Note: The information contained in this class syllabus is subject to change without notice. Students are expected to be aware of any additional course policies presented by the instructor during the course.

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%

Instructor's lecture note

Recommended Texts:
• Differential Equations, Dynamical Systems and Linear Algebra by M. Hirsch and S. Smale (available at Amazon or in the library)
• Ordinary Differential Equations by V. I. Arnold, M.I.T press, 1998 (paperback)
• Geometrical Methods in the Theory of Ordinary Differential Equations by V. I. Arnold, Springer Verlag, 2nd Edition 1988.
• Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields by J. Guckenheimer and P. Holmes (Applied Mathematical Sciences Vol 42) Springer Verlag.
• Mathematical Methods of Classical Mechanics by V. I. Arnold, Springer Verlag, 2nd

It is not necessary to buy anything of the references, though Ordinary Differential Equations by V. I. Arnold and Differential Equations, Dynamical Systems and Linear Algebra by M. Hirsch and S. Smale would be most useful to own. The books are useful for reference and lecture notes will be based on these texts and a variety of sources.

This course is an introduction to differential equations. We cover linear theory: existence and uniqueness for autonomous and non-autonomous equations; stability analysis; stable and unstable manifolds and floquet theory. We will also cover topics such as quasiperiodic motion; normal form theory; perturbation theory and classical mechanics.

Assessment: There will be one midterm (worth 20 points), a final exam (30 points) as well as 2 to 4 take-home problem sheets (to make up 50 points in total).

This course treats topics related to the solvability of various types of equations, and also of optimization and variational problems. The first half of the semester will concentrate on introductory material about norms, Banach and Hilbert spaces, etc. This will be used to obtain conditions for the solvability of linear equations, including the Fredholm alternative. The main focus will be on the theory for equations that typically arise in applications. In the second half of the course the contraction mapping theorem and its applications will be discussed. Also, topics to be covered may include finite dimensional implicit and inverse function theorems, and existence of solutions of initial value problems for ordinary differential equations and integral equations

Instructor's course website:
http://www.math.uh.edu/~gorb/math6360_f14.html

L.D. Berkovitz; Convexity and Optimization in R n . Wiley-Interscience, New York, 2001
I. Ekeland and R. Temam; Convex Analysis and Variational Problems. SIAM, Philadelphia, 1999

A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, 2nd edition, Texts in Applied Mathematics, V.37, Springer, 2010.

Recommended Texts:
• Mathematical Statistics with Applications, Sixth Edition, by Wackerly, Menden-
hall III and Scheaffer, Duxbury Press
• Elementary Probability for Applications (Cambridge U. Press, 2009) by Richard
Durrett.
• A First Look at Rigorous Probability Theory by Jeffrey Rosenthal, 2000.
• An Introduction to Probability Theory and Its Applications, Vol 1, 3rd edition, 1968 by William Feller (any edition would be fine).
• Probability by Leo Breiman, 1968, Addison- Wesley.
• An Introduction to Stochastic Modeling, 3rd Edition, by H Taylor and SKarlin, Academic Press.
• A First Course in Probability, Sixth Edition by Sheldon Ross, 2002, Prentice Hall

There is no core text and the lectures are based on a variety of sources Mathematical Statistics by Wackerly, Mendenhall and Scheaffer cover some of the material but at a more elementary level. Durrett is a good resouce for probability theory. Ross also is a standard text that treats many of these topics. An Introduction to Stochastic Modeling is useful for Markov Chain theory. Rosenthal’s book is good as an introduction to measure theory for students of probability. Feller’s book is a classic text and worth looking at. Breiman’s book is slightly more advanced than the other texts. There are many other texts that may be useful for reference and the library has a wide selection of books on probability.

Emphasis will be placed on a thorough understanding of the basic concepts of modern probability as well as developing problem solving skills.
Topics covered include: combinatorial analysis, independence and the Markov property, introduction to Markov chain theory, the major discrete and continuous distributions, joint distributions and conditional probability, modes of convergence. These notions will be examined through examples and applications.
Topics covered:
• Combinatorial analysis (sampling with, without replacement etc)
• Introduction to measure theory probability spaces, random variables, random variables. Expectation and integration of random variables. Markov and Chebychev’s inequality.
• Distribution of a random variable, distribution function, probability density function
• Major discrete distributions- Bernoulli, Binomial, Poisson, Geometric. Modeling with the major discrete distributions.
• De Moivre-Laplace limit theorem. Normal approximation to the Binomial, Poisson approximation (‘law of rare events’). Modeling with continuous distributions.
• Modes of convergence: in probability, in L p , almost surely, in distribution. Weak and strong law of large numbers, Moment generating functions and central limit theorem.
• Conditional probability- Bayes theorem. Discrete conditional distributions, continuous conditional distributions, conditional expectations and conditional probabilities. Conditional expectations, Martingales.

Required Texts:
1. Introduction to mathematical Finance: Discrete-Time Models, by Stanley R. Pliska, Blackwell, 1997.
2. Investment Science, 2nd edition, by David Luenberger, Oxford University Press, 2014.

Some initial questions for a student before joining this class may be:
- How do waves propagate and how is this phenomenon modelled and analyzed?
- What is an inverse problem in the context of wave propagation?
- How are inverse problems related to imaging, i.e., how do we image with waves?

It is the aim of this class to introduce the students to the mathematical and physical principles behind wave propagation and to discuss how the understanding of the associated inverse problems may be used to address questions from seismic imaging, radar/sonar imaging, or medical imaging.

An inverse problem is the “dual” to a direct problem associated to the same model. The simplest example of such dual pair, i.e., direct/inverse problems, is associated to a linear system: here the direct problem will be, for a given mxn matrix A and a given nx1 vector v determine the product Av while the inverse problem will be, for known Av, determine v. Thus, every time you solved a linear system you solved in fact an inverse problem! In another example, in various imaging modalities the direct problem is the analysis of propagation of input waves in the known host media while the inverse problem will be the determination of an unknown host media from the characteristics of an echo (reflected wave).

For medical imaging, the practical question will be to detect cancerous tumors, or other abnormalities present in a part of the human body by non-invasive methods, i.e., by using only data collected without direct contact with the patient. Thus, from the acquired data the scientist would like to invert and obtain information about the source of this data. The same scenario is valid for the problem of mapping the inner structure of the earth or the problem of oil exploration and recovery problems, where the engineer would like to be able to map the inner structure of the earth (thus detecting potential oil resources and their size) only from inexpensive data acquired at the surface of the earth.

For the radar/sonar imaging arena one needs first to understand that a radar or sonar form an image based on echoes received after initial impulses are generated towards a target. Then an interesting question is: how can one make certain objectives (aircrafts, buildings, ships) invisible to radar or sonar. A possible inverse problem approach to this question will inquire into the possibility to design suitable antennas to achieve the desired nulling around the object of interest while maintaining its communication capability.

Complete typed notes will be provided.

None

Monte Carlo Statistical Methods, by Christian P. Roberts and George Casella, Springer, 2004