm AB = BA$. Prove that there is a basis for $\mathbb{C}\,^{n}$ that simultaneously diagonalizes ${A}$ and ${B}$.

Problem 19   Let $\mbox{\bf {F}}$ be a field. Prove that every finite subgroup of the multiplicative group of nonzero elements of $\mbox{\bf {F}}$ is cyclic.

Problem 20   Evaluate

$$I = \int_{0}^{\pi}\frac{\cos 4\theta}{1+{\cos}^2 \theta} \,d\theta\, .$$