APPM 5450, Applied Analysis 2, Spring 2019
Announcements
First day of class Monday January 14, 2019.
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Course Information
This is the second semester of the applied analysis series. Analysis is the foundation of applied mathematics. This course will pick up where we left off last semester, covering some topics in Ch7-Ch12 (more or less) It will cover basic applications of orthonormal basis in Hilbert spaces, such as Fourier series. We will develop the theory for Hilbert spaces, in particular bounded linear operators and their spectrum. We will also discuss some topics related to linear differential operators and Green’s functions (Ch 10) and some distribution theory (Ch 11). The course will end measure theory and integration.This course is designed to prepare students for the Analysis qualifying exam.
You can find the topics covered in the exam here: http://tinyurl.com/juhaw5k.
Text: Applied Analysis by by J. Hunter and B. Nachtergaele, World Scientific Publishing, 1st ed., 2001, ISBN 978-9812705433.
Syllabus: You can find the syllabus here.
Prelim exam: this course is designed to partially prepare you for the Analysis prelim exam (Analysis prelim exam, old website) for graduate students in the applied math department (you should also know advanced calculus and the first semester of this course; in addition, independent studying is recommended).
Outline
| Section | Date | Topics |
|---|---|---|
| Sec 7.1 | 1/14 | Fourier basis |
| Sec 7.1-7.2 | 1/16 | Fourier basis, convolutions, derivatives |
| Sec 7.2 | 1/18 | weak derivatives, distributions |
| 1/21 | MLK day | |
| Sec 7.2-7.3 | 1/23 | Sobolev Embeddings and Heat Equation |
| Sec 8.1 | 1/25 | Projections |
| Sec 8.2 | 1/28 | Dual Spaces of Hilberts Spaces |
| Sec 8.3 | 1/30 | Adjoint Operators |
| Sec 8.4 | 2/1 | Self-Adjoing Operators |
| Sec 8.5 | 2/4 | Weak Convergence in Hilberts Spaces |
| Sec 8.5 | 2/6 | Weak Convergence in Hilberts Spaces |
| CH 9 | 2/8 | Spectrum of Bounded Operators |
| CH 9 | 2/11 | Spectrum of Bounded Operators |
| CH 9 | 2/13 | Spectrum of Bounded Operators |
| CH 9 | 2/15 | Spectrum of Bounded Operators |
| CH 9 | 2/18 | Spectrum of Bounded Operators |
| Exam | 2/20 | Exam 1 |
| CH 9 | 2/22 | Spectrum of Bounded Operators |
| CH 9 | 2/25 | Spectrum of Bounded Operators |
| CH 9 | 2/27 | Spectrum of Bounded Operators |
| CH 9 | 03/01 | Spectrum of Bounded Operators |
| 10.1 | 03/04 | Unbounded operators |
| Sec 10.1 | 03/06 | Unbounded operators |
| No class | 03/08 | Recruitment week |
| Sec 10.2 | 03/11 | Adjoint of differential Opertors |
| Sec 10.2 | 03/13 | Adjoint of differential Opertors |
| Sec 10.3 | 03/15 | Green's functions |
| Sec 10.4 | 03/18 | Weak derivative theory |
| Sec 10.4 | 03/20 | Weak derivative theory |
| Exam 2 | 03/22 | Exam 2 |
| No class | 03/25 | Spring Break |
| No class | 03/27 | Spring Break |
| No class | 03/29 | Spring Break |
| Sec 10.5 | 04/01 | Sturm-Liouville Problem |
| Sec 12.1 | 04/03 | Measures |
| Sec 12.1/12.2 | 04/05 | Measures/measurable functions |
| Sec 12.2 | 04/08 | Measurable functions |
| NO CLASS | 04/10 | Class cancelled |
| Sec 12.3 | 04/12 | Lebesgue Integral |
| Sec 12.4 | 04/15 | Convergence Theorems |
| Sec 12.5 | 04/17 | Fubini's Theorem |
| Sec 12.6 | 04/19 | Lp Spaces |
| Sec 12.7 | 04/22 | Basic inequalities |
| Sec 12.8 | 04/24 | Dual of Lp |
| Sec 12.9 | 04/26 |
Lecture Times and Location
| Instructor | Room Number | Time |
|---|---|---|
| Rodriguez | ECCR 257 | MWF 12:00 to 12:50 |
Office Hours
| Instructor/TA | Room Number | Office Hours |
|---|---|---|
| Rodriguez | ECOT 235 | W: 2-4 pm and Th: 2-3 pm |
Homeworks
| Homework | Due Date | Notes |
|---|---|---|
| Homework 1 | Friday, Jan 18, 2019 by 4 pm | solutions in canvas |
| Homework 2 | Friday, Jan 25, 2019 by 4 pm | solutions in canvas |
| Homework 3 | Friday, Feb 1, 2019 by 4 pm | solutions in canvas |
| Homework 4 | Friday, Feb 8, 2019 by 4 pm | solutions in canvas |
| Homework 5 | Friday, Feb 15, 2019 by 4 pm | |
| Homework 6 | Friday, Feb 22, 2019 by 4 pm | solutions in canvas |
| Homework 7 | Friday, March 1, 2019 by 4 pm | solutions in canvas |
| Homework 8 | Friday, March 8, 2019 by 4 pm | solutions in canvas |
| Homework 9 | Friday, March 15, 2019 by 4 pm | solutions in canvas |
| Homework 10 | Friday, March 22, 2019 by 4 pm | solutions in canvas |
| Homework 11 | Friday, April 4, 2019 by 4 pm | |
| Homework 12 | Friday, April 12, 2019 by 4 pm | |
| Homework 13 | Friday, April 19, 2019 by 4 pm |
Exams
There will be two midterm. The tentative date for this are Friday, February 15, 2019 and March 22, 2019.. The final exam is in accordance with the standard CU Final Exam schedule and is Tuesday, Sunday, May 5, 2019 from 7:30-10:00 pm.
Notes
A total of 10 quizzes (about 10 minutes each) will be given at random during the semester.