MATH 4317
Analysis I
Fall 2016
TR 12:05-13:25 Skiles 255
INSTRUCTOR
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Stavros Garoufalidis
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Room 105, Skiles
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Telephone: 404-894-6614
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email
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Office hours: Wed 3:30-4:30 or by appointment.
GRADER
TEXTBOOK
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Maxwell Rosenlicht, Introduction to Analysis, Dover publications.
COURSE DESCRIPTION
We plan to give a self-contained course in analysis, covering chapters 1-6
of the book. The goal of the course, aside from analysis, is to
learn what is a mathematical proof, and how to present it.
For more details, see
here .
WHAT IS A PROOF?
See
a note by Chris Heil.
PREREQUISITES
MATH 4317.
HOMEWORK
Homework will be due on Fridays, collected in the beginning of the class,
and returned back to you the following Monday. I will not be assuming that
you know how to think, write and present a mathematical proof, but achieving
this by the end of the semester, will be an important gain for the class.
You are welcomed to work together, but please write down your homework
separately . Please staple your homework, and write clearly, each
problem on a new page. The two lowest homework grades will be dropped. Any
missed homework will be counted as zero.
MIDTERM-EXAMS
There are two in-class midterm exams on Thursday October 13 and November 17.
I do not accept make-up exams unless there is a serious
documented reason.
FINAL EXAM
Conflict in final exams means having 3 final exams in the same day. In that
case, you can reschedule the one in the middle. In case of a final exam
conflict, I need a two week advance notice to accomodate your
request.
ATTENDANCE
Although attendance is not mandatory, you are strongly encouraged to attend.
Students who do not attend
the lectures statistically do below average--and usually end up with a C or
lower grade. Attendance is an important part of learning. During the
lectures, you should turn all powered devices (especially phones, laptops,
music players) off.
GRADES
The final course grade will be calculated according to the
following scheme:
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Homework : 15 %
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Midterm : 2 x 25 %
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Final Exam : 35 %
If the average course grade (out of 100) is
- at least 80, then you will get an A,
- at least 70, then you will get a B or A,
- at least 60, then you will get a C, B or A,
- at least 50, then you will get a D, C, B or A,
- if your final exam grade is less than 40, then you will
get an F.
ACADEMIC HONOR CODE
All students are required to review and accept the Honor Code
of Georgia Tech, which may be found
here.
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You are not allowed to use calculators, formula sheets, notes or texts
during the midterms and final exams.
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Identification is required in taking tests.
ACADEMIC CALENDAR
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August 22, First day of class.
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September 5, School Holiday: Labor Day.
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October 29, Last day to withdraw with W.
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October 10-11 Fall recess.
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October 13, Midterm I.
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November 17, Midterm II.
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November 23-25 School Holiday: Thanksgiving.
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December 6, Last Day of classes.
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December 15, (Thu) 11:30am - 2:20pm
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December 20, Grades available.
HOMEWORK ASSIGNEMENTS
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Homework 1 due Thursday Sep 1 in the beginning of the class.
Chapter I, p.12-13. Problems 7, 8, 9, 10.
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Homework 2 due Thursday Sep 8 in the beginning of the class.
Chapter II, p.30-31. Problems 7, 10, 11, 12, 13, 14.
Problem: Show that the set of 0-1 sequences of real numbers (i.e.,
the set of binary sequences) is not countable.
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Homework 3 due Thursday Sep 22.
Chapter III: 1, 2, 6, 11, 12, 13, 14, 15, 16.
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Homework 4 due Thursday Sep 29.
Chapter III: 17, 18, 19, 20. Problem: (a) Show that 7^n converges to 0 in the
7-adic topology of Z. Recall d_7(n,m)=7^{-v(n-m)} where 7^v(k) is the biggest
power of 7 that divides an integer k, and v(0)=infinity. (b) Describe the open
and closed ball with center 0 and radius 1 in the 7-adic topology of the
integers.
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Homework 5 due Thursday Oct 6.
Chapter III: 22, 24, 26, 27, 28.
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Homework 6 due Thursday Oct 13.
Chapter III: 29, 30, 31, 33, 35.
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Homework 7 due Thursday Oct 20.
Chapter III: 36, 37, 38.
Problem: Define what is a non-open subset S of a metric space (E,d).
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Homework 8 due Tuesday Oct 25.
Chapter IV: 3, 6, 9, 10.
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Homework 9 due Thursday Nov 3.
Chapter IV: 13, 15, 24, 31, 32.
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Homework 10 due Thursday Nov 10.
Chapter IV: 34, 37. Chapter V: 1, 2, 4, 6.
Redo problem 31 of chapter IV using x=0.aa1b10a2b20... where ai,bi are
dinary digits (ie 0 or 1) to construct an explicit curve f: [0,1]->[0,1]^2.
Compute the values of f(x) for x = 0,1/8,2/8,3/8,4/8,5//8,6/8,7/8,8/8,
and join the corresponding points of [0,1]^2 with line segments.
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Homework 11 due Thursday Nov 17.
Chapter V: 9, 10, 11, 14.
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Homework 12 due Tuesday Nov 29.
Chapter VI: 2, 4, 8, 14, 21, 22.