MATH 490
Senior Seminar
Class Policies
Mike OLeary
Ofce: YR 307A
Ofce Phone: 410-704-4757
Email:[email protected]
Fall 2013
Class: TuTh 5:00 - 6:15
Room: YR 122
Section: 101
Ofce Hours: Tu 2-3, Th 3-4 and by appointment
Prerequisites: Senior standing.
Ca
f, wt. r. Lc
Lor
tr
- r/yz +3
+ 2yt -x3
t( (v,=x,\, (J,!r\ ).:
a
a) *s t.;s ,. [*
trLru
\$
r? t4 I p1*,h
VJte.l-f V
.'
k.
n1.
n
oA n
lu\"n-,s g ?
>J v'2
!- | cfw_*,
2,
,
xl, v,1.I
t) Le
A->
,f
(s,2,?) .r, b=(
I
oi,).o-.
\
8,2,a!io,r)\:
r
\,
,
l- ,t!r"rcfw
hpi)r 4q0
-w/1 2-o s
e [arJ.
s l2
t.
Nktk
uS[ol rS
oa #V"n
R"
^ ,(a ,ol"
Scocfw_Ct r-v-J/frpl.6a11g.
6r-JJ,'t'r
o
Vfcfo *S th
nJ
,R
2.
ccfw_
a \dctnZ'
A n'r^f P,i* tu.). X $ cfw_( f-cfw_ f,a* ?
w\-ot-
rl
Cl.o"te
r
Sr
Lr
#
Y ,7
(+ o r.ffi4,1 x 6 rflr.fu. ]
MATH 490
Senior Seminar
Assignment Sheet #1
For each of the following, dene the necessary operations to make the space into a vector space,
or explain why that is impossible.
For those that are vector spaces, determine the dimension of the space. If it is
MATH 490
Senior Seminar
Assignment Sheet #2
8. Read the papers
Adrian Rice and Eve Torrence, Lewis Carrolls Condensation Method for Evaluating Determinants, Math Horizons, Vol. 14, No. 2 (November 2006), pp. 12-15, Mathematical
Association of America. UR
MATH 490
Senior Seminar
Assignment Sheet #3
10. Let T : P3 P3 be given by T y = (1 x2 )y + 2xy . Find the eigenvectors and eigenvalues
of T .
Comments: We can repeat the problem in PN for any positive integer N . An eigenvector
p(x) normalized so that p(1
MATH 490
Senior Seminar
Assignment Sheet #4
16. Prove that f (x) =
x is continuous for all x > 0.
17. Let
f ( x) =
exp(1/x2 )
0
if x > 0,
if x 0.
Prove that f (x) is dierentiable for x = 0.
18. Show that, if f is dierentiable at x that
f (x + h) f (x h)
h
MATH 490
Senior Seminar
Assignment Sheet #5
23. Let
fn (x) =
x
.
1 + nx2
Show that there is a function f (x) so that fn (x) f (x) uniformly. Does fn f ?
Hint: To show the uniform convergence, handle the set cfw_x : |x| and the set cfw_x : |x| >
dierentl
MATH 490
Senior Seminar
Assignment Sheet #6
33. Consider the function
f
u
v
=
1
2
2 (u
v2 )
.
uv
(a) What are the curves of constant u? What are the curves of constant v ?
(b) Graph f .
(c) Find Df .
(d) Does f have a (global) inverse?
(e) Where, if anyw
MATH 490
Senior Seminar
Assignment Sheet #7, Update 2
Note: Section 4.5 will be skipped.
41. Let (X, U ) and (X, V ) be topological spaces. Show (X, U V ) is a topological space.
42. Let (X, T ) be a topological space, and let A X . Dene TA = cfw_U A : U
MATH 490
Senior Seminar
Assignment Sheet #8
52. Let T be the range of the function f : R2 R3 given by
f (, ) = a cos , sin , 0 + b cos cos , cos sin , sin .
for a > b. Is T a dierentiable manifold in R3 ? Prove your claim.
53. Let S be the range of the fu
MATH 490
Senior Seminar
Final Project
Version 1.0
Choose any one of the papers below to be the focus of your nal paper and your
nal presentation. Start by reading the paper carefully. Be sure you understand the
results presented- not only their signicance