Continuation of Math 4331. Covers Chapters 7, 9, 11 of Rudin: Sequences and series of functions, functions of several variables, Lebesgue Theory on R^1.

The focus of the course is on investigating the qualitative properties of the solutions of ordinary differential equations: Where they go, and what they do once they get there. One of the strengths of the book we will be using is the great number of applications that are discussed. All abstract concepts will be illustrated in pertinent examples taken from biology, chemistry, engineering, physics and a number of other disciplines. I will keep this focus on applications in the course. I plan to cover most of Strogatz' book during the semester, and will use some supplementary materials.

We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. This is an introductory course and will be a mix of mathematics and computing.

The emphasis this semester will be on interpolation, numerical integration, direct and iterative methods for solving linear systems of algebraic equations and numerical methods for solving nonlinear algebraic equations.

N.B. This is the first semester of a two semester course. The emphasis the second semester will be in particular on numerical methods for ordinary differential equations and partial differential equations.

Syllabus: First five Chapters of the book:
1. Fields and Linear Equations
2. Vector Spaces over fields, subspaces, bases and dimension,.
3. Linear Transformations and Matrix Representations
4. Polynomials, ideals and prime factorization
5. Determinants (if time permits)
Grading: There will be three Tests and the Final. I will take the two highest test scores (60%) and the mandatory final (40%).

Option contracts, asset price dynamics, binomial pricing model, Ito's calculus, Black-Scholes pricing model, hedging and arbitrage.

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Thomas W. Hungerford, Algebra, Springer Verlag Graduate Texts in mathematics # 73

• Measures.
• Integration.
• Signed Measures and Differentiation.
• Point Set Topology.
• Elements of Functional Analysis.
• Lp Spaces.
• Radon Measures.
• Elements of Fourier Analysis.
• Elements of Distribution Theory.
• Topics in Probability Theory.
• More Measures and Integrals.

The content will follow the description for the ODE:
http://www.math.uh.edu/Matweb/PhDsyllabi/SyllabusODE.pdf

The course will be essentially self-contained but some of the material to be discussed can be found in:

K.ATKINSON and W. HAN, Theoretical Numerical Analysis: A Functional Analysis Framework, 2nd Edition, Springer-Verlag, New York, NY, 2005 R.

GLOWINSKI, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, NY, 1984.

Following Part I of the course where various functional spaces of practical importance have been introduced, we will focus on the solution of variational problems from Image Processing (L^1 fitting, in particular). The following topics will be systematically discussed:
1)Existence and uniqueness of solutions to the variational problem
2)Finite Element approximations
3)Iterative solution by a variety of algorithms, including over-relaxation ones.

The methodology to be discussed can be applied to a variety of variational problems from Mechanics, Physics and Image Processing.

• L.D. Berkovitz; Convexity and Optimization in $ \bf R^{n} $. Wiley-Interscience, New York, 2001
• I. Ekeland and R. T\'{e}mam; Convex Analysis and Variational Problems. SIAM, Philadelphia, 1999

We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. The topics this semester will include: systems of nonlinear equations, eigenvalue problems, iterative methods for systems of linear equations, and initial value problems for ordinary differential equations.

This course is an introduction to mathematical statistics. Topics covered include random samples, data reduction and clustering, maximum likelihood estimators and their asymptotic behaviour, confidence intervals, regression and classification.

Students will have the possibility to base part of their grade on projects realization involving the use of scientific softwares such as R or Mathlab. Extra reading material for potential PhD students will be suggested by the instructor.

This course will cover a wide range of topics in stochastic processes and applied probability. The emphasis will be on understanding the main ideas with a view to applications. Some group projects involving simulations will be given, but ni computer programming experience will be assumed.

1. Brief review of discrete time Markov chains. Continuous time Markov chains: birth-death processes; Poisson processes; birth and death with absorbing states. Applications.
2. Martingales and martingale convergence theorems. Stopping times. Brownian motion, properties of Brownian paths and applications.
3. Stochastic differential equations and applications.
4. Diffusion processes, backward and forward equations, diffusion models with killing, semigroup formulation of continuous time Markov processes.
5. Stationary processes, ergodic theorems, prediction of mean square error and covariance, applications of ergodic theory.
6. Brief introduction to branching processes, generating functions, extinction probabilies.

The purpose of this course is to introduce the student to the mathematical techniques that are useful in the modelling and analysis of active membranes, neurons and neuronal networks. The course will start with a brief review of the biology, and an introduction to applied dynamical systems theory. This will be followed by the derivation and analysis of the fundamental equations of neuroscience - the Hodgkin-Huxley equations - and their various reductions. The course will continue with a description of the dynamics of small networks, including central pattern generators. Finally analytically tractable models of large scale networks will be considered. Time permitting, I will discuss information theoretic techniques and their applications in data analysis.

Brief course description: This course will be about properties of measure preserving transformations of a probability space (including mixing, weak mixing and
ergodicity). Our first main result will be Birkoff's ergodic theorem - "time average = space average"'. This result is a far reaching generalization of Borel's strong law of large numbers and has some rather amazing applications to number theory (as we
shall illustrate).

We will then specialize to the class of topological Markov chains (or subshifts of finite type). These spaces can be viewed as modelling coin tossing of a multi-faceted coin with quite variable statistics. After setting up the spaces and map (there
is just one --- the shift map), we use functional analytic techniques based on Ruelle's transfer operator (a generalization of the Perron-Frobenius operator) to determine ergodic, mixing and other statistical properties of topological Markov chains. Although a topological Markov chain is, topologically speaking, a Cantor set, we can use topological Markov chains to model a wide range of differentiable dynamics. We indicate how.

None required. Texbooks that will be used are:

• W. Strauss's: "Partial Differential Equations"
• R. Glowinski: "Numerical Methods for Fluids (Part 3)"
• Chorin and Marsden: "Mathematical Introduction to Fluid Mechanics"
• Y.C. Fung: "Circulation"
• Y.C. Fung: "Biomechanics: Mechanical properties of living tissues
• R. LeVeques: "Conservation Laws", Research Papers

Review of basic linear PDEs. Analysis of quasilinear PDEs with concentration to hyperbolic conservation laws.

Introduction to fundamentals of fluid mechanics(basic equations of motion: continuity, momentum, energy, vorticity). Incompressible/compressible flow examples (derivation of the incompressible, viscous Navier-Stokes equations).

A brief introduction to Sobolev spaces. Fluid-structure interaction arising in blood flow modeling(effective models). Energy estimates.

Special topics related to the study of blood flow through compliant blood vessels.

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