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Preliminary Exam - Summer 1977
Problem 2
Let f be continuous on
![% latex2html id marker 731
$\mathbb{C}\,^{}$](img12-5.gif)
and analytic on
![$\{z \;\vert\; \Im z \neq 0 \}$](img13-5.gif)
. Prove that
![$f$](img14-5.gif)
must be analytic on
![% latex2html id marker 737
$\mathbb{C}\,^{}$](img15-5.gif)
.
Problem 3
Prove that
![$\alpha = \sqrt{3} +\sqrt{5}$](img16-5.gif)
is algebraic over
![% latex2html id marker 759
$\mathbb{Q}\,^{}$](img17-5.gif)
,
by explicitly finding a polynomial with coefficients in
![% latex2html id marker 761
$\mathbb{Q}\,^{}$](img18-5.gif)
of
which
![$\alpha$](img19-5.gif)
is a root.
Problem 4
Let
![$A$](img20-5.gif)
be an
![$r\times r$](img21-5.gif)
matrix of real numbers. Prove that the infinite sum
of matrices converges (i.e., for each
![$i,j$](img23-5.gif)
, the sum of
![$(i,j)^{th}$](img24-5.gif)
entries converges), and hence that
![$e^{A}$](img25-5.gif)
is a well-defined matrix.
Problem 5
Write all values of
![$i^i$](img26-5.gif)
in the form
![$a+bi$](img27-5.gif)
.
Problem 6
Show that
![$0\leq k < 1$](img29-5.gif)
, is an increasing function of
![$k$](img30-5.gif)
.
Problem 7
Let
![% latex2html id marker 835
$A : \mbox{$\mathbb{R}^{6}$} \to \mbox{$\mathbb{R}^{6}$}$](img31-5.gif)
be a linear transformation such that
![$A^{26} = I$](img32-5.gif)
. Show that
![% latex2html id marker 841
$\mbox{$\mathbb{R}^{6}$} = V_1\oplus V_2\oplus V_3$](img33-5.gif)
,
where
![$V_1$](img34-5.gif)
,
![$V_2$](img35-5.gif)
, and
![$V_3$](img36-5.gif)
are two-dimensional invariant subspaces for
![$A$](img37-5.gif)
.
Problem 8
Prove that the initial value problem
has a solution
![$x(t)$](img39-5.gif)
defined for all
![% latex2html id marker 874
$t \in \mbox{$\mathbb{R}^{}$}$](img40-5.gif)
.
Problem 9
Show that every rotation of
![% latex2html id marker 889
$\mathbb{R}^{3}$](img41-5.gif)
has an
axis; that is, given a
3
![$\times $](img42-5.gif)
3 real matrix
![$A$](img43-5.gif)
such that
![$A^t=A^{-1}$](img44-5.gif)
and
![$\det A > 0$](img45-5.gif)
,
prove that there is a nonzero vector v such that Av = v. _matrix,>axis
Problem 10
Suppose that
![$f(x)$](img46-5.gif)
is defined on
![$[-1,1]$](img47-5.gif)
, and that
![$f'''(x)$](img48-5.gif)
is
continuous. Show that the series
converges.
Problem 11
Let
![$f(x,t)$](img50-5.gif)
be a
![$C^1$](img51-5.gif)
function such that
![$\frac{\partial f}{\partial x}
= \frac{\partial f}{\partial t}$](img52-5.gif)
. Suppose that
![$f(x,0)>0$](img53-5.gif)
for all
![$x$](img54-5.gif)
.
Prove that
![$f(x,t)>0$](img55-5.gif)
for all
![$x$](img56-5.gif)
and
![$t$](img57-5.gif)
.
Problem 12
Let
![$V$](img58-5.gif)
be the vector space of all polynomials of degree
![$\leq 10$](img59-5.gif)
, and
let
![$D$](img60-5.gif)
be the differentiation operator on
![$V$](img61-5.gif)
(i.e.,
![$Dp(x)=p'(x)$](img62-5.gif)
).
- Show that
.
- Find all eigenvectors of
and
.
Problem 13
Let
![$f$](img66-5.gif)
be an analytic function such that
![$f(z)=1+2z+3z^2+\cdots$](img67-5.gif)
for
![$\vert z\vert<1$](img68-5.gif)
. Define a sequence of real numbers
![$a_0,a_1,a_2,\ldots$](img69-5.gif)
by
What is the radius of convergence of the series
Problem 15
Let
![% latex2html id marker 1048
$A\subset\mbox{$\mathbb{R}^{n}$}$](img75-5.gif)
be compact,
![$x\in A$](img76-5.gif)
; let
![$(x_i)$](img77-5.gif)
be a
sequence in
![$A$](img78-5.gif)
such that every convergent subsequence of
![$(x_i)$](img79-5.gif)
converges to
![$x$](img80-5.gif)
.
- Prove that the entire sequence
converges.
- Give an example to show that if
is not compact, the
result in Part 1 is not necessarily true.
Problem 16
Use the Residue Theorem to evaluate the integral
where
![$a$](img84-5.gif)
is real and
![$a>1$](img85-5.gif)
. Explain why the formula obtained for
![$I(a)$](img86-5.gif)
is also valid for certain complex (nonreal) values of
![$a$](img87-5.gif)
.
Problem 17
In the ring
![% latex2html id marker 1094
$\mbox{$\mathbb{Z}^{}$}[x]$](img88-4.gif)
of polynomials in one variable over the integers,
show that the ideal
![$\mathfrak{I}$](img89-4.gif)
generated by
![$5$](img90-4.gif)
and
![$x^2+2$](img91-4.gif)
is a maximal ideal.
Problem 18
Let
![$\hat{a}_0+\hat{a}_1z+\cdots+\hat{a}_nz^n$](img92-4.gif)
be a polynomial having
![$\hat{z}$](img93-4.gif)
as a simple root. Show that there is a continuous function
![% latex2html id marker 1131
$r : U \to \mbox{$\mathbb{C}\,^{}$}$](img94-4.gif)
, where
![$U$](img95-4.gif)
is a neighborhood of
![$(\hat{a}_0,\ldots,\hat{a}_n)$](img96-4.gif)
in
![% latex2html id marker 1137
$\mathbb{C}\,^{n+1}$](img97-4.gif)
, such that
![$r(a_0,\ldots,a_n)$](img98-4.gif)
is always a root of
![$a_0+a_1z+\cdots+a_nz^n$](img99-4.gif)
, and
![$r(\hat{a}_0,\ldots,\hat{a}_n)=\hat{z}$](img100-4.gif)
.
Problem 19
Let p be an odd prime. If the congruence
![$x^2\equiv -1 \pmod{p}$](img101-4.gif)
has
a solution, show that
![$p\equiv 1 \pmod{4}$](img102-4.gif)
.
Problem 20
Determine all solutions to the following infinite system of linear
equations in the infinitely many unknowns
![$x_1,x_2,\ldots$](img103-2.gif)
:
How many free parameters are required?
Previous: Spring77
Next: Fall77
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10