Preliminary Exam - Fall 1994

Problem 1   For which values of the real number $a$ does the series

$$\sum_{n=1}^{\infty} \left( \frac{1}{n}- \sin \frac{1}{n} \right)^{a}$$

converge?

Problem 2   Prove that the matrix

$$\left( \begin{array}{ccc} 1 & 1.00001 & 1 \\ 1.00001 & 1 & 1.00001 \\ 1 & 1.00001 & 1 \end{array}\right)$$

has one positive eigenvalue and one negative eigenvalue.

Problem 3   Evaluate the integrals

$$\int_{-\pi}^{\pi} \frac{\sin n \theta}{\sin \theta} d \theta, \quad n=1, 2, \ldots .$$

Problem 4   Suppose the group $G$ has a nontrivial subgroup $H$ which is contained in every nontrivial subgroup of $G$. Prove that $H$ is contained in the center of $G$.

Problem 5  

1. Find a basis for the space of real solutions of the differential equation
   
   $$(*)\: \: \: \: \:\:\:\:\: \sum_{n=0}^{7} \frac{d^{n}x}{dt^{n}}=0.$$
   
   

2. Find a basis for the subspace of real solutions of $(*)$ that satisfy
   
   $$\lim_{t \to + \infty} x(t) =0.$$
   
   

Problem 6   Let $A= \left( a_{ij} \right)_{i,j=1}^{n}$ be a real $n \times n$ matrix such that $a_{ii} \geq 1$ for all $i$, and

$$\sum_{i \neq j} a_{ij}^{2} < 1.$$

Prove that $A$ is invertible.

Problem 7   Let $f$ be a continuously differentiable function from $\mathbb{R}^{2}$ into $\mathbb{R}$. Prove that there is a continuous one-to-one function $g$ from $[0,1]$ into $\mathbb{R}^{2}$ such that the composite function $f \circ g$ is constant.

Problem 8   Let $\mathbb{Q}$ be the field of rational numbers. For $\theta$ a real number, let $\mbox{\bf {F}}_{\theta}=\mathbb{Q}(\sin \theta)$ and $\mbox{\bf {E}}_{\theta}=\mathbb{Q}\left(\sin \frac{\theta}{3}\right)$. Show that $\mbox{\bf {E}}_{\theta}$ is an extension field of $\mbox{\bf {F}}_{\theta}$, and determine all possibilities for ${\dim}_{\mbox{\scriptsize\bf F}_{\theta}} \mbox{\bf {E}}_{\theta}$.

Problem 9   Evaluate

$$\int_{0}^{\infty} \frac{(\log x)^2}{x^{2}+1}\, dx\, .$$

Problem 10   Let the function $f: \mathbb{R}^{n} \to \mathbb{R}^{n}$ satisfy the following two conditions:

(i)

$f(K)$ is compact whenever $K$ is a compact subset of $\mathbb{R}^{n}$.

(ii)

If $\{K_{n}\}$ is a decreasing sequence of compact subsets of $\mathbb{R}^{n}$, then

$$f \left( \bigcap_{1}^{\infty} K_{n} \right) = \bigcap_{1}^{\infty} f \left( K_{n} \right).$$

Prove that $f$ is continuous.

Problem 11   Write down a list of 5$\times$5 complex matrices, as long as possible, with the following properties:

1. The characteristic polynomial of each matrix in the list is $x^{5}$;
2. The minimal polynomial of each matrix in the list is $x^{3}$;
3. No two matrices in the list are similar.

Problem 12   Suppose the coefficients of the power series

$$\sum_{n=0}^{\infty} a_{n}z^{n}$$

are given by the recurrence relation

$$a_{0}=1, \; a_{1}=-1, \; 3a_{n}+4a_{n-1}-a_{n-2}=0, \;\;n=2, 3, \ldots .$$

Find the radius of convergence of the series and the function to which it converges in its disc of convergence.

Problem 13   Let $p$ be an odd prime and $\mbox{\bf {F}}_{p}$ the field of $p$ elements. How many elements of $\mbox{\bf {F}}_{p}$ have square roots in $\mbox{\bf {F}}_{p}$? How many have cube roots in $\mbox{\bf {F}}_{p}$?

Problem 14   Find the maximum area of all triangles that can be inscribed in an ellipse with semiaxes $a$ and $b$, and describe the triangles that have maximum area.

Note: See also Problem .

Problem 15   Let $M_{7\times 7}$ denote the vector space of real 7$\times$7 matrices. Let $A$ be a diagonal matrix in $M_{7\times 7}$ that has $+1$ in four diagonal positions and $-1$ in three diagonal positions. Define the linear transformation $T$ on $M_{7\times 7}$ by $T(X)=AX-XA$. What is the dimension of the range of $T$?

Problem 16   Let ${\cal D}$ denote the open unit disc in $\mathbb{R}^{2}$. Let $u$ be an eigenfunction for the Laplacian in ${\cal D}$; that is, a real valued function of class $C^{2}$ defined in $\overline{\cal D}$, zero on the boundary of ${\cal D}$ but not identically zero, and satisfying the differential equation _Laplacian,>eigenfunction

$$\frac{\partial ^{2}u}{\partial x^{2}} + \frac{\partial ^{2}u}{\partial y^{2}}= \lambda u \,,$$

where $\lambda$ is a constant. Prove that

$$(*)\qquad \int\!\!\int_{\cal D} \left\vert\mathrm{grad}\:u \right\vert^{2}dxdy + \lambda \int\!\!\int_{\cal D} u^{2}dxdy=0\,,$$

and hence that $\lambda < 0$.

Problem 17   Let $R$ be a ring with identity, and let $u$ be an element of $R$ with a right inverse. Prove that the following conditions on $u$ are equivalent:

1. $u$ has more than one right inverse;

2. $u$ is a zero divisor;

3. $u$ is not a unit.

Problem 18   Let the function $f$ be analytic in the complex plane, real on the real axis, $0$ at the origin, and not identically $0$. Prove that if $f$ maps the imaginary axis into a straight line, then that straight line must be either the real axis or the imaginary axis.