This is a first semester of a two semester course.

The emphasis in MATH 4331 will be on 1-variable theory and results from "classical analysis". Topics covered will include infinite series, sequences, functions (continuous, analytic, smooth), uniform convergence, Weierstrass Approximation theorem, Fourier series, the Gamma-function and the Euler-Maclaurin formula. There will also be a fairly extensive introduction about the real number system.

For detailed information, visit
http://www.math.uh.edu/~mike/4331-4332/

Partial Differential Equations are used to model fundamental physical phenomena such as heat conduction, wave propagation, quantum mechanics, and electrostatics. In this course we will use the separation of variables method, and Fourier Series, to construct solutions. Boundary value problems with eigenvalues will prove to be important. We will introduce the Fourier Transform, especially as it applies to the equation for heat conduction. In addition we will stress how different pde's have very different properties.

This is a more theoretical and more mathematically rigorous course than 3363.

If a second semester (4336) is approved (this is not guaranteed), we will be able to study Green's functions, the Schroedinger equation, the wave equation in several space dimensions, and Distributions, Fourier transforms, and Laplace Transforms.

This is a first semester of a two semester course.

We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. The emphasis this semester will be on solving nonlinear equations, interpolation, numerical integration, initial value problems of ordinary differential equations, and direct methods for solving linear systems of algebraic equations. This is an introductory course and will be a mix of mathematics and computing.

Finite State Machines and Turing Machines serve as models of computations. We will study Kleene’s Theorem for a concise description of finite state machines. . We will also discuss the halting problem for Turing machines.

Course Organization and Grading: Students will receive on a regular basis homework assignments. There will be a midterm and a final. Homework counts for 30%, midterm for 30% and final for 40% of the final grade.

This course is an introduction to complex analysis. It will cover the theory of holomorphic functions, Cauchy theorem and Cauchy integral formula, residue theorem, harmonic and subharmonic functions, and other topics.

On-line course is taught through Blackboard Vista, visit http://www.uh.edu/webct/ for information on obtaining ID and password.

The course will be based on my notes and the textbook.

In each week, some lecture notes will be posted in Blackboard Vista, including homework assignment.

Homework will be turned in by the required date through Blackboard Vista. And the solution will be posted.

This is a first semester of a two semester course.

The emphasis in MATH 6312 will be on 1-variable theory and results from "classical analysis". Topics covered will include infinite series, sequences, functions (continuous, analytic, smooth), uniform convergence, Weierstrass Approximation theorem, Fourier series, the Gamma-function and the Euler-Maclaurin formula. There will also be a fairly extensive introduction about the real number system.

For detailed information, visit
http://www.math.uh.edu/~mike/4331-4332/

This is the first semester of a 2 semester sequence. This semester we will be developing the basic principles of measure and integration. This body of knowledge is essential to most parts of mathematics (in particular to analysis and probability) and falls within the category of "What every graduate student has to know". The one test and the final exam will be based on the notes given in class, and on the homework. After each chapter we will schedule a problem solving workshop, based on the homework assigned for that chapter. The most important part of your task as a graduate student in this course is simply to reread the class notes making sure you understand everything. Please ask me about anything you don't follow.

Final grade is approximately based on a total score of 400 points consisting of homework (100 points), a semester test (100 points), and a final exam (200 points). The instructor may change this at his discretion. The syllabus for the first semester will cover some but not all of the following topics: Measures. Measurable functions. Integration. Convergence of sequences of functions. The Lp spaces. Signed and complex measures. Product measures and Fubini's theorem. Differentiation and integration.

References:

• Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Volume 19, American Mathematical Society.
• Robert McOwen, Partial Differential Equations, 2nd edition, Prentice-Hall Inc.

Topics to be covered include basic analysis and calculus of weak and classical derivatives. The relevant Hilbert-Sobolev spaces will be described with some of properties proved. Then weak formulations of linear elliptic equations, the Lax-Milgram theorem and Garding’s inequality. Variational characterizations, spectral results and the spectral representation of solutions for self-adjoint problems. Regularity results, inequalities and maximum principles for classical solutions.

Results for linear parabolic equations will include energy inequalities and a priori bounds. Maximum principles and regularity results. Spectral and Galerkin methods for approximating solutions with various types of boundary conditions.

Students should know the basic definitions and results of metric spacetopology, matrices and finite dimensional linear algebra.

Reproducing kernel Hilbert spaces play an important role in a number of areas of mathematics, including statistics, approximation theory, harmonic analysis, complex analysis and operator theory.

In this course we will develop their basic properties, study special spaces, including the Hardy and Bergman spaces. We will then look at multipliers on these spaces and use their theory to prove the classical Nevanlinna-Pick interpolation theorem. Finally, we will develop the theory of vector-valued reproducing kernel Hilbert spaces, linear fractional maps on matrix balls and use these results to prove the matrix-valued version of the Nevanlinna-Pick theorem.

This course will provide an introduction to the theory of wavelets and itsapplications in mathematics and signal processing. Some of the topics I will address are:

• Orthonormal bases and frames: A basic problem in mathematics andengineering is to represent a function or a signal as superposition ofelementary components. I will introduce the theory of frames and show thatit provides the general framework to address this problem. Orthonormalbases are a special example of frames.
• Wavelet bases: The first wavelet basis, the Haar basis, was discovered in1909 before wavelet theory was born. Unfortunately, the elements of thisbasis are not continuous. The success of the wavelet theory is due to theability to construct a variety of wavelet bases with very nicemathematical properties such as smoothness, compact support, vanishmoments, etc. I will present several examples of wavelet bases anddescribe what kind of features are desirable in such a basis.
• Multiresolution Analysis: Multiresolution analysis is a general method forconstructing wavelet bases. I will describe how to use this approach toconstruct the Shannon wavelets, the Daubechies wavelets and the splinewavelets.
• Wavelets and approximation theory: One striking feature of wavelets istheir ability to represent function with discontinuities. In fact waveletshave optimal approximation properties for several classes of functions andsignals. I will introduce linear and nonlinear approximations, examine theapproximation properties of wavelets and compare them to Fourier methods.
• Wavelets and signal processing: Wavelets appear today in a variety ofadvanced signal processing applications, including analysis anddiagnostics, quantization and compression, transmission and storage, noisereduction and removal. I will describe the connection between wavelettheory and filter banks theory in signal processing. I will presentapplications of wavelets to data/image compression and denoising. Some ofthis applications will be further explored by the students as individualor group projects.

While deterministic models of biological systems can offer valuable insights into their function and behavior, they do not fully capture the effects of randomness and variability which are fundamental features of nearly all biological systems. In this course we will apply the theory of probability and stochastic processes to models of biological systems. Students taking the course should be comfortable with multivariate calculus, differential equations and linear algebra.

Topics to be covered include: a review of probability, including numerical techniques for generating random samples, Markov processes with discrete and continuous space vari-ables, diffusion processes, Wiener and Ornstein-Uhlenbeck processes, point processes, Gillespie's algorithm and other algorithms for simulating stochastic processes and their application in biology, statistical analysis of time series, power spectra of random processes. A portion of the course will be devoted to numerical simulations of stochastic systems using MATLAB.

Stochastic differential equations arise when some randomness is allowed in the coefficients of a differential equation. They have many applications, including mathematical biology, theory of partial differential equations, differential geometry and mathematical finance.

This is an introduction to the theory and applications of stochastic differential equations. A knowledge of measure theory is strongly recommended but not required. First we will review measure theory, probability spaces, random variables and stochastic processes. Brownian motion will be discussed in some detail before we introduce the Ito integral and relevant aspects of martingale theory as a method to formulate and solve stochastic differential equations. Numerical schemes will be also discussed. Applications will include mathematical finance (arbitrage and option pricing) and PDE's.

Deformable mathematical models of smooth curves and surfaces in R3 have important applications to the modeling of soft organs by analysis of 3D medical images. We will present the widely used NURBS (non uniform rational B-spline) models and show how the minimization of deformation energy leads to the quantitative comparison of "soft shapes".