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 and exciting area of applied mathematics which deals the representation of signals and images. It addresses fundamental and challenging questions like: how to efficiently and robustly store or transmit an image or a voice signal? how to remove unwanted noise and artifacts from data? how to identify features of interests in a signal? Students will learn the basic theory of Fourier series and wavelets which are omnipresent in a variety of emerging applications and technologies including image and video compression, electronic surveillance, remote sensing and data transmission. Some specific applications will also be discussed in the course.

This course is an introduction to Analysis. It will cover limit, continuity, differentiation and integration for functions of one variable and functions of several variables, and some selected applications. More precisely, it will cover the textbook from the chapter 3 to the chapter 7 (skip the section 15 and the section 24).

On-line course is taught through Blackboard Learn, visit https://accessuh.uh.edu/login.php for information on obtaining ID and password.

Homework: Homework will be submitted through Blackboard Learn by pdf file. The deadline for each homework assignment can be found in Blackboard Learn. No late homework assignments accepted.

Exams: There are two exams. The mid-term exam, and the comprehensive final exam. The dates are to be dertermined

The instructor has written lecture notes for this course that will be available to students in the class. He may not follow the lecture notes closely, but the course will cover most of the topics in the notes.

References: Good texts that cover most of the material include
L.D. Berkowitz, "Convexity and Optimization in Rⁿ", John Wiley, (2002).
D.G. Luenberger, "Linear and Nonlinear Programming", 2nd edition, Addi-son Wesley (1984).

Graduate standing or consent of the instructor.

Students are expected to have a good grounding in advanced calculus, introductory probability theory, and matrix-vector algebra. Having prior knowledge on dynamic systems theory,  control, optimization, or operations research is useful but not mandatory.

This course will cover a wide range of topics in stochastic processes and applied
probability. Main emphasis will be on applied topics in continuous-time stochastic
processes and stochastic differential equations (SDEs).

The following topics will be covered - continuous time Markov chains, averaging of fast
sub-system in Markov chains, estimation of transition probability matrix from data,
application of Markov chains to particle systems and cellular automata, multiscale SDEs,
avergaing and homogenization for multiscale SDEs.

Lecture notes will be comprehensive and no texts are required, though Billingley's or Durrett's book would be a worthwhile purchase.

Reference Texts:

- Probability: Theory and Examples, Richard Durrett, Any edition will do.
  The most recent edition is the 4th edition but 2nd or 3rd would also be fine.
- Large Deviations, Techniques and Applications, A. Dembo and O. Zeitouni, 2nd edition
- Extremes and related properties of random sequences and processes, M. R. Leadbetter, G. Lindgren and H. Rootzen, 1983 (Springer-Verlag)
- Probability and Measure, P. Billingsley, 3rd edition

This is a graduate course topics of current research in probability. It will assume a background in graduate level real analysis. First we will review classical limit theorems such as: the central limit theorem, stable laws, law of the iterated logarithm and ergodic theorems. Then we will consider extreme value theory and large deviations.

Extreme value theory plays a central role in applied statistics, and estimates the probability of unusual events such as oods, hurricanes or high animal population levels. It is concerned with the behavior of the sequence of successive maxima of a time series.

Large deviation results estimate the probability of outliers in the convergence of probability distributions.

Such estimates are useful in a variety of applications, including statistical mechanics,  information theory and engineering. Like the theory of extreme values, large deviations are used to estimate risk. In fact the pioneering work of the statistician Cramer in large deviations theory was motivated by applications to the insurance industry. This course will develop the fundamentals of extreme value and large deviation theory, motivated by applications.

Topics will include:
1. Characteristic functions and classical limit theorems
2. Large deviations and applications
3. Extreme value theory and applications

Automatic Learning of unknown functional relationships Y = f(X) between an output  Y and high-dimensional inputs X , involves algorithms dedicated  to the intensive analysis of large finite "training sets"  of "examples" of inputs/outputs pairs (X,Y) to discover efficient approximations of the function f. Automatic learning was first applied  to emulate intelligent tasks involving identification  of complex patterns such as shapes, sounds and speech, handwriting, texts, etc.  and has then been widely extended to the analysis of classification / proteomics and genes interactions data, as well as to smart mining of massive data sets gathered on the Internet. 

The course will study  major machine learning algorithms such as Support Vector Machines, Decision Trees, Random Forests , Artificial Neural Nets, etc. with emphasis on  key conceptual features  of automatic learning such as generalisation capacity and functional approximation.  We will present selected applications to proteomics and to automated pattern classification .

Homework and exams :  Students familiar with Mathlab or equivalent scientific softwares will have the possibility to replace a large part of the homework assignments and of the exams by applied projects involving computer simulations and algorithms implementation .