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 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.

"Matrix Analysis and Applied Linear Algebra" by Carl D. Meyer, published by SIAM,
ISBN 978-0-898714-54-8

This is a two semester sequence. The first semester will cover chapters 1-5 of the text. Topics include:

- Linear Systems of Equations and Gaussian Elimination
- Matrix algebra
- Vector Spaces
- Norms, Inner Products, and Orthogonality

These topics will be covered with more depth and difficulty than in Math 2331.

We will discuss various applications including two-point boundary value problems, Electrical Circuits, Least-Squares approximation, the Discrete Fourier Transform, and the Fast Fourier Transform.

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%

"Matrix Analysis and Applied Linear Algebra" by Carl D. Meyer, published by SIAM,
ISBN 978-0-898714-54-8

This is a two semester sequence. The first semester will cover chapters 1-5
of the text. Topics include:

- Linear Systems of Equations and Gaussian Elimination
- Matrix algebra
- Vector Spaces
- Norms, Inner Products, and Orthogonality

These topics will be covered with more depth and difficulty than in Math 2331.

We will discuss various applications including two-point boundary value problems, Electrical Circuits, Least-Squares approximation, the Discrete Fourier Transform, and the Fast Fourier Transform.

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.

Convex Optimization, Stephen Boyd and Lieven Vandenberghe, Cambridge University Press, 2004 (available on the web at http://www.stanford.edu/ ̃boyd/cvxbook.html) 

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

We will begin with the notes of L. C. Evans (UC Berkeley), that used to be available on his web-page. Additional material will be handed out or placed on reserve in the library during the course.

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. We begin by reviewing measure theory, probability spaces, random variables and stochastic processes. We discuss Brownian motion and its properties, then introduce the Ito integral and relevant aspects of martingale theory. We formulate and solve stochastic differential equations, including numerical schemes. Applications will include mathematical finance (arbitrage and option pricing) and connections to PDE's.

Convexity is a simple idea that is used in very many parts of mathematics, sometimes in surprising ways. The field has a very rich structure and theory, with numerous powerful applications.    We will be touching on several topics, namely a subset of the list below (this list will be pruned to fit the needs and mathematical maturity of the class).  We also note that some topics in this list may be out of order.  For each of these we will develop the basic theory, and illustrate it with selected applications. We will begin with convex sets in finite dimensional spaces, and the theorems of Caratheodary, Radon, and Helly.  Supporting and separating hyperplanes.  Polyhedra and Polytopes. Applications: Schur-Horn and Birkhoff-von Neumann theorems. Blaschke selection theorem.  Duality.  Krein-Milman and Milman theorems. Convex functions.  The affine function space $A(K)$.  Cones and ordered spaces. Application: Positive definite functions and Bochner's theorem.  Facial structure of convex sets.   Fixed point theorems. Representing measures and maximal measures.  Application: Solution of the one-dimensional moment problem.   The barycenter formula.

The Bishop-Phelps and Choquet-Bishop-deLeeuw theorem.  The Choquet and Shilov boundary.  The noncommutative Choquet boundary.  Structure of compact convex sets and separation theorems. Choquet (and other) simplexes.  Applications e.g. to measures and ergodic theory.  Some basics of potential theory.

This course is designed for graduate students who have been exposed to basic probability and statistics and would like to learn more advanced topics of design and analysis of experiments with applications to biological studies, public health and industries. The selected topics will include review of linear regressions, completely randomlized design, randomized block design, factorial designs, etc.. The instructor reserves the right to exclude certain topics from the textbook and add other topics not covered in the textbook.

Grading. Final grades will be based on class attendance and in-class discussion (10%), assignment (30%), midterm exams (30% each), the final research project (written report,30%).

R software
R is a open source statistical analysis software, and can be downloaded for free
At http://www.r-project.org/ .
SAS is an industry-standard software that are used and recommended by pharmaceutical companies for analzing clinical trial data for reports to the FDA.

Recommended book(s):
Maria Rizzo: Statistical Computing with R (Chapman & Hall/CRC The R Series) 2007
ISBN-13: 978-1584885450 ISBN-10: 1584885459 Edition: 1st

References:
Efron, B and Tibshirani, R. An Introduction to the Bootstrap, Chapman Hall / CRC

This course is designed for graduate students who have been exposed to basic probability and statistics and would like to learn more advanced statistical computing techniques in modelling data. The selected topics will include basic sampling techniques from known probability distributions, Monte Carlo estimation and testing, bootstrapping and jackknife, permutation methods for testing, etc. The instructor reserves the right to exclude certain topics from the textbook and add other topics not covered in the textbook.

Grading. Final grades will be based on class attendance and in-class discussion (10%), assignment (30%), midterm exams (30% each), the final research project (written report,30%).

R software
R is a open source statistical analysis software, and can be downloaded for free
At http://www.r-project.org/ .