Preliminary Exam - Spring 1978

Problem 1   Let $k\geq 0$ be an integer and define a sequence of maps

$$% latex2html id marker 675 f_n :\mbox{$\mathbb{R}^{}$} \to ... ... f_n(x) = \frac{x^k}{x^2+n}, \hspace{.2in} n = 1, 2, \ldots .$$

For which values of $k$ does the sequence converge uniformly on $\mathbb{R}^{}$? On every bounded subset of $\mathbb{R}^{}$?

Problem 2   Prove that a map $g:\mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{n}$}$ is continuous only if its graph is closed in $\mbox{$\mathbb{R}^{n}$}\times\mbox{$\mathbb{R}^{n}$}$. Is the converse true?

Note: See also Problem .

Problem 3   Let $f:\mbox{$\mathbb{C}\,^{}$} \to \mbox{$\mathbb{C}\,^{}$}$ be a nonconstant entire function. Prove that $f(\mbox{$\mathbb{C}\,^{}$})$ is dense in $\mathbb{C}\,^{}$.

Problem 4   Evaluate

$$\int_{-\infty}^{\infty}\frac{\sin^2x}{x^2}\,dx\, .$$

Problem 5   let $\mbox{$\mathbb{Z}^{}$}_n$ denote the ring of integers modulo n. Let $\mbox{$\mathbb{Z}^{}$}_n[x]$ be the ring of polynomials with coefficients in $\mbox{$\mathbb{Z}^{}$}_n$. Let $\mathfrak{I}$ denote the ideal in $\mbox{$\mathbb{Z}^{}$}_n [x]$ generated by $x^2+x+1$.

1. For which values of n, $1 \leq n \leq 10$, is the quotient ring $\mbox{$\mathbb{Z}^{}$}_n [x]/\mathfrak{I}$ a field?

2. Give the multiplication table for $\mbox{$\mathbb{Z}^{}$}_2/\mathfrak{I}$.

Problem 6   Prove that the sum of two algebraic numbers is algebraic. (An algebraic number is a complex number which is a root of a polynomial with rational coefficients.)

Problem 7   What is the volume enclosed by the ellipsoid

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1?$$

Problem 8   Consider the differential equation

$$\frac{dx}{dt} = x^2 + t^2, \quad x(0)=1.$$

1. Prove that for some $b>0$, there is a solution defined for $t \in [0,b]$.
2. Find an explicit value of $b$ having the property in Part 1.
3. Find a $c>0$ such that there is no solution on $[0,c]$.

Problem 9   Determine the Jordan Canonical Form of the matrix

$$A = \left( \begin{array}{ccc} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 4 \end{array} \right).$$

Problem 10   Suppose $A$ is a real $n\times n$ matrix.

1. Is it true that $A$ must commute with its transpose?

2. Suppose the columns of $A$ (considered as vectors) form an orthonormal set; is it true that the rows of $A$ must also form an orthonormal set?

Problem 11   Show that there is a complex analytic function defined on the set $U=\{z \in \mbox{$\mathbb{C}\,^{}$}\;\vert\;\vert z\vert>4\}$ whose derivative is

$$\frac{z}{(z-1)(z-2)(z-3)}\cdot$$

Is there a complex analytic function on $U$ whose derivative is

$$\frac{z^2}{(z-1)(z-2)(z-3)}\;?$$

Problem 12   Prove that the uniform limit of a sequence of complex analytic functions is complex analytic. Is the analogous theorem true for real analytic functions?

Problem 13  

1. For which real numbers $\alpha >0$ does the differential equation
   
   $$(*) \qquad \frac{dx}{dt}=x^{\alpha}, \hspace{.2in} x(0)=0\,,$$
   
   
   have a solution on some interval $[0,b],\; b>0$?

2. For which values of $\alpha$ are there intervals on which two solutions of $(*)$ are defined?

Problem 14   Let $G$ be a group of order $10$ which has a normal subgroup of order $2$. Prove that $G$ is abelian.

Problem 15   Is $x^4+1$ irreducible over the field of real numbers? The field of rational numbers? A field with $16$ elements?

Problem 16   Let $A$ and $B$ denote real n$\times$n symmetric matrices such that $AB = BA$. Prove that $A$ and $B$ have a common eigenvector in $\mathbb{R}^{n}$.

Problem 17   Evaluate

$$\int_{\cal A}e^{-x^2-y^2}\,dxdy\,,$$

where ${\cal A} = \{(x,y) \in \mbox{$\mathbb{R}^{2}$} \;\vert\; x^2+y^2\leq 1\}$.

Problem 18   Let $M$ be a matrix with entries in a field $\mbox{\bf {F}}$. The row rank of $M$ over $\mbox{\bf {F}}$ is the maximal number of rows which are linearly independent (as vectors) over $\mbox{\bf {F}}$. The column rank is similarly defined using columns instead of rows. _matrix,>row rank _matrix,>column rank

1. Prove row rank = column rank.

2. Find a maximal linearly independent set of columns of
   
   $$\left( \begin{array}{cccc} 1 & 0 & 3 & -2 \\ 2 & 1 & 2 & 0 \... ... -4 & 4 \\ 1 & 1 & 1 & 2 \\ 1 & 0 & 1 & 2 \end{array} \right)$$
   
   
   taking $\mbox{\bf {F}} =\mbox{$\mathbb{R}^{}$}$.

3. If $\mbox{\bf {F}}$ is a subfield of $\mbox{\bf {K}}$, and $M$ has entries in $\mbox{\bf {F}}$, how is the row rank of $M$ over $\mbox{\bf {F}}$ related to the row rank of $M$ over $\mbox{\bf {K}}$?

Problem 19   Let $f:[0,1] \to \mbox{$\mathbb{R}^{}$}$ be Riemann integrable over $[b,1]$ for all b such that $0<b\leq 1$.

1. If f is bounded, prove that f is Riemann integrable over $[0,1]$.

2. What if f is not bounded?

Problem 20   Consider the system of equations

$$\begin{array}{r} 3x + y - z + u^4 = 0 \\ x - y + 2z + u = 0 \\ 2x + 2y - 3z + 2u = 0 \end{array}$$

1. Prove that for some $\varepsilon>0$, the system can be solved for $(x,y,u)$ as a function of $z \in [-\varepsilon,\varepsilon ]$, with $x(0)=y(0)=u(0)=0$. Are such functions $x(z)$, $y(z)$ and $u(z)$ continuous? Differentiable? Unique?

2. Show that the system cannot be solved for $(x,y,z)$ as a function of $u \in [-\delta,\delta]$, for all $\delta >0$.