MATH 347 – Advanced Real Analysis

Summer 2007

Section 01: 8:55-10:15 a.m., MTWThFInstructor: Lucian M. Ionescu

Contact Info: Office 303G, 438-7167; ;

Office Hours: TBA or by appointment at other times.

Text: Real analysis and foundations, Steven G. Krantz, 2nd edition, 2005.

Content: Math 347 is a course in advanced real analysis, concerned with topology of metric spaces, sequences, continuous functions, uniform convergence, differentiation, Taylor's Theorem, Riemann integration, the fundamental theorem of calculus, infinite series, power series. As opposed to MAT 247, the emphasis is placed on metric spaces. Applications to differential equation are considered, with a brief introduction to harmonic analysis and wavelets for graduate credit.

Prerequisites: C or better in MAT 247 and MAT 336; or MAT 337 or consent of instructor required.

Objectives: The general goals of the course are 1) to introduce the basic ideas and concepts of real analysis with proofs, 2) to teach students how to design mathematical proofs, thinking conceptually, while reading and writing “math-code”, 3) to help students develop a rigorous understanding of the theorems of calculus and differential equations, and 4) to enhance their ability to communicate abstract ideas in written and oral form.

Course Format & Learning Cycle: You are responsible for reading assignments, and this is not a trivial task. Browsing assignments before the next scheduled meeting of the class is meant to prepare you to assimilate the explanations during the lecture (you need a map before going on the field trip). During the class, we will focus on understanding the concepts and methods. After the class read the assignment. After you are done and before attempting the homework, write one page outline of the corresponding section (include major concepts, methods and relations, with brief explanations). Check you have acquired the corresponding knowledge by reading the outline (keywords known? relations?) Attempt all the homework problems before the next class period, and build your list of questions. Come to class, when you should turn in the outline for one point per submission. Before discussing the new assigned material, we will discuss your difficulties or questions regarding the theory and homework problems. You may choose to present solutions to the homework problems on transparencies, for “class participation” points, as time permits. Attendance and active participation are expected. The sections marked with * are for graduate credit.

Homework: There will be daily homework assignments. Homework will be collected on Mondays. Please show all the work (a list of answers is not a homework assignment).

Exams: Attendance at exams is mandatory. It is highly recommended that you contact me before the exam, if an emergency has arisen.

Evaluation: Grades will be based on the following points:

Homework 100

Class activities (Attendance 50, outlines 30, and participation 20): 100

In-class exams (2)200

Final exam200

Total 600 points.

The grading scale is based on: A [90-100%], B [80-90%), C [70-80%), D [60,70%), F [0,60%).

N.B.: Students who believe they may need accommodations in this class/program are encouraged to contact the DisabilityAccessCenter as soon as possible to better ensure that such accommodations are implemented in a timely fashion.

“Warning: Plagiarism and cheating are serious offenses. Penalties can range from a minimum of a zero grade on the invalid instrument to expulsion from the University”

Homework Assignments

The “problems assigned” for a “section covered” on a specific “date” constitutes the “HW No.” (solutions to the problems of the previous week), which is “due” Mondays.

“R&p”= review and problems.

HW#
/Due / Date / Section covered / Problems assigned / HW#
/Due / Date / Section covered / Problems assigned
1 / M / 6/18 / Introd. / 9 / M / 7/26 / 5.1 / 1,2,3
T / 1.1,1.2,1.3 / 1,2,3 / T / 5.2 / 4,5
W / 1.4,1.5,1.6 / 4,5,6 / W / 5.3 / 17,18
R / 1.7 / 7 / R / 5.4 / 20
F / 1,8, R&p / 8 / F / 5.5 / 31
2 / M / 6/25 / 2.1,2.2,2.3 / 2,7 / 10 / M / 7/23 / 6.1 / 1,2
T / 2.4 / 12,14 / T / 6.2 / 3,4
W / 2.5 / 19,20 / W / 6.3 / 7,9
R / 2.6 / 25,26 / R / 6.4 / 14,15
F / Q, R&p / F / R&p
3 / M / 7/2 / 3.1,3.2 / 1,2,3,4 / 11 / M / 7/30 / Exam 2
T / 3.3 / 12,14,19 / T / 7.1 / 1,2
W / July 4th / No class / Holiday / W / 7.2 / 9,14
R / 7/5 / 3.4(Online) / R / 7.3 / 23,25
F / 7/6 / R&p (Online) / F / R&p
4 / M / 7/9 / Exam 1 / 12 / M / 8/6 / 8.1 / 1,2
T / 4.1 / 1,2 / T / 8.2 / 8,10
W / 4.2,4.3 / 14 / W / 8.3 / 14,15
R / 4.4 / 18 / R / R&p / Review
F / 4.5, R&p / 19 / F / 8/10 / Final Exam

H1: Outlines & distilling knowledge

H2: Data structures and what/how to learn

H3: Concepts (keywords) and math code (compiling);

H4: Black boxes and user’s interface

H5: How to write proofs

Q: Quaternions and R3

L: Logic: classical, probabilistic, quantum