5/15/15: The mean on the Final is 56.5 (or 70.6%) and the median is 57
out of 80. Good job! I am entering the grades shortly. Thank you,
folks, for the great semester!
5/15/15: The mean on the Final is 56.5 (or 70.6%) and the median is 57
out of 80. Good job! I am entering the grades shortly. Thank you,
folks, for the great semester!
5/8/15: I am working on solutions to the Sample Final. You can see the
current version here: Solutions to Sample
Final, which is constantly being updated. You may need to reload
it in your browser to see the latest draft.
5/7/15: Please note that the final exam will last 2 hours, or 120
minutes. There was a place on the class web site where it was
scheduled as if it was a one-hour exam. I have now corrected
that. Thus, the Final will be from 1:30 to 3:30 p.m. on Tuesday,
May 12.
5/7/15: I have to change my office hours for Monday, May 11, to
1:00-1:50 p.m. because of a doctor's appointment.
5/6/15: In addition to my regular office hours on Friday, May 8,
11:15-12:05, I will hold extra office hours the same day 1:30 to 2:20
p.m., and on Monday, May 11, 1:00-1:50.
5/6/15: I have finished writing the Sample
Final Exam and posted it. Please think which problems you prefer
to discuss on Friday.
5/5/15: A Sample Final Exam is partially
posted. We will discuss that part (the first 4 problems) tomorrow. I
will post the rest by tomorrow (Wednesday) night.
5/3/15: Coverage for the 1:30-3:30 p.m., Tuesday, May 12,
Final Exam (in our regular classroom) is basically all we have
studied through Monday, April 27. This will be Chapter 6 (6.1-6, 6.8,
6.12(a-d), 6.20-22), the Riemann sums material from your class notes,
Chapter 7, Chapter 8 (pp. 172-174, 178-185), Chapter 9 (9.4, 9.6(a,c),
9.10-18, 20-21, 24-29, 39, 41) and Chapter 11 (11.1-8, 11.10-25,
28-30). How to get ready for the test? The best way would be
to try to solve as many problems as you can on your own (with an open
textbook, if needed, and no timing) from
the sample final exam, the homework,
and the textbook. For those problems you cannot solve, it may be a
good idea to discuss them with your friends,
i.e., your study group. Come to my office hours to get hints or
other help. Go over homework problems again without consulting your
solutions and look for problems from exams in similar classes on the
web: search for Analysis tests, exams, midterms, finals, etc.,
using such keywords as outer measure, inverse function
theorem and so on, but make sure the test you find is for
undergraduates. When you know you can get an idea of how you might
approach most of the problems right away, you may feel more sure that
you will do well on our exam. Good luck!
5/3/15: The mean on Midterm II is 30.2 and the median is 31 out of
40. Good job!
4/30/15: Now the Sample Midterm Exam II
has solutions. Good luck on the test tomorrow!
4/24/15: Sample Midterm Exam II is
posted.
4/21/15: I have reduced the Coverage for the Friday
Midterm to something more realistic. See below and wait for a
sample test sometime late this night.
4/21/15: Coverage for the Friday, May 1, Midterm Exam II
(in our regular classroom) is basically all we will have studied since
Midterm I through Monday, April 27. This will be Chapter 9 (9.15-18,
20-21, 24-29, 39, 41) and Chapter 11 (11.1-8, 11.10-25, 28-30). How
to get ready for the test? The best way would be to try to solve
as many problems as you can on your own (with an open textbook, if
needed, and no timing) from the sample
midterm exam, which will be posted soon, the homeworks, and the
textbook. For those problems you cannot solve, it may be a good idea
to discuss them with your friends, i.e., your study group. Come to my
office hours to get hints or other help. Go over homework problems
again without consulting your solutions and look for problems from
exams in similar classes on the web: search for Analysis tests, exams,
midterms, finals, etc., but make sure the class you find is for
undergraduates. When you know you can get an idea of how you might
approach most of the problems right away, you may feel more sure that
you will do well on our exam. Good luck!
4/19/15: The new homework is complete. Sorry for the delay.
4/18/15: New homework is posted (partially). I will add a couple more
problems on Sunday morning.
4/15/15: I need to reschedule my office hours on Monday, May 4, to
10:10-11:00. As always, if you need to see me other time, make an
appointment, drop me a line, or just stop by to see me.
4/13/15: Another message from Frieda, who made a presentation on Peace
Corps in class the other Friday: "Tomorrow, Tuesday, April 14, we will
host the final Peace Corps Information Session of the year at 6 PM in
Humphrey School of Public Affairs, Room 289. We will go over the
basics of Peace Corps, hear from a Returned Peace Corps Volunteer
(RPCV), and students can ask a panel of RPCVs any questions they may
have."
4/12/15: I have added two problems to the homework. Sorry, my morning
started late, as I allowed myself to sleep in to be refreshed for
reaching new heights of mathematical knowledge for the coming
workweek.
4/11/15: New homework is posted (just 5 problems, so far). I will add
a couple more problems on Sunday morning.
4/6/15: New homework is posted (just 5 problems). I have already
reworded the problem on the Cantor staircase function c(x)
since the original version, so as not to require proving that c(x)
+ x is a homeomorphism, and corrected inf to sup in the last
equation in Problem 4.
4/5/15: Folks, I am sorry, I have totally forgot to post the
homework. I will post a short one the first thing in the morning on
Monday.
4/1/15: Improved notation for the face of a rectangle in Problem 7.
4/1/15: Updated the Class Outlines page so as to correspond with what
has been covered in class today. Removed Sections 11.9-11 from the
reading assignment on HW 8.
3/31/15: One more update of Homework 8. I did not realize that Problem
2 was practically the same as Problem 3. So I replaced Problem 2 with
a following mundane question: is the Cantor set a Borel set
on R? I do apologize for the confusion. However, what you might
have done for Problem 2 will be useful for Problem 3. In particular, I
moved the hint from Problem 2 to Problem 3 (in an adjusted form).
3/31/15: Another update of Homework 8. I reworded Problem 2 to be less
vague: Prove that the σ-algebra generated by the collection of
open intervals in R is the same as the σ-algebra of Borel
sets.
3/30/15: Updated Homework 8: reworded Problem 1, as discussed in
class, and added the definition of a face to Problem 7.
3/30/15: Updated Homework 8: corrected a typo in Problem 1, added
hints to Problems 2 and 4 and a couple of minor clarifications.
3/28/15: I will post the homework (#8) before 1 p.m. on Sunday. Sorry
for the delay.
3/24/15: I have changed the wording of Problem 4 to be more clear.
3/24/15, 12:05 a.m.: I have just completed HW 7: 5 problems, due this
Friday.
3/20/15: I have added a little clarification to Problem 9 on HW 6,
because a few of you seem to be confusing connected sets for convex
ones. Problem 9 does not obviously follow from Problem 8, because a
connected set is not necessarily convex.
3/13/15: The American Math Society invites you to celebrate Pi Day
with us on 3/13/14 here in Vincent Hall! We know you all want to come
in the first Saturday of Spring Break, but we have decided to
celebrate a day early.
Prof. Paul Garrett will be giving a talk at 3:30 in VinH 113 (title and abstract below). Following his talk (4:45 pm), we invite you to join us for pizza (courtesy Mesa), pie (courtesy Sarah Jane's), games and socializing.
Title: "How to fail to prove the Riemann Hypothesis"
Abstract: Yes, everyone knows the basic approach: "do whatever you want... and fail to prove RH". With that trenchant observation out of the way, we also recall Hilbert's and Polya's throwaway quip: if we can find a self-adjoint operator whose eigenvalues, parametrized as s(s-1), are given by zeros s of zeta, then, since s(s-1) is real, s must be on the critical line (or the real axis, but nevermind). With no hint of what operator to consider, this is not much help. Our story begins in 1977, when H. Haas attempted numerical computation of eigenvalues of the non-Euclidean Laplacian on the modular curve. H. Stark and D. Hejhal observed that the list of eigenvalues s(s-1) contained zeros "s" of zeta and another L-function. Hejhal determined that these were spurious, due to a flaw in the numerical procedure. However, the flaw itself suggested possibilities, pursued tentatively by Y. Colin de Verdiere in 1982-4. For many people, the significance of Colin de Verdiere's speculations was apparently difficult to understand, and things were left in a somewhat confused state until 2011, when some of the muddle began to be resolved. This talk will be at a technical level accessible to math grad students and faculty.
3/12/15: Added a hint to Problem 5 on HW 6.
3/11/15: In Problem 4 on HW 6, the function is continuous, even though
I suggested at first to prove it is not. I have corrected that and
added a hint on how to show it's not differentiable at 0, as well as a
hint to Problem 2.
3/9/15: The mean on Midterm I is 31.3 and the median is 33 out of
40. Good job!
3/9/15: HW 6 is posted.
3/3/15: I need to change my office hours for the week of March 9-13
(after the midterm and before the Spring Break) to 10:10 - 11:00
MWF. I will be substituting for a colleague from 11:15 to 12:05 that
week.
3/2/15: I have adjusted the coverage for Midterm I below slightly
(for Chapter 9) to reflect what has been covered in class today.
2/28/15: I have posted a Sample Midterm
Exam I. Enjoy it but do not go overboard!
2/28/15: I have corrected pages in the coverage from Chapter 7 for the
Midterm Exam in the paragraph below: changed 172-179 to 172-174.
2/26/15: Coverage for the Friday, March 6, Midterm Exam
I (in our regular classroom) is basically all we will have studied
through Monday, March 2. This will be Chapter 6 (6.1-6, 6.8,
6.12(a-d), 6.20-22), the Riemann sums material from your class notes,
Chapter 7, Chapter 8 (pp. 172-174, 178-185), and Chapter 9 (9.4,
9.6(a,c), 9.10-14). How to get ready for the test? The best
way would be to try to solve as many problems as you can on your own
(with an open textbook, if needed, and no timing) from
the sample midterm exam, the homeworks,
and the textbook. For those problems you cannot solve, it may be a
good idea to discuss them with your friends, i.e., your study
group. Come to my office hours to get hints or other help. Go over
homework problems again without consulting your solutions and look for
problems from exams in similar classes on the web: search for Analysis
tests, exams, midterms, finals, etc., but make sure the class
you find is for undergraduates. When you know you can get an idea of
how you might approach most of the problems right away, you may feel
more sure that you will do well on our exam. Good luck!
2/26/15: There was a typo in the hint below (2/25/15): a division
sign, /, was missing at the very end. Sorry. This is now corrected. I
have also added a couple of parentheses to avoid ambiguity.
2/25/15: Here are some more hints to Problem 3 on Homework 5. Let me
use the formula and notation at the bottom of page 176 for b_m (The
notation and setting in the HW problem is just slightly different, but
you cannot use the remaining part of the proof of that theorem in the
textbook, because it uses the double series material, which we have
never covered.): b_m = \sum_{n=m}^infty (n choose m) c_n a^{n-m} =
\sum_{n=0}^infty (n+m choose m) c_{m+n} a^n. Note that despite limsup
sqrt[n]{|c_n|} = 1/R looks scary, it just implies that for only
finitely many n's |c_n| > 1/R^n. Thus, for large enough n, we have
|c_n| <= 1/R^n. Thus, |b_m| <= \sum_{n=0}^infty (n+m choose m)
1/R^{m+n} r^n = 1/R^m \sum_{n=0}^infty (n+m choose m) r^n/R^n = 1/(R^m
(1-r/R)^{m+1}) = R/(R-r)^{m+1} Now look at limsup of the mth root of
that.
2/21/15: I have just posted Homework 5. I had hard time deciding upon
the material for next week, whence the delay. Originally I did not
plan to go over the exponential and trigonometric functions, but
changed my mind yesterday night.
2/19/15: For Problem 7 on Homework 4, I am adding a hint: To show
closedness over R, I suggest to estimate the distance between
fn and fm by a constant, say, 1/2. This would
imply that this set is discrete, i.e., each point of it being open,
thereby making all subsets open and closed.
2/16/15: If on Homework 2, you have got a point taken off for failing
to prove that a step-function was not continuous, please show me your
homework, so that I add that point back. Apparently, the grader was
not aware that we are beyond these things.
2/14/15: I have just posted Homework 4 for you. Sorry for the delay: I
got totally exhausted yesterday night and started falling asleep at my
computer while I was writing outlines of classes for next week, even
before starting on the homework. Happy Valentine's Day, my favorite
class!
2/12/15: A hint for Problem 4 on the homework which is due
tomorrow: Here is how you can prove that the subset U = {f in B(X,R) |
f(1) < 2} is open in the metric space B(X,R) of bounded functions,
where X is a set. Given g in B(X,R) such that g(1) < 2, take r = 2 -
g(1), which is > 0. Then the ball of radius r about g will be
contained in U, because if sup |f(x) - g(x)| < r, then |f(1) - g(1)| <
r and thereby f(1) < 2. Explaining why Un is Problem 4 is
open is similar.
2/7/15: An updated (since last night) version of Homework 2 is
posted.
2/6/15: If you do not mind, I will change my Friday office hours to
11:15-12:05 for the rest of the semester.
2/6/15: Next week CSE will be welcoming more than 135 employers to
campus for our largest-ever Spring Science and Engineering Career
Fair. More than 2,000 undergraduate and graduate students are expected
to attend the one-day fair to explore opportunities for jobs,
internships and co-ops:
Science and Engineering Career Fair
Tuesday, Feb. 10, 2015
10 a.m.-4 p.m.
TCF Bank Stadium (use Benton County entrance)
Employers attending the Career Fair include many top national and regional companies such as Aerotek, Amazon, Boston Scientific, BuzzFeed, Cummins, Dell, Eaton Corporation, General Mills, Graco, Honeywell, Hormel, Intel, Medtronic, Pearson, Schlumberger, Siemens, Target, Unisys, Xcel Energy, and many others. To see a full list of employers attending the Science and Engineering Career Fair, visit z.umn.edu/employers2015.
2/6/15: It has turned out your graded homework had been in my mailbox
before today's class, but I did not check my mailbox just before
class. I will bring it to class on Friday.
2/4/15: Added a hint to Problem 7 on Homework 2.
2/4/15: Added a hint to Problem 5 on Homework 2.
2/2/15: Corrected the interval in Problem 3 on Homework 2 from (0,1)
to [0,1].
1/31/15, 2:18 a.m.: I have just posted Homework 2 for you. Sorry for
the delay: I totally forgot about that until 10 minutes before
midnight on Friday.
1/30/15: I need to reschedule my office hours once again, now for
Friday, February 6, to 11:15-12:05. Just for that particular day,
February 6.
1/30/15: I incorrectly assumed that this class would get the same
grader we had the last term. We have got a new grader. His name is
Harris, and his e-mail address is moham189 at umn.edu
1/28/15: I need to reschedule my office hours on Friday, January 29,
to 11:15-12:05. Just for that particular day, January 29.
1/24/15: Updated Homework 1 with a hint.
1/24/15: Homework 1 is now posted online. Sorry for the delay.
1/23/15: I have added a more detailed description of coverage for this
semester to the Syllabus: Rigorous treatment of Riemann-Stieltjes
integration with an emphasis on Riemann integration; Sequences/series
of functions, uniform convergence, equicontinuous families,
Stone-Weierstrass Theorem, power series; Rigorous treatment of
differentiation/integration of multivariable functions, Implicit
Function Theorem, Stokes' Theorem.
1/23/15: The midterm exam dates are March 6 and May 1 (both are
Fridays). No homework will be due on those weeks.
1/23/15: The syllabus handed out in class contained incorrect coverage
for the course. The correct coverage is Chapters 6-11 (selected
topics).
1/23/15: I recommend the following way to study for this class. Attend
each class, take notes, participate in class actively. After each
class review your notes and study the corresponding part of the
text. You can find out which part of the text at
the
Class Outlines page. Then do the assigned homework problems
pertinent to that material.
1/22/15: If you need to register for this class, please, send a
message to Ms. Lawson at ugrad@math.umn.edu and ask for a permission number. If
permission is granted, go to OneStop and register. See more
instructions
at www.math.umn.edu/undergrad/registration.
1/22/15: I am sorry, I said and wrote "infimum" instead of "supremum"
in the proof of the fact that the lower integral does not exceed the
upper integral in the first class meeting. Thanks to Anthony who
pointed that out to me.
Last modified: (2015-10-05 17:03:50 CDT)