M 383C/CSE 386C Methods of Applied Mathematics I.

M 383C, Unique #53650     and     CSE 386C, Unique #65155
Fall 2015

Instructor:

Prof. Todd Arbogast
Office: RLM 11.162, Phone: 512-471-0166
Office: POB 5.334, Phone: 512-475-8628
E-Mail: arbogast@ices.utexas.edu
Office hours: MW 12:30-1:50 p.m. Also, unless the instructor has pressing business, he will be available to help students who find him in one of his offices.

Teaching Assistant:

Mr. Sam Krupa
Office: RLM 11.138, Phone: 512-475-8827
E-Mail: skrupa@math.utexas.edu
Office hours: M 5:00-6:30 p.m. and Tu 3:45-5:15 p.m.

Textbook:

A bound set of lecturer-prepared notes (2010-2013 version) will be available for purchase from the UT Copy Center, McCombs location, GSB 3.136 (21st and Speedway) shortly before class begins. A recommended supplemental text is E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1978.

Meeting:

MWF 11:00-12:00 p.m., RLM 11.176.

Class Web Sites:

We will use the University's CLIPS and Canvas web site. Please check that your scores are recorded correctly in Canvas.

Homework, Exams, and Grades:

Homework will be assigned regularly. Students are encouraged to work in groups; however, each student must write up his or her own work. Two mid-term exams will be given in approximately weeks seven and twelve. The final exam will be comprehensive and given Saturday, December 12, 7:00-10:00 p.m. Grades will be recorded in the University Canvas system, so students can check their scores directly. The final grade will use the plus/minus system and be based on the homework and the three exams, with somewhat greater emphasis on the final exam.

Course Description:

This is the first semester of a course on methods of applied mathematics. It is open to mathematics, science, engineering, and finance students. It is suitable to prepare graduate students for the Applied Mathematics Preliminary Exam in mathematics and the Area A Preliminary Exam in CSEM. The first semester is an introduction to functional analysis.

Semester I.

  1. Preliminaries (0 weeks)
    1. Elementary Topology
    2. Lebesgue Measure and Integration
    3. Complex Contour Integration
  2. Normed Linear Spaces and Banach Spaces (6 weeks)
    1. Basic Concepts and Definitions
    2. Some Important Examples
    3. Hahn-Banach Theorems
    4. Applications of Hahn-Banach
    5. The Open Mapping Theorem
    6. Uniform Boundedness Principle
    7. The Embedding of X into its Double Dual X**
    8. Compactness and Weak Convergence in a NLS
    9. The Dual of an Operator
  3. Hilbert Spaces (2 weeks)
    1. Basic Properties of Inner-Products
    2. Best Approximation and Orthogonal Projections
    3. The Dual Space
    4. Orthonormal Subsets
    5. Weak Convergence in a Hilbert Space
  4. Spectral Theory and Compact Operators (4 weeks)
    1. Definitions of the Resolvent and Spectrum
    2. Basic Spectral Theory in Banach Spaces
    3. Compact Operators on a Banach Space
    4. Bounded Self-Adjoint Linear Operators on a Hilbert Space
    5. Compact Self-Adjoint Operators on a Hilbert Space
    6. The Ascoli-Arzela Theorem
    7. Sturm-Liouville Theory
  5. Distributions (2 weeks)
    1. The Notion of Generalized Functions
    2. Test Functions
    3. Distributions
    4. Operations with Distributions
    5. Convergence and Approximations to the Identity
    6. Some Applications to Linear Differential Equations
    7. Local Structure of D'
Semester II. (Generally, the following topics are covered.)
  1. The Fourier Transform
  2. Sobolev Spaces
  3. Boundary Value Problems
  4. Differential Calculus in Banach Spaces
  5. The Calculus of Variations

The University of Texas at Austin Student Honor Code:

"As a student of The University of Texas at Austin, I shall abide by the core values of the University and uphold academic integrity."

The University of Texas at Austin Code of Conduct:

The core values of The University of Texas at Austin are learning, discovery, freedom, leadership, individual opportunity, and responsibility. Each member of the university is expected to uphold these values through integrity, honesty, trust, fairness, and respect toward peers and community.

Students with Disabilities:

The University provides upon request appropriate academic accommodations for qualified students with disabilities. Contact the Office of the Dean of Students at 471-6259, 471-4641 TTY, and notify your instructor early in the semester.

Religious Holidays:

Appropriate academic accommodation for major religious holidays is provided upon request.

Emergency Classroom Evacuation:

Occupants of University of Texas buildings are required to evacuate when a fire alarm is activated. Alarm activation or announcement requires exiting and assembling outside. Familiarize yourself with all exit doors of each classroom and building you may occupy. Remember that the nearest exit door may not be the one you used when entering the building. Do not re-enter a building unless given instructions by the Austin Fire Department, the University Police Department, or the Fire Prevention Services office.

Counselling and Mental Health Services:

Available at the Counseling and Mental Health Center, Student Services Building (SSB), 5th floor, M-F 8:00 a.m. to 5:00 p.m., phone 512-471-3515, web site www.cmhc.utexas.edu.