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 year-long course will introduce the theory of the geometry of curves and surfaces in three-dimensional space using calculus techniques, exhibiting the interplay between local and global quantities.

Topics include: curves in the plane and in space, global properties of curves and surfaces in three dimensions, the first fundamental form, curvature of surfaces, Gaussian curvature and the Gaussian map, geodesics, minimal surfaces, Gauss' Theorem Egregium, Gauss-Bonnet theorem etc.

This course is designed to provide a college-level experience in history of mathematics. Students will understand some critical historical mathematics events, such as creation of classical Greek mathematics, and development of calculus; recognize notable mathematicians and the impact of their discoveries, such as Fermat, Descartes, Newton and Leibniz, Euler and Gauss; understand the development of certain mathematical topics, such as Pythagoras theorem, the real number theory and calculus.

Aims of the course: To help students
to understand the history of mathematics;
to attain an orientation in the history and philosophy of mathematics;
to gain an appreciation for our ancestor's effort and great contribution;
to gain an appreciation for the current state of mathematics;
to obtain inspiration for mathematical education,
and to obtain inspiration for further development of mathematics.

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. The textbook is used for extra reading, do homework or do project (write essays).

In each week, three chapters of my notes will be posted on Monday, Wednesday and Friday in Blackboard Vista. Weekly homework and reading assignment may be posted in Blackboard Vista in Friday, including projects (essays). Turn all your homework  by next Friday through Blackboard Vista.

All homework, essays or exam paper, handwriting or typed,  should be turned into PDF files and be submitted through Blackboard Vista..

There is one final exam in multiple choice.

Grading: 30% homework, 50% projects, 20 % Final exam.

By petition, this course will count toward  major or minor requirements in mathematics.

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 three tests and a Final.

Grading: Tests count for 60%, final for 40%

Students will be given an introduction to mathematical modeling with exposure to empirical modeling, discrete and continuous dynamical systems, simulation and optimization. No previous Excel experience us required.

Students will be expected to have access to Excel and a high speed internet connection to accommodate online meetings.

MATH 5397: Selected Topics in Mathematics: (section# 35171)
- Mult. Var. Calculus & Geometry -

Multivariate Calculus and Geometry, by Sean Dineen, 2nd edition
ISBN 978-1852334727, Springer-Verlage

Multivariate calculus links together in a non-trival way, perhapes for the first time in student's experience, four important subject areas: analysis, linear algebra, geometry and differential calculus.

This course covers: Differential calculus on open sets and surfaces (chapter 1-4), Integration theory (chapter 6, 9, 11-15), Goemtry of curves and surfaces (chapter 5, 6-8, 10, 16-18). The course provides a review and further development of the calculus III (Math 2333, Multivariate calculus and vector calculus). It also gives a nice preparation for students who intend to take my differential geometry online course later.

MoWeFr 10:00AM - 11:00AM - Room : SEC 103

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 focuses on the basic principles of measure and integration, which is essential in many areas of mathematics (in particular in analysis and probability). The syllabus for the first semester will cover most of the following topics: Measures. Measurable functions. Integration. Convergence of sequences of functions. Elementary Hilbert space theory. Banach spaces, e.g. the L^p spaces. Complex measures. Differentiation. Product measures and Fubini's theorem. Fourier transforms.

The midterm and the final exam will be based on the notes given in class, and on the homework. The 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).

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 Edition.

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; floquet theory and elementary bifurcation 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 is the first semester of a two-semester introductory graduate course in topology (the second semester is largely devoted to differential geometry and I probably won't teach that). This is a central and fundamental course and one which graduate students usually enjoy very much! This semester we cover point-set topology. 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.

You do not need a textbook, although I recommend the Munkres or the Runde books. You are expected to read the classnotes carefully each week, and bring to me things you don't understand there. 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 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. We may also move the class time to a time thats more convenient for all; if the latter is possible ;

MATH 4331 or equivalent or consent of instructor.

The focus is on key topics in optimization that are connected through the themes of convexity, Lagrange multipliers, and duality. The aim is to develop a analytical treatment of finite dimensional constrained optimization, duality, and saddle point theory, using a few of unifying principles that can be easily visualized and readily understood. The course is divided into three parts that deal with convex analysis, optimality conditions and duality, computational techniques. In Part I, the mathematical theory of convex sets and functions is developed, which allows an intuitive, geometrical approach to the subject of duality and saddle point theory. This theory is developed in detail in Part II and in parallel with other convex optimization topics. In Part III, a comprehensive and up-to-date description of the most effective algorithms is given along with convergence analysis.

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.

We will begin with basic notions used in describing properties of dynamical systems and continue with tools for understanding certain chaotic systems.

The first part includes: basic examples (circle and interval maps, toral automorphisms, expanding maps, subshifts of finite type); ergodicity, mixing and their spectral characterization; recurrence, transitivity, minimality; the Ergodic Theorems of Birkhoff and von Neumann.

We will then use functional analysis (operators on Banach spaces) to describe a class of invariant measures for subshifts of finite type, and the mixing properties of these measures. Through Markov partitions, subshifts of finite type model uniformly hyperbolic dynamical systems, as we will describe.

These methods extend in a non-trivial way to Young towers, which model a class of non-uniformly hyperbolic systems.

Books that contain some of these topics are "Introduction to Ergodic Theory" by Peter Walters, "Dynamical Systems and Ergodic Theory" by Mark Pollicott and Michiko Yuri, "Introduction to the Modern Theory of Dynamical Systems" by A. Katok and B. Hasselblatt, "Introduction to Dynamical Systems" by M. Brin and G. Stuck, "Zeta functions and the periodic orbit structure of hyperbolic dynamics" by William Parry and Mark Pollicott. Additional notes will be provided.

The first topic in this course is an introduction to Sobolev spaces of functions in one or more variables. A variational analysis of some classes of ordinary differential equations will then be described. This will include the use of variational principles to characterize solutions, obtain existence-uniqueness results, prove conservation laws and derive inequalities for these systems. Results for both boundary value problems and initial value problems will be treated. Results from convex analysis will be introduced as needed.

If time permits, extensions to classes of boundary value problems for PDEs and related evolution equations will be started.

The course will cover Hilbert spaces, Banach spaces and elementary properties of Linear Operators acting on these spaces. A number of applications will also be presented in class.

This course is part of a two semester sequence. The second semester will be a more technical development of the theory of linear operators on Hilbert spaces; unbounded operators; topics from Fourier Analysis. The selection of topics for the second semester will be based, in part, on the interest and feedback from interested students

MATH 6382-6383

Required text: Monte Carlo Statistical Methods, by Christian P. Roberts, and George Casella Springer, Second Edition, 2010, ISBN 978-0-387-21239-5

Recommended text: Introducing Monte Carlo Methods with R" by Christian Robert and George Casella, Springer, 2009