An Introduction to Stochastic Modeling" by Mark Pinsky, Samuel Karlin. Academic Press, Fourth Edition.
ISBN-10: 9780123814166
ISBN-13: 978-0123814166

We study the theory and applications of stochastic processes. Topics include discrete-time and continuous-time Markov chains, Poisson process, branching process, Brownian motion. Considerable emphasis will be given to applications and examples.

This first course in the sequence Math 4331-4332 provides a solid introduction to deeper properties of the real numbers, continuous functions, differentiability and integration needed for advanced study in mathematics, science and engineering. It is assumed that the student is familiar with the material of Math 3333, including an introduction to the real numbers, basic properties of continuous and differentiable functions on the real line, and an ability to do epsilon-delta proofs.

Topics: Open and closed sets, compact and connected sets, convergence of sequences, Cauchy sequences and completeness, properties of continuous functions, fixed points and the contraction mapping principle, differentiation and integration.

MATH 3331 or equivalent, and three additional hours of 3000-4000 level Mathematics. Previous exposure to Partial Differential Equations (Math 3363) is recommended.

"Partial Differential Equations: An Introduction (second edition)," by Walter A. Strauss, published by Wiley, ISBN-13 978-0470-05456-7

Description:Initial and boundary value problems, waves and diffusions, reflections, boundary values, Fourier series.

Instructor's Description: will cover the first 6 chapters of the textbook. See the departmental course description.

MATH 2331, In depth knowledge of Math 3331 (Differential Equations) or Math 3321 (Engineering Mathematics)

*Ability to do computer assignments in FORTRAN, C, Matlab, Pascal, Mathematica or Maple.

This is an one semester course which introduces core areas of numerical analysis and scientific computing along with basic themes such as solving nonlinear equations, interpolation and splines fitting, curve fitting, numerical differentiation and integration, initial value problems of ordinary differential equations, direct methods for solving linear systems of equations, and finite-difference approximation to a two-points boundary value problem. This is an introductory course and will be a mix of mathematics and computing.

MATH 2331, In depth knowledge of Math 3331 (Differential Equations) or Math 3321 (Engineering Mathematics)

*The students in this online section will be introduced to scientific computing in Excel. All computations in this course will be done either directly in the Excel spreadsheet, or via VBA programming. Consequently, the course will also include instruction associated with the use of Excel and programming in VBA. Students are expected to have access and basic familiarity with Excel, but they are not expected to know advanced spreadsheet functionality or have programming experience with VBA. Excel and VBA will be incorporated into nearly every lesson and assignment

This is an one semester course which introduces core areas of numerical analysis and scientific computing along with basic themes such as solving nonlinear equations, interpolation and splines fitting, curve fitting, numerical differentiation and integration, initial value problems of ordinary differential equations, direct methods for solving linear systems of equations, and finite-difference approximation to a two-points boundary value problem. This is an introductory course and will be a mix of mathematics and computing.

MATH 2331 , or equivalent, and six additional hours of 3000-4000 level Mathematics.

Conditioning and stability of linear systems, matrix factorizations, direct and iterative methods for solving linear systesm, computing eigenvalues and eigenvectors, introduction to linear and nonlinear optimization 

Graduate standing.

Linear Algebra Using MATLAB, Selected material from the text Linear Algebra and Differential Equations Using Matlab by Martin Golubitsky and Michael Dellnitz)
The text  will made available to enrolled students free of charge.

Software: Scientific Note Book (SNB) 5.5  (available through MacKichan Software, http://www.mackichan.com/)

Syllabus: Chapter 1 (1.1, 1.3, 1.4), Chapter 2 (2.1-2.5), Chapter 3 (3.1-3.8), Chapter 4 (4.1-4.4), Chapter 5 (5.1-5.2, 5.4-5-6), Chapter 6 (6.1-6.4), Chapter 7 (7.1-7.4), Chapter 8 (8.1)

Project: Applications of linear algebra to demographics. To be completed by the end of the semester as part of the final.

Course Description: Solving Linear Systems of Equations, Linear Maps and Matrix Algebra, Determinants and Eigenvalues, Vector Spaces, Linear Maps, Orthogonality, Symmetric Matrices, Spectral Theorem

Students will also learn how to use the computer algebra portion of SNB for completing the project. 

Homework: Weekly assignments to be emailed as SNB file.

There will be two tests and a Final.

Grading: Tests count for 90% (25+25+40), HW 10%

MATH 2331, In depth knowledge of Math 3331 (Differential Equations) or Math 3321 (Engineering Mathematics)

*The students in this online section will be introduced to scientific computing in Excel. All computations in this course will be done either directly in the Excel spreadsheet, or via VBA programming. Consequently, the course will also include instruction associated with the use of Excel and programming in VBA. Students are expected to have access and basic familiarity with Excel, but they are not expected to know advanced spreadsheet functionality or have programming experience with VBA. Excel and VBA will be incorporated into nearly every lesson and assignment

Numerical Analysis (9th edition), by R.L. Burden and J.D. Faires, Brooks-Cole Publishers, 9780538733519

This is an one semester course which introduces core areas of numerical analysis and scientific computing along with basic themes such as solving nonlinear equations, interpolation and splines fitting, curve fitting, numerical differentiation and integration, initial value problems of ordinary differential equations, direct methods for solving linear systems of equations, and finite-difference approximation to a two-points boundary value problem. This is an introductory course and will be a mix of mathematics and computing.

Required Text: Abstract Algebra by David S. Dummit and Richard M. Foote, ISBN: 9780471433347

This book is encyclopedic with good examples and it is one of the few books that includes material for all of the four main topics we will cover: groups, rings, field, and modules.  While some students find it difficult to learn solely from this book, it does provide a nice resource to be used in parallel with class notes or other sources.

Graduate standing. MATH 2331 and minimum of 3 semester hours of 3000 level mathematics.

This first course in the sequence Math 4331-4332 provides a solid introduction to deeper properties of the real numbers, continuous functions, differentiability and integration needed for advanced study in mathematics, science and engineering. It is assumed that the student is familiar with the material of Math 3333, including an introduction to the real numbers, basic properties of continuous and differentiable functions on the real line, and an ability to do epsilon-delta proofs.

Topics: Open and closed sets, compact and connected sets, convergence of sequences, Cauchy sequences and completeness, properties of continuous functions, fixed points and the contraction mapping principle, differentiation and integration.

Graduate Standing and MATH 4331. In depth knowledge of Math 3333 required.

 No textbook required. Lecture notes provided

This is a year-long course. The first semester part is an introduction to complex analysis. This two semester course will cover the theory of holomorphic functions, residue theorem, harmonic and subharmonic functions, Schwarz's lemma, Riemann mapping theorem, Casorati-Weterstrass theorem, infinite product, Weierstrass' (factorization) theorem, little and big Picard Theorems and compact Riemann surfaces theory

Required: The instructor will provide notes on much of the material for the course and there is no prescribed text.

Recommended Texts: The course will cover the material in the first three chapters of ”Hilbert Space Methods in Partial Differential Equations” by Ralph E. Showalter (Dover or free online) and some of the text ”Elliptic Equations: An Introductory Course”,by Michel Chipot, Birkhauser, 2009. The Universitext ”Functional Analysis, Sobolev Spaces and Partial Differential Equations” by Haim Brezis, Springer 2011 provides a thorough treatment of the functional analysis used. These three texts may be good reference texts for the material treated in the course.

Course Outline: This course will provide an introduction to the modern theory of elliptic partial differential equations using Sobolev space methods. The prerequisite is competence in multivariable calculus and real analysis. Ideally a student should have done well in M6320- 21 and having a working knowledge of Lebesgue integration and some Fourier analysis. The basic constructions of linear analysis in Hilbert and Banach spaces will be assumed known.

(Recommended) Topology, A First Course, J. R. Munkres, Second Edition, Prentice-Hall Publishers.

V. Runde A taste of topology , Springer Universitext (paperback, inexpensive).

Topology is the perfect course to take as a first year graduatestudent, since it does not contain too much material, or material that is too sophisticated(the typed notes for the course are about 47 pages). It is a central and  fundamental course and one which graduate students usually enjoy very much!

Thetopic is basically point-set topology, we will discuss a little algebraic topology at the end. We begin by discussing a little set theory, the basic definitions of topology and basis, and go on to discuss separation properties, compactness, connectedness, nets, continuity, local compactness, Urysohn's lemma, local compactness, Tietze's theorem, the characterization of separable metric spaces, paracompactness, partitions of unity, and basic constructions such as subspaces, quotients, and products and the Tychonoff theorem. At the end we will discuss a little algebraic topology, like simple connectedness and the fundamental group. 

You do not need a textbook, although I recommend the Munkres or the Rundebooks. You are expected to read the classnotes carefully each week, line by line, and bring to me the things you don't understand there. Classnotes will be put on the web. You are also expected to do most of the homework sets, and turn in selected homework problems for grading.

You are encouraged to work with others, form study groups, and so on, however copied turned in homework will not help you assimilate the material, and will not be graded. The final grade is aproximately based on a total score of 300 points consisting of homework and other assignments (100 points), a semester test (100 points), and a final exam (100 points). That is, the final exam will count as much as the semester test; and it may be a take-home exam due a different date than the officially scheduled time listed on the UH website for the final exam for this course. In the final week of the semester you will be given time to work on a project, which you will be able to choose.

Graduate standing. 

Two years of Calculus, MATH 6308 (Advanced Linear Algebra I), and MATH 6383 (Probability Statistics), or consent of instructor

Applied Linear Statistical Models 5th Edition, by Michael Kutner, Christopher Nachtsheim, John Neter, William Li 

Graduate standing.

Being familiar with at least one programming language. An undergraduate-level understanding of probability, statistics, calculus and linear algebra is assumed.

- Introductory Statistics with R, 2nd ed., Authors: Dalgaard, Peter, ISBN: 9780387790534

- Statistical Computing with R, Maria L. Rizzo, ISBN: 9781584885450

• use R statistical programming as a tool for conducting research and data analysis

• comprehend and implement the main statistical computing techniques for their research goals

• writing their own R functions and potentially developing new computational methods in the future

J.K. Hunter and B. Nachtergaele, Applied Analysis, World Scientific, (2005). ISBN: 9789812705433

 A.W. Naylor and G.R. Sell, Linear Operator Theory in Engineering and Science,  Springer. ISBN: 9780387950013

Introduction to Nonlinear Optimization Theory, Algorithms, and Applications with MATLAB; by Amir Beck, SIAM. ISBN: 9781611973648

Recommended Texts :
- A First Look at Rigorous Probability Theory by Jeffrey Rosenthal, 2000..
- An Intermediate Course in Probability Theory by Allan Gut, Springer 2009 (any edition)

Review of Undergraduate Probability:
- A First Course in Probability, 6th Edit. by Sheldon Ross, 2002, Prentice Hall

Complementary Texts for further reading:
- Probability: theory and Examples, 3rd Edit., Richard Durrett, Duxbury Press
- An Introduction to Probability Theory and Its Applications, Vol 1,  by William Feller
- Probability by Leo Breiman, 1968, Addison-Wesley 

General Background (A).
(1) Combinatorial analysis and axioms of probability
(2) Elementary random variables theory: expectation, variance, moments, distribution
function, probability density functions, impact of change of variable on density functions
(3) Major discrete probability distributions: Bernoulli, Binomial, Poisson, Geometric
Major continuous probability distributions: Uniform, Normal, Exponential
(4) Basic Modelling Applications
(5) Conditional probability: Bayes formula, Independence, Conditional Expectation,
Conditional density function, Conditional Probability distribution, Independent identically distributed random variables
(6) Joint distributions, joint density functions, marginal distributions, marginal densities, covariances and correlation coefficients
(7) Moment generating functions, Characteristic functions,

Measure theory (B).
(1) Elementary measure theory : Boolean algebras, probability spaces , continuity of probabilities, Borel-Cantelli lemma, Chebychevs inequality,
(2) Convergence of random variables: Almost sure convergence, Convergence in distribution, Law of Large Numbers, Central Limit theorem

Markov chains and random walks (C).
Markov chain theory for finite or countable state spaces
(1) Markov property and Transition matrix, Irreducibility
(2) First hitting times, Transience, Recurrence ,
(3) Stationary distributions : existence theorems and computation
(4) Random walks on Z and Z2 as Markov chains; Gambler's ruin problem

Introduction to Mathematical Finance: Discrete-time Models, by Stanley Pliska, Blackwell, 1997.

Graduate standing.

There is no official textbook for this course. Relevant references for each topic will be suggested during lectures

Topological groups; continuous unitary representations; positive definite functions; representations of compact groups; Peter-Weyl theorem; unitary representations of discrete groups; weak containment; amenability; property T.

Stochastic differential equations arise when some randomness is allowed in the coefficients of a differential equation. They have many applications, including mathematical finance, mathematical biology, theory of partial differential equations and differential geometry. One emphasis of the course is understanding how the theory of stochastic integration can be used in modeling processes that have randomness.

This course is an introduction to the theory and applications of stochastic differential equations. Special attention will be devoted to their applications in mathematical finance.

We will review probability spaces, random variables and stochastic processes. We introduce Brownian motion, Ito integration and relevant aspects of martingale theory, Ito's change of variable formula. We then formulate and solve stochastic differential equations.

As applications to mathematical finance we will discuss the Girsanov theorem, arbitrage and option pricing (including the Black-Scholes equation), portfolio optimization, stochastic integrals with jumps, the Heston model. We will also discuss relations between SDE's and PDE's.

Graduate standing.  

It is strongly encouraged that students have a background in Linear Algebra and Real Analysis. Some programming experience or some willingness to learn. Prior knowledge of Matlab not required.

• There is no official textbook.
• I will select material from: A basis theory primer, by C. Heil, 2011,  A wavelet tour of signal processing, by S. Mallat, Third edition, 2009 A First Course on Wavelets , by E. Hernandez and G. Weiss, 1996, Fourier analysis and applications, by C. Gasquet and P. Witomski
• Notes and papers will be provided by the instructor.

We live in a data-intensive age which is bringing significant changes in the process of scientific discovery. During the last decade, sparsity has emerged as a leading theme in connection with the goal to produce faster and simpler algorithms for a wide range of signal processing applications. By enabling to accurately approximate functions in a certain class using a relatively small number of nonzero coefficients, sparse representations are able to reveal the essential information we are looking for in the data. Therefore, sparsity implies not only data compression. Understanding the sparsity of a given data type entails a precise knowledge of the modelling and approximation of that data type. This knowledge is essential to design the most efficient algorithms for a tasks such as classification, denoising, interpolation, and segmentation. Multiscale techniques based on wavelets and their generalizations have emerged in the last decade as the most successful approach for sparse signal representations, as testified, for example, by their use in the new FBI fingerprint database and in JPEG2000, the new standard for image compression. Multiscale techniques were also extended beyond the traditional setting of physical spaces allowing for the efficient analysis of general structures, such as manifolds, graphs and point clouds in Euclidean space. The aim of this course is to provide the mathematical tools to understand multiscale representations starting from the setting of traditional wavelets up to more advanced and emerging constructions such as curvelets, shearlets and diffusion wavelets. Applications of these ideas will also be presented.

This course will provide an introduction to the theory of wavelets and its applications in mathematics.

Background: Orthonormal bases and frames: A basic problem in mathematics and engineering is to represent a function or a signal as superposition of elementary components. I will introduce the theory of frames and show that it provides the general framework to address this problem. Orthonormal bases are a special example of frames. 

Background: Elements of Fourier analysis: I will review basic elements of Fourier analysis, including Fourier series and Fourier transforms,

Wavelet bases: The first wavelet basis, the Haar basis, was discovered in 1909 before wavelet theory was born. Unfortunately, the elements of this basis are not continuous. The success of the wavelet theory is due to the ability to construct a variety of wavelet bases with very nice mathematical properties such as smoothness, compact support, vanish moments, etc. I will present several examples of wavelet bases and describe what kind of features are desirable in such a basis. 

Multiresolution Analysis: Multiresolution analysis is a general method for constructing wavelet bases. I will describe how to use this approach to construct Shannon wavelets, Daubechies wavelets and Meyer wavelets.

Sparse compression and approximation theory: One striking feature of wavelets is their ability to represent function with discontinuities. In fact wavelets have optimal approximation properties for several classes of functions and signals. I will introduce linear and nonlinear approximations, examine the approximation properties of wavelets and compare them to Fourier methods.

Wavelets and signal processing: Wavelets appear today in a variety of advanced signal processing applications, including analysis and diagnostics, quantization and compression, transmission and storage, noise reduction and removal. I will describe the connection between wavelet theory and filter banks theory in signal processing. I will present applications of wavelets to data/image compression and denoising. Some of this applications will be further explored by the students as individual or group projects.

1. Basic anatomy and physiology of the cardiovascular system;

2. Elements of fluid dynamics and modeling blood flow

3. Elements of elasticity theory and modeling arterial walls;

4. Fluid-structure interaction;

5. 1D reduced models of blood flow;

6. 2D and 3D fluid-structure interaction;

7. Overview of numerical schemes for FSI modeling cardiovascular flow.