Sequences and series of functions and uniform convergence. Functions of several variables and the inverse and implicit function theorems. Measure theory, in particular Lebesgue measure for the real line and Euclidean n-space.

We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. The emphasis this semester will be on the iterative methods for solving linear systems, approximation theory, numerical solutions of nonlinear equations, iterative methods for approximating eigenvalues, and elementary methods for ordinary differential equations with boundary conditions and partial differential equations. This is an introductory course and will be a mix of mathematics and computing.

Fields, vector spaces, linear systems, and matrices.

This course is a continuation of MATH 4377 which was taught from the same text.

Topics to be covered include Determinants, Elementary Canonical Forms, and the Rational and Jordan Forms.

Text book: "An Introduction to Hilbert Space", by N. Young, Cambridge paperback. ISBN 0-521-33717-8.

Reference book: "Real Analysis, Measure Theory, Integration and Hilbert Spaces", by Elias M. Stein and Rami Shakarchi, Princeton Lectures in Analysis III, Princeton U Press. ISBN 978-7-5062-8238-3.

Topics to be covered are:

1. Introduction to Hilbert and Banach spaces.
2. Orthonormal bases, Fourier series and the geometry of Hilbert spaces,
3. Dual spaces and the Hahn-Banach theorem, weak convergence and completeness.
4. L^p spaces over abstract measure spaces.
5. Analysis of continuous linear operators on H or X.
6. Weak differentiation and Hilbert-Sobolev spaces.

Graduate standing or consent of instructor.
Students should have had a course in Linear Algebra (for instance Math 4377-4378) and an introductory course in Analysis(for instance Math 4331-4332).

This is the second semester of a two semester course. We will develop and analyze numerical methods for approximating the solutions of common mathematical problems. The focus in this semester is on approximation theory and numerical methods for initial and boundary-value problems of ordinary differential equations. The applications of approximation theory to interpolation, numerical differentiation and numerical integration will be addressed. The concepts of consistency, convergence, stability for the numerical solution of ODEs will be discussed.

Further Literature:

• A. Quarteroni, R. Sacco, and F. Saleri; Numerical Mathematics. Springer-Verlag, 2000
• U.M. Ascher, R.M.M. Mattheij, and R.D. Russell; Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM, Philadelphia, 1995
• U. Ascher and L. Petzold; Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia, 1998
• E. Hairer, F.P. Norsett, and G. Wanner; Solving Ordinary Differential Equations. I. Nonstiff Problems. Springer, Berlin- Heidelberg-New York, 1987
• E. Hairer and E. Wanner; Solving Ordinary Differential Equations. II. Stiff Problems. Springer, Berlin-Heidelberg-New York, 1991

The course provides an introduction to the development, analysis, and implementation of finite difference and finite element methods for partial differential equations:

1. Foundations of the theory of PDEs
   1.1 Classification and characteristics
   1.2 Sobolev spaces
2. Numerical methods for elliptic PDEs
   2.1 Finite difference methods
   2.2 Finite element methods
   2.3 Other methods
3. Numerical methods for parabolic PDEs
   3.1 Finite difference methods
   3.2 Finite element methods
   3.3 Other methods
4. Numerical methods for hyperbolic PDEs
   4.1 Numerical methods for systems of conservation laws
   4.2 Finite difference and finite element methods for the wave equation
   4.3 Other methods

Recommended Texts:

• Mathematical Statistics with Applications, 6th Edition by Dennis Wackerly, William Mendenhall and Richard Scheaffer, 2002, Duxbury Press.
• Probability and Statistical Inference by Hogg and Tannis, 6th Edition, 1997.
• Mathematical Statistics by Jun Shao, 2nd Edition, 2003, Springer Verlag.

Of the books listed Statistical Inference by Casella and Berger is required reading. The others are useful for reference and lecture notes will be based on a variety of sources.

Topics for this course will be taken from

1. Mathematical neuroscience Wilson-Cowan equations and rate models
2. Pattern formation -- Turing bifurcations Murray's work --- patterns of spots and stripes. Phyllotaxis (Duoady, Gole, Newell --- why Fibonacci sequences appear)
3. Genetics and evolution --- Nowak
4. Networks theory --- scale-free, etc. models. gene transcription networks, central pattern generators.

Possibly some low-dimensional models as well.

Students will be expected to read papers or books and present material in class once during the semester.

Reference books : selected chapters in

• Pierre Bremaud : Markov Chains, Gibbs Fields, Monte Carlo Simulation, Springer TAM Vol 31
• Bernard Chalmond : Modeling and inverse problems in image analysis , Springer Vol 155, 2003

Gibbs-Markov fields are probabilistic models designed to analyze large systems of "particles" or "processors" in interaction and were introduced by physicists to modelize networks of small magnets.

We give a self contained presentation for the basics of Gibbs- Markov fields theory and develop several applications :

• Texture identification and segmentation for image analysis
• Simulated annealing for stochastic minimization of "hard" combinatorics problems

The main probabilistic topics in this course involve the dynamics of discrete time Markov chains on (huge) finite spaces, Gaussian random vectors, Monte-Carlo simulations.

Students familiar with Mathlab or equivalent scientific softwares will have the possibility to replace a large proportion of homework assignments and exams by the realization of applied projects

Recommended Texts: A First Course in Stochastic Processes, Karlin and Taylor, Second Edition, Academic Press.

Reference Book: A Second Course in Stochastic Processes, Karlin and Taylor, 1981, Academic Press.

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 no computer programming experience will be assumed.

1. 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. Renewal processes and the renewal equation. Applications.
4. Stochastic differential equations and applications.
5. Diffusion processes, backward and forward equations, diffusion models with killing, semigroup formulation of continuous time Markov processes.
6. Stationary processes, ergodic theorems, prediction of mean square error and covariance, applications of ergodic theory.
7. Gibbs Fields and Monte Carlo simulations.

Lecture notes will be based on A First Course in Stochastic Processes, (Karlin and Taylor) supplemented by a variety of other texts.

Recommended: John M. Lee, Introduction to Smooth Manifolds.
Other relevant books will be placed on reserve in the library