Lecture Note

Recommended Text:
"Graph Theory with Applications" by Bondy and Murty (available online)

This course is a continuation of MATH 4335. The following topics will be covered: PDEs and boundary value problems in multi-dimensions, Green's functions, Fourier Transform, Spectral methods, Nonlinear conservation laws.

Content:

Chapter 7: Green's Identities and Green's Functions

7.1 Green's First Identity
7.2 Green's Second Identity
7.3 Green's Functions
7.4 Half-Space and Sphere

Chapter 9: Waves in Space

9.1 Energy and Causality
9.2 The Wave Equation in Space-Time
9.3 Rays, Singularities, and Sources

Chapter 10: Boundaries in the Plane and in Space

10.1 Fourier's Method, Revisited
10.2 Vibrations of a Drumhead
10.3 Solid Vibrations in a Ball

Chapter 11: General Eigenvalue Problems

11.1 The Eigenvalues Are Minima of the Potential Energy
11.2 Computation of Eigenvalues
11.3 Completeness
11.4 Symmetric Differential Operators
11.5 Completeness and Separation of Variables
11.6 Asymptotics of the Eigenvalues

Chapter 12: Distributions and Transforms
12.1 Distributions
12.2 Green's Functions
12.3 Fourier Transform
12.4 Source functions
12.5 Laplace Transform Techniques

Chapter 14(optional) Nonlinear PDE

This course is a self-contained introduction to a very active area of applied mathematics, the representation of signals and images. The study of such representations is motivated by challenging questions: How can we efficiently and robustly store or transmit an image or a voice signal? How do we remove unwanted noise and artifacts from data? How can we identify features of interests in a signal? Students will learn the basic theory of Fourier series and wavelets which are present in a variety of applications and technologies including image and video compression, electronic surveillance, remote sensing and data transmission. 

Topics include: Inner product spaces, Schwarz and triangle inequalities, orthogonal projections and the least squares fit, Fourier series and transform, computation of Fourier series, convergence of Fourier series, convolutions, linear filters, the sampling theorem, analog to digital and digital to analog conversions. analog and digital filters, the Discrete Fourier transform (DFT), FFT, its use for the approximate computation of integral Fourier transforms, the Haar wavelet, Multiresolution Analysis, properties of the scaling function, decomposition and reconstruction algorithms, wavelet design in the frequency domain, the Daubechies wavelet.

This is the second semester of a two-semester sequence in graduate real analysis. After covering measure theory and Lebesgue integration during the first semester, this semester will be devoted mostly to the following topics: Lp spaces and Hilbert spaces, convolutions, Fourier transform and Fourier series, differentiation.

Course outline:
 Lp spaces and Hilbert spaces
 Abstract measures
 Convolution
 Fourier transform and Fourier series
 Differentiation

1. Numerical Methods in Scientific Computing, Volume 1
Society for Industrial Mathematics (September 4, 2008)
Germund Dahlquist, Ake Bjorck
ISBN 978-0-898716-44-3

2. A First Course in the Numerical Analysis of Differential Equations, 2nd Edition
Cambridge University Press; 2nd edition (December 29, 2008)
Arieh Iserles
ISBN 978-0-521734-90-5

Lecture Notes will be handed out for the first 16 lectures

Complementary Reading Assignments will refer to the contents of  specific chapters extracted from

Undergraduate level :
- "Introductory Probability" by Sheldon Ross
- "Statistics" by David Freemann, Robert Pisani, Roger Purves 

graduate level
- "Intermediary course in probability " by Allan Gut
- " Statistics ..."  by G. Casella

M.A.Olshanskii "Lecture notes on multigrid methods" Moscow: INM RAS, 2012(available free online)

Optional text: W. Hackbusch: Multi-Grid Methods and Applications, Springer, 1985

Nowadays, every researcher working with the numerical solution of partial differential equations should at least be familiar with this powerful approach. This course introduces to multigrid methods and their applications in computational physics. We shall study geometric multigrid methods, including classical V- and W-cycles as well as additive multigrid methods and algebraic multigrids. Applications will be considered to basic PDEs as well as to various fluids models, and Maxwell equations.

The course complements standard numerical analysis (Math 4364), Partial Differential Equations (Math 4335), Advanced Linear Algebra I & II (Math 4377 - 4378). The latter two courses are desirable, but not pre-requested though, since elementary introduction to necessary concepts will be given.

The final grade will be based on accomplishment of homework assignments.

This course will start with the representation theory of finite groups:
induced representations, irreducible representations, the left regular representation, group algebras and character theory. We will continue with the representations of compact topological groups and some classical Lie groups (e.g., the 2 x 2 invertible matrices). We will discuss the theory behind the finite dimensional representation of Lie groups: Cartan sub-algebras, root systems, highest weight representations, the Weyl character formula.

It will be assumed that the student has a good foundation in linear algebra. Familiarity with the basics of topology and measure theory is needed for the second part of the course.

View additional information :
http://www.math.uh.edu/%7etorok/math_6397_Groups/