# Preliminary Exam - Spring 1978

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

$$
f _ { n } : \mathbb { R } \to \mathbb { R } , \quad f _ { n } ( x ) = { \frac { x ^ { k } } { x ^ { 2 } + n } } , \quad n = 1 , 2 , \ldots .
$$

For which values of k does the sequence converge uniformly on R? On every bounded subset of R?

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

Note: See also Problem ??.

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

Problem 4 Evaluate

$$
\int _ { - \infty } ^ { \infty } { \frac { \sin ^ { 2 } x } { x ^ { 2 } } } d x .
$$

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

1. For which values of n, $1 \leqslant n \leqslant 1 0$ , is the quotient ring $\mathbb { Z } _ { n } [ x ] / { \mathfrak { I } }$ a field?

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

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

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

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 $\textit { a c } > 0$ such that there is no solution on $[ 0 , c ]$

Problem 9 Determine the Jordan Canonical Form of the matrix

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

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 $U = \{ z \in \mathbb { C } \mid | z | > 4 \}$ 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

$$
\frac { d x } { d t } = x ^ { \alpha } , \quad x ( 0 ) = 0 ,
$$

have a solution on some interval [0, b], $b > 0 \%$

2. For which values of α are there intervals on which two solutions 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×n symmetric matrices such that $A B = B A$ . Prove that A and B have a common eigenvector in Rn.

Problem 17 Evaluate

$$
\int \int _ { A } e ^ { - x ^ { 2 } - y ^ { 2 } } d x d y ,
$$

where $\mathcal { A } = \{ ( x , y ) \in \mathbb { R } ^ { 2 } \ | \ x ^ { 2 } + y ^ { 2 } \leqslant 1 \}$

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

1. Prove row rank = column rank.

2. Find a maximal linearly independent set of columns of

$$
\left( \begin{array} { c c c c } { { 1 } } & { { 0 } } & { { 3 } } & { { - 2 } } \\ { { 2 } } & { { 1 } } & { { 2 } } & { { 0 } } \\ { { 0 } } & { { 1 } } & { { - 4 } } & { { 4 } } \\ { { 1 } } & { { 1 } } & { { 1 } } & { { 2 } } \\ { { 1 } } & { { 0 } } & { { 1 } } & { { 2 } } \end{array} \right)
$$

taking $\mathbf { F } = \mathbb { R }$

3. If F is a subfield of K, and M has entries in F, how is the row rank of M over F related to the row rank of M over K?

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

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

$$
3 x + y - z + u ^ { 4 } = 0
$$

$$
x - y + 2 z + u = 0
$$

$$
2 x + 2 y - 3 z + 2 u = 0
$$

1. Prove that for some $\varepsilon > 0$ , the system can be solved for $( x , y , u )$ as a function $o f \ z \ \in \ [ - \varepsilon , \varepsilon ]$ ], 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 $o f u \in$ $[ - \delta , \delta ]$ , for all $\delta > 0$