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Nonlinear Dynamics Courses - Fall 2002




Aerospace Engineering

ASE 346 - Viscous Fluid Flows (Professor D. Dolling). Navier-Stokes equations, laminar and turbulent boundary layers; effects of pressure gradients; compressibility, and heat transfer. Boundary layer transition. (Next offering to be determined)

ASE 382R - Advanced Computational Methods (Professor John Kallinderis). The state of the art on finite-volume numerical methods for solution of equations in fluid mechanics is presented. Numerical issues that are shared by all schemes will be investigated. Those include grid generation, stability, boundary conditions, and artificial dissipation. Emphasis is given on coding the schemes and applying them to realistic situations. The class will be useful not only to students interested in Fluid Mechanics but also to anyone else who is interested in solving similar equations computationally. (Next offering to be determined)

ASE 382R (Topic 6) (same as M E 381Q, Topic 4) - Molecular Gas Dynamics (Professor P. Varghese). This course focuses on the molecular description of physical and chemical processes in gases. The molecular viewpoint is essential to understand processes that occur at very high or low temperature and pressure, non-equilibrium phenomena, etc. Additionally, the microscopic viewpoint provides insight into the behavior of fluids at ordinary conditions. The course provides a basic working knowledge of the kinetic theory of gases, chemical thermodynamics, and statistical mechanics. These analytical tools are then applied to the study of equilibrium gas properties over wide temperature ranges, the kinetics of physical and chemical reactions, and the interaction of matter with radiation. The course material is useful in studies of high temperature and rarefied gas dynamics, plasma processing (e.g. in semiconductor manufacturing), combustion chemistry, laser diagnostics of high temperature gases, hypersonics, etc. (Next offering to be determined)

ASE 382Q (Topic 8) - Langrangian Methods in Computational Fluid Dynamics (Professor D. Goldstein). The computational modeling of fluid dynamics through the use of moving particles is discussed. Applications involve viscous/inviscid, incompressible and continuum/rarefied flows. Topics include: point vortex methods, vortons, direct simulation Monte Carlo techniques, particle in cell schemes, immersed boundary schemes, cellular automata, smoothed particle hydrodynamics and molecular dynamics. Part of the material is presented as formal lectures and part as student led seminars discussing their application of a Lagrangian method of their own choice (Offered Fall 2002, U# 12655, TTH 11-12:30P, WRW 312)

ASE 384P (Topic 4) - E M 394F - Finite Element Methods (see for description) and CAM 394F.

ASE 388P - Hamiltonian Mechanics Contents of the course will include the following: 1) The Lagrangian and Hamiltonian in dynamics. Forms of equations of motion in celestial mechanics. 2) Coordinate transformations in hamiltonian mechanics. Canonical transformations. Conditions for Canonicity. Different canonical variables in celestial mechanics. 3) Perturbation methods. Solution by expansions in a small parameter. Applications to celestial mechanics: Satellite theory. 4) Stability of dynamics systems. Stability of orbits. Poincaré's characteristic exponents. Application to the J2-problem. Frozen orbits. Orbits around an ellipsoid of revolution. 5) Chaos theory; mappings. Chaotic orbits in celestial mechanics. Long-term stability of orbits. Applications of Poincaré surfaces of section (Next offering to be determined).

ASE 388P - Regularization in Space Dynamics Transformations of the independent and dependent variables are treated in considerable analytical details. Advantages of increased accuracy and reduced integration time are emphasized. Contributions by Levi-Civita, Steifel, Tisserand and Birkoff are emphasized (Next offering to be determined).

ASE 388P (Topic 5) - Theory of Orbits I Description and equations of the restricted and general problems of three bodies. Reductions by means of the integrals of motion. Streamline analogy after Poincaré and Lanczos. Jacobian integral and the totality of solutions. Linear and non-linear dynamics near equilibrium points. Linear and non-linear stability of the Lagrangian points. Transformations of the dependent and independent variables to eliminate singularities. Regularization, linearization, uniformization and smoothing with numerical and analytical advantages. Families of periodic orbits and range of chaotic motion (Next offering to be determined).

Astronomy

AST 381 (also PHY 394T) - The Formation of Galaxies and Large-Scale Structure in the Universe (Professor P. Shapiro) Our universe at large is well-described, on average, by the standard Big Bang model of cosmology, in which a homogeneous distribution of mass-energy expands uniformly and isotropically from an initial space-time singularity, in accordance with the expectations of Einstein's General Theory of Relativity. Shortly after its birth, our observable universe was extraordinarily smooth on all scales. Today, however, it is highly structured, populated by galaxies, clusters, and superclusters of galaxies surrounded by huge regions devoid of galaxies, with most of the volume permeated by a clumpy intergalactic medium of diffuse gas. How this cosmic structure arose over the last 10 or 20 billion years from an otherwise smooth, featureless beginning at the initial moment of the Big Bang will be the subject of this course. We will start by reviewing the basic observed characteristics of cosmic structure -- the galaxies and intergalactic matter, their clustering in space and peculiar motions with respect to the universal expansion, and their evolution, including the evidence for dark matter. A brief self-contained summary of the current standard cosmology framework -- the Big Bang model and the properties of the evolving homogeneous background universe -- will be included. The formation of galaxies and large-scale structure by gravitational instability which results when this expanding homogeneous universe is perturbed by initially small-amplitude density fluctuations will then be studied in depth. We will trace the growth of such primordial fluctuations from linear to nonlinear amplitude and its dependence on the mass-energy content of the universe and on the nature of the dark matter which is believed to dominate the present matter density. The current status of theories of galaxy and large-scale structure formation in comparison with data will be assessed, with a special focus on the well-known Cold Dark Matter model. Topics to be addressed will range from the formation of galaxies from dwarfs to giants to the formation of X-ray clusters of galaxies to the cosmic history of star formation to the reionization of the intergalactic medium, from the origin of galactic rotation to the origin of the Lyman alpha forest of quasar absorption-lines, to the origin of the statistical pattern of large-scale structure observed today in the spatial distribution and motions of galaxies in the universe. (Next offering to be determined)

AST 381 - Astrophysical Gas Dynamics. (Professor P. Shapiro). This course will survey a wide range of basic gas dynamics in an astrophysical context. The conservation equations of hydrodynamics and magnetohydrodynamics will be derived and studied as in any basic physics course in fluid mechanics. Hydrostatics, magneto-statics, virial equilibrium, polytropes, sound and MHD waves, shocks, flux-freezing, kinetic theory, viscosity, and thermal conduction will be included. Following this, the application to astronomical flows will focus on the equations of compressible flow at high Reynolds number. We will discuss steady flows (e.g. stellar winds, accretion, de Laval nozzles as twin-exhaust radio jets, thermal evaporation of interstellar clouds, steady-state radiative shocks), self-similar but non-steady flows (e.g. Sedov blast waves for supernova remnants, cosmological blast waves in an expanding universe, thermal conduction fronts, interstellar-wind-driven bubbles), and non-steady, non-self-similar flows (e.g. ionization fronts and H II regions, cosmological pancakes, the heating and ionization of a cosmological expanding intergalactic medium). We will discuss instabilities such as the Rayleigh-Taylor, Kelvin-Helmholtz, Jeans, and thermal instabilities, and the gravitational growth of cosmological density fluctuations, using linear perturbation analysis. Additional topics include hydromagnetic dynamos and the generation of cosmical magnetic fields, turbulence, relativistic hydrodynamics, and an introduction to numerical hydrodynamics. Requirements: Homework Problem sets and one end- of-semester student lecture. A Complete and self-contained set of classnotes will be handed out in place of a textbook. No exams. Prerequisites: Suitable for all beginning and advanced graduate students in physics and astronomy. Otherwise consent of instructor. (Next offering to be determined)

AST 381 - Astrophysical Gas Dynamics II (Professor P. Shapiro). This course will survey a wide range of basic gas dynamics in an astrophysical context. It will begin where Semester I (i.e. AST 382C: Astrophysical Gas Dynamics, Fall 2000) ended, including a brief review of the essentials of Semester I. The basic conservation equations derived in Semester I will be generalized to include the effects of thermal conduction and viscosity. We will then apply the basic equations to describe a number of astrophysical flows.

Course outline:
  1. The Conservation Equations of Gas Dynamics
    1. Review essentials of Astrophysical Gas Dynamics I
    2. Viscosity and thermal conductivity; Navier-Stokes equation
    3. Kinetic equilibrium and relaxation times
  2. Steady-State Flows
    3.
    1. e.g. Thermal evaporation of clouds, photoevaporation of clouds, MHD shocks
  3. Self-Similar Flows
    4.
    1. e.g. Stellar wind-driven interstellar bubbles, superbubbles, conduction fronts
  4. Nonsteady, Non-Self-Similar Flows
    5.
    1. e.g. Strömgren spheres, ionization fronts, and the dynamics of H II regions
  5. Relativistic Hydrodynamics
    6.
    1. The conservation equations of relativistic hydrodynamics
    2. Special relativistic simple waves and Riemann invariants
    3. Relativistic shock waves
    4. Relativistic blast waves and gammas ray bursts
  6. Cosmological Gas Dynamics
    7.
    1. The newtonian approximation and the conservation equations of cosmological gas dynamics
    2. Cosmological pancakes and the gravitational collapse of cosmological density fluctuations
    3. Cosmological H II regions
    4. Galaxy and large-scale structure formation
  7. Hydromagnetic Dynamos and the Generation of Cosmical Magentic Fields
    8.
  8. Instabilities
    9.
    1. Jeans instability
    2. Thermal instability
    3. Rayleigh-Taylor instability
    4. Kelvin-Helmholtz
    5. Parker instability
    6. Convection and the Schwarzschild criterion
  9. Turbulence
    10.
  10. Introduction to Numerical Hydrodynamics
    11.
    1. The Reimann Problem
    2. Brief Overview of Finite-Difference Methods
    3. Riemann Solvers
    4. Smoothed Particle Hydrodynamics (SPH)
Prerequisites: suitable for all beginning and advanced graduate students in physics and astronomy. AST 381: Astrophysical Gas Dynamics I (Fall 2000) or equivalent background helpful but not required for Part II. Copies of Part I lecture notes available for students who take Part II without having taken Part I. Textbook: A complete and self-contained set of class notes will be handed out. Requirements: Homework problem sets and one end of semester student lecture. No exams. (Next offering to be determined)

AST 382C -Astrophysical Gas Dynamics. (Professor P. Shapiro). This course will survey a wide range of basic gas dynamics in an astrophysical context. The conservation equations of hydrodynamics and magnetohydrodynamics will be derived and studied as in any basic physics course in fluid mechanics. Hydrostatics, magneto-statics, virial equilibrium, polytropes, sound and MHD waves, shocks, flux-freezing, kinetic theory, viscosity, and thermal conduction will be included. Following this, the application to astronomical flows will focus on the equations of compressible flow at high Reynolds number. We will discuss steady flows (e.g. stellar winds, accretion, de Laval nozzles as twin-exhaust radio jets, thermal evaporation of interstellar clouds, steady-state radiative shocks), self-similar but non-steady flows (e.g. Sedov blast waves for supernova remnants, cosmological blast waves in an expanding universe, thermal conduction fronts, interstellar-wind-driven bubbles), and non-steady, non-self-similar flows (e.g. ionization fronts and H II regions, cosmological pancakes, the heating and ionization of a cosmological expanding intergalactic medium). We will discuss instabilities such as the Rayleigh-Taylor, Kelvin-Helmholtz, Jeans, and thermal instabilities, and the gravitational growth of cosmological density fluctuations, using linear perturbation analysis. Additional topics include hydromagnetic dynamos and the generation of cosmical magnetic fields, turbulence, relativistic hydrodynamics, and an introduction to numerical hydrodynamics. Requirements: Homework Problem sets and one end- of-semester student lecture. A Complete and self-contained set of classnotes will be handed out in place of a textbook. No exams. Prerequisites: Suitable for all beginning and advanced graduate students in physics and astronomy. Otherwise consent of instructor. (Offered Fall 2002, U# 46630, TTH 12:30-2P, RLM 15.216B)

Civil Engineering

C E 358 -Introductory Ocean Engineering. (Professor S. A. Kinnas). Topics include equations of ideal fluid motion, wave theory, random waves and wave spectra, wind wave generation, and forces on onshore or offshore structures. Prerequisite: C E 319F or consent of the teacher (Offered Fall 2002, U# 14430, TTH 11-12:30P, ECJ 7.208).

C E 380P (Topic 3) - Principles of Hydrodynamics. (Professor S. A. Kinnas). Topics include motion of a viscous or ideal fluid, waves and wave/body interactions, lifting surfaces, cavitating flows, and computational hydrodynamics. Prerequisite: Graduate standing (Next offering to be determined)

C E 380P (Topic 4) - Boundary Element Methods. (Professor S. A. Kinnas). Topics to be covered are formulation and numerical implementation of boundary element methods, applications to problems in fluid mechanics, structural analysis, and solid mechanics. Prerequisite: Graduate standing. (Next offering to be determined)

Computational and Applied Mathematics

CAM 386K - same as M 383G (see for description).

CAM 385 - same as M 383D (see for description).

CAM 393C - same as M 393C (see for description)

CAM 393D - same as M 393D (see for description).

CAM 393M - same as M 393N (see for description).

CAM 394F - same as E M 394F (see for description).

CAM 395T - same as C S 395T (see for description).

Computer Science

C S 342 - Neural Networks (Professor Miikkulainen). An undergraduate introduction to Biological information processing; architectures and biological algorithms for supervised learning, self-organization, reinforcement learning, and neuro-evolution; hardware implementations and simulators; applications in engineering, artificial intelligence, and cognitive science. More information at more (Offered Fall 2002, U# 52085, T 3:30-6:30P, WEL 1.316).

C S 367 - Numerical Methods (Dr. David Kincaid). The objectives of the course are to 1) acquaint students of science and engineering with the potentialities of high-performance computers and modern mathematical software for solving the numerical problems that will arise in their profession; 2) to give students an opportunity to hone their skills in programming and in problem solving; and 3) to help students understand errors that inevitably accompany scientific computing and to arm them with methods for detecting, predicting, and controlling these errors. Topics will include: systems of linear equations, numerical integration, ordinary differential equations, and nonlinear equations; construction and use of large numerical systems; influence of data representation and computer architecture on algorithm choice and development. (Next offering to be determined)

C S 378 - Parallel Scientific Computing. (Professor D. Kincaid) The main objective is to introduce students to the use of parallel computers and mathematical methods used in science and engineering. This is done by way of a survey of techniques for solving systems of linear equations on parallel computers. Programming projects are done on actual parallel computers such as the Cray T3E or IBM PS2 to obtain hands-on experience. (Offered Fall 2002, U# 52265, TTH 2-3:30P, RLM 7.120)

CS 383C - Numerical Analysis: Linear and Nonlinear Algebra (Professor Cline). Same as M 383C. Review of computer arithmetic, stability, conditioning; solution of dense linear systems; sensitivity of linear systems; least squares; eigenvalues and eigenvectors. Text: Applied Numerical Linear Algebra by J. Demmel. (Offered by Professor Dhillon, Fall 2002, U# 52330, MW 11-12:30P, ECH 1.204)

C S 386K - same as M 383G (see for description).

C S 393D - same as M 393D (see for description).

C S 393N - same as M 393N (see for description).

C S 394N - Neural Networks (Professor Miikkulainen). A graduate research class in neural networks. The first part of the course is an introduction to neural computation, including biological motivation, neural network architectures and algorithms, and applications in engineering, artificial intelligence, and cognitive science. The second part of the course consists of student presentations on selected advanced topics of interest. The third part involves conducting an original research project and reporting and discussing them in class. More information at CS 394N (Next offering to be determined)

C S 395T- Parallel Techniques for Numerical Algorithms (Professor R. van de Geijn). The course deals with highly practical issues related to programming high performance parallel architectures. Background required to take the course consists of a strong background in numerical algorithms and a strong programming background (C and/or FORTRAN). Topics covered will be: communication on parallel architectures; design of scalable algorithms; and case studies drawn from practical applications. Grading will be on the basis of two projects (Next offering to be determined).

Electrical Engineering

E E 380L - Advanced Topics in Neural Networks (Professor J. Ghosh). An advanced course on artificial neural networks (ANNs) and their applications. It is meant for students who are interested in doing active research in this area. Students taking the course should have an understanding of undergraduate-level calculus, linear algebra and probability theory. A series of lectures that concentrate on three or four topics will comprise the first half of the course. The second half will consist of presentations by students enrolled in the course, followed by active discussions. Topics to be covered are links between ANNs and statistical pattern recognition; ANNs for spatio-temporal processing; modular neural networks; and hybrid connectionist-symbolic architectures (tentative). Prerequisites are a first course (undergrad or grad) in neural networks OR consent of the instructor. (Next offering to be determined)

E E 380L (Topic 9) - Artificial Neural Systems (Professor J. Ghosh). This is an introductory course on neural networks. The primary emphasis is on the theory, modeling/analysis and representative applications of artificial neural networks rather than their neurophysiological plausibility. We shall look at the computational capabilities and limitations of several popular neural networks. Students taking this course should have an understanding of undergraduate-level calculus, linear algebra and probability theory including the following topics: Ordinary differential equations, Linear vector spaces: inner products; linear independence; orthogonality; sub-spaces; eigenvectors and eigenvalues; Bayes theorem; Poisson and Gaussian distributions; expectation of random variables; mean and variance. Prerequisites: graduate standing or consent of instructor and working knowledge of a computer language: C/PASCAL/C++. Grading: Final exam - 30 points; Term paper and presentation - 25+5 points; Mid-term exam - 15 points; Homework and LAB assignments - 25 points. For the term paper you are encouraged to work in pairs. A list of possible topics shall be given, but you can choose almost any topic related to ANS provided you have not already used this for another term paper/thesis. Each group is expected to make a 25-30 minute presentation towards the end of the course. Text: Elements of Artificial Neural Networks, by K. Mehrotra, C.K. Mohan and S. Ranka, MIT Press, 1996 (MANDATORY). (Next offering to be determined)

E E 380L - Neural Networks for Pattern Recognition This course will focus on feedforward neural networks for pattern recognition and learning/generalization problems, and compare them with classical techniques. I will follow the textbook quite closely, so its contents will give you a fairly good idea of the course topics and sequencing. Matlab based software will be used to study and reinforce a variety of neural network and pattern recognition concepts. You do not need a previous course on neural networks to follow and benefit from this course. Prerequisite: Graduate standing or consent of instructor. Grading: Final exam - 30 points; Term paper + presentation - 25+5 points; Midterm exam - 20 points; Homework - 20 points. Text (mandatory): Neural Networks for Pattern Recognition by C. Bishop, Oxford University Press, 1995. (Next Offering to be determined)

E E 381L - Digital Time Series Analysis and Application to Nonlinear Systems (Professor E. J. Powers). The discrete Fourier transform; classical power spectral analysis and applications; wavenumber- frequency spectra; higher-order spectra (e.g., bispectra and trispectra); nonlinear system modeling and determination of V~olterra nonlinear transfer functions from experimental data via higher-order spectra; nonlinear spectral energy transfer; compensation and linearization of nonlinear systems; wavelet-based higher-order spectra for detection of short-time duration nonlinear interactions; and applications of higher order spectra to various nonlinear problems in science and engineering. (Next offering to be determined)

Engineering Mechanics

E M 386M - Functional Analysis in Theoretical Mechanics (Professor L. Demkowicz). Same as Computational and Applied Mathematics 386M. An introduction to modern concepts in functional analysis and linear operator theory, with emphasis on their application to problems in theoretical mechanics; topological and metric spaces, norm linear spaces, theory of linear operators on Hilbert spaces, applications to boundary value problems in elasticity and dynamical systems. Prerequisite: graduate standing, Engineering Mechanics 386L, and Mathematics 365C. (Offered Fall 2002, U# 13030, MWF 11-12P, WRW 312)

E M 393N - Numerical Methods in Flow and Transport Problems (Professor G. Carey). An introductory course to intermediate graduate level course on approximate methods for solutions of flow and transport problems. Finite element, finite difference, and residual methods will be covered. Both linear and nonlinear problems are examined. (Next offering to be determined)

E M 394F (same as ASE 384P,Topic 4 and CAM 394F) - Finite Element Methods (Professor Demkowicz). A basic course covering the fundamentals of the formulation and computer implementation of finite element methods for the solution of boundary value problems. The emphasis is on linear, scalar problems in one and two dimensions. (Offered by Professor Dawson, Fall 2002, U# 13045, MWF 9-10,WRW 113)

E M 394G - Comp Techs in Finite Element (Professor E. Becker). Techniques for formulation and solution by finite element methods of linear and nonlinear problems in continuum mechanics. Emphasis is on coding strategies for large problems. This course should be useful to those who will develop finite element methods. Prerequisite: knowledge of basic principles of finite element methods as in E M 394F. (Next offering to be determined)

E M 397 (Topic 4) - Grid Generation and Adaptive Grids (Professor G.F. Carey). In this course, a treatment of grid generation, adaptive refinement and moving grid techniques is presented together with supporting mathematical, algorithmic, and software concepts. Efficient solution strategies such as multigrid methods that exploit nested grid hierarchies are also described and analyzed. The emphasis is on the fundamental ideas but the presentation includes practical guidelines as well as material of current research interest related to parallel computing, superconvergence, accuracy and moving boundaries. (Next offering to be determined)

E M 397 - Advanced Computational Flows/Transport (Professor G. Carey). The focus of this course is methodology and algorithms for solving problems in fluid flow and heat or mass transfer. The course assumes some exposure of the student to basic methods and the underlying phenomena and equations. The following main topics will be covered: Numerical methods and solution techniques: for potential flows, compressible flows, viscous incompressible flows, and transport processes. The emphasis will be on Galerkin type methods but there will be some discussion of global expansion techniques, collocation, spectral methods, high order compact differencing and other strategies. The treatment will include coverage of such phenomena as "locking" and spurious modes in solving incompressible flow problems numerically, solution of coupled nonlinear problems, treatment of bifurcations and related nonlinear dynamical behavior. Part of the work on algorithms will consider smart adaptive strategies for treating grids and parallel computing using cell or edge based methods and domain decomposition techniques. We will also discuss emerging research issues such as stabilized methods and space time methods. Representative engineering and scientific applications from a variety of areas (semiconductor manufacturing, aerospace, environmental modeling, etc.) will be discussed. This course is offered only alternate Fall semesters. Inquiries should be addressed to Professor Carey. (Next offering to be determined)

E M 397 - Special Topics - Numerical Methods in Advanced Transport Modeling (Professor G. Carey). Assumes some exposure to standard numerical methods and deals with nonlinear diffusion-convection-reaction processes plus other selected nonlinear transport problems. (Next offering to be determined)

E M 397S - Numerical Methods in Advanced Transport Modeling (Professor G. F. Carey). Topics covered will include: diffusion, linear and nonlinear processes, finite difference schemes, Galerkin finite elements, descent and gradient algorithms; convec-tion and SUPG diffusion analysis of oscillations, upwind techniques,Petrol-Galerkin, Taylor-Galerkin, SUPG higher-order schemes; reaction-diffusion problems, limit and bifurcation points, nonlinear solution algorithms, nonlinear multi-grid aspects; diffusion and Monte Carlo techniques, single and multi-species diffusion, application to semi-conductor doping, Ion implantation; viscous flow and transport, Navier Stokes problems, consistent elements, LBB condition, h, p and adaptive strategies, coupled problems (heat and species transport), applications to Rayleigh-Benard-Marangoni and silicon oxidation. Time (daily 1-1/1 hours) TBA on first class day. Grading: homework-20%, quizzes - 30%, two paper presentations - 20%, project - 30%. Office hours will be daily - TBA first class day. (Next offering to be determined)

Mathematics

M 348 - Scientific Computation in Numerical Analysis (Professor T. Arbogast). Solving scientific, engineering, and other problems often requires the use of numerical methods and computers. This course presents various basic numerical methods, discusses their mathematical properties, and provides practice in computer programming. We will cover chapters 1-6 and 10 of the text by Burden and Faires, Numerical Analysis, 7th ed., 2001. For additional information, see the M 348 website. (Offered Fall 2002, U# 56910, MWF 11-12P, RLM 6.120)

M 367L - The Topology of Chaotic Dynamical Systems (Professor Gordon). Topics will include topology of sets which arise under iteration of smooth mappings; the logistics map, and study period doubling and the Sarkovsky theorem; symbolic dynamics; strange attractors; and branched manifolds or "train tracks." Text to be used is "An Introduction to Chaotic Dynamical Systems" by Robert Devaney. Prerequisite: Topology (M 367K or equivalent) or consent of instructor. (Next offering to be determined)

M 368K - Numerical Mathematics for Applications (Professor Cheney). The course covers computer arithmetic, control of errors, solution of nonlinear equations, linear systems of equations, ordinary differential equations, quadrature, linear programming, simulation and Monte Carlo techniques, minimization of functions, partial differential equations, data smoothing, and spline functions. Computer problems are assigned to be done in whatever programming language the student prefers. (Fortran, C, C++, Pascal, Maple, Mathematica, Matlab, Octave,..) Instruction in and use of some high-level system such as Maple, Mathematica, Matlab, or Octave is incorporated in the course. Prerequisites: students should be familiar with differential equations, elementary linear algebra, and basic computer use. (Next offering to be determined)

M 375- Mathematical Modeling in Biology (Professor Uhlenbeck)
Mathematical modeling in biology follows a similiar course designed for upper division and graduate students in biology at Harvard who have had some calculus. It assume some sophistication in science, but not in mathematics. We cover the basics of solving the important ordinary differential equations and systems with a selection of biological applications. A section on discrete modeling and numerical iterations is included. Emphasis is on equilibrium solutions and stability, although a short discussion of chaotic behavior will be included. Conceptual material in biology will be an important part of the course. Some statistical models will be covered if there is interest in the class. A portion of the grade will be determined by work on a group project. Also, elementary computer exercises will be designed to illustrate the material. This course is designed for biology students, and the calculus which is used will be reviewed. This might make it useful for chemistry and geology students as well. Biology students may receive graduate credit under special circumstance. Mathematics majors who are interested in applications may substitute this course for 427K, which is designed for engineers rather than math majors or biologists. The course will not cover Laplace transforms and Fourier series, but it will cover phase plane anaylsis (systems). 427K Students interested in rigor and proofs should not take this course. Text: C. Taubes, "Introductory Lectures on Differential Equations and Their Applications in the Biological Sciences" to be published in August, 2000 (Prentice Hall). Students who are interested can consult with Professor Uhlenbeck's secretary and look at a copy of the text. For a more detailed description, please see M 375. (Offered by Professor Sadun, Fall 2002, U# 57100, TTH 11-12:30P, RLM 6.118)

M 376C - Methods of Applied Mathematics I (Professor R. Showalter)
Prerequisite: M427K with grade of at least C, and some acquaintance with linear algebra. Course Description: Variational methods and related concepts from classical and modern applied mathematics are introduced. Models of conduction and vibration lead to systems of linear equations and ordinary differential equations, eigen-value problems, initial and boundary value problems for partial differential equations. Topics include a selection from diagonalization of matrices, eigenfunctions and minimization, asymptotics of eigenvalues, separation of variables, generalized solutions, approximation methods. More information at M 376C (Next offering to be determined)

M 381C- Introduction to Real Analysis (Professor Beckner) We will develop the theory of Lebesgue measure and integration: Lebesgue measure and outer measure, integration, convergence theorems, repeated integration, Lebesgue differentiation and covering theorems, Lpclasses, abstract integration. Text: Wheeden and Zygmund, Introduction to Real Analysis (Next offering to be determined)

M 383C - same course description as C S 383C

M 383C (also CAM 385C) - Methods of Applied Mathematics I (Semester I) (Professors T. Caffarelli). Course Description: This is the first semester of a course on methods of applied mathematics. It is open to mathematics, science, engineering, and finance students. It is suitable to prepare graduate students for the Applied Mathematics Preliminary Exam in mathematics and the Area A Preliminary Exam in CAM. The first semester is an introduction to functional analysis.

I. Banach Spaces
1. Normed linear spaces and convexity
2. Convergence, completeness, and Banach spaces
3. Continuity, open sets, and closed sets
4. Continuous Linear Transformations
5. Hahn-Banach Extension Theorem
6. Linear functionals, dual and reflexive spaces, and weak convergence
7. The Baire Theorem and uniform boundedness
8. Open Mapping and Closed Graph Theorems
9. Closed Range Theorem
10. Compact sets and Ascoli-Arzel\ 'a Theorem
11. Compact operators and the Fredholm alternative
II. Hilbert spaces
1. Basic geometry, orthogonality, bases, projections, and examples
2. Bessel's inequailty and the Parseval Theorem
3. The Riesz Representation Theorem
4. Compact and Hilbert-Schmidt operators
5. Spectral theory for compact, self-adjoint and normal operators
6. Sturm-Liouville Theory
III. Distributions
1. Seminorms and locally convex spaces
2. Test functions and distributions
3. Calculus with distributions
(Next offering to be determined)

M383D (also CAM 385D) - Methods of Applied Mathematics I (Semester II) (Professor Showalter). Course Description: This is the second semester of a course on methods of applied mathematics. It is open to mathematics, science, engineering, and finance students. It is suitable to prepare graduate students for the Applied Mathematics Preliminary Exam in mathematics and the Area A Exam in CAM

IV. The Fourier Transform and Sobolev Spaces
1. The Schwartz space and tempered distributions.
2. The Fourier transform.
3. The Plancherel Theorum.
4. Convolutions.
5. Fundamental Solutions of PDE's.
6. Sobolev spaces.
7. Imbedding Theorums.
8. The Trace Theorum.
V. Variational Boundary Value Problems (BVP)
1. Weak Solutions to elliptic BVP's
2. Variational forms.
3. Lax-Milgram Theorum.
4. Galerkin approximations
5. Green's functions
VI. Differential Calculus in Banach Spaces and Calculus of Variations
 1. The Fr'echet derivatives.
2. The Chain Rule and Mean Value Theorums.
3. Higher order derivatives and Taylor's Theorum.
4.Banach's Contraction Mapping Theorum and Newton's Method.
5. Inverse and Implicit Function Theorums, and applications to nonlinear functional equations.
6. Extremum problems, Lagrange equation.
7. The Euler-Lagrange equation.
8. Applications to classical mechanics and geometry.
VII. Asymptotic Analysis
1. Definitions and fundamental properties.
2. Examples of transcendental equations and initial-value problems.
3. Boundary layers in regular and singular perturbations.
4. Perturbation methods for linear eigenvalue problems.
(Spring 2002, U# 55645, TTH 11-12:30p, RLM 10.176 )

M 383G (also C S 386K and CAM 386K) - Numerical Treatment of Differential Equations The course is designed primarily for first-year graduate students. Some prior course in numerical analysis at the undergraduate level is desirable, but not essential. Some background in ordinary differential equations (e.g., M 427K) and in linear algebra (e.g., M 311 or M 340L) would be helpful. Topics covered:

Textbook: D. Young and R. Gregory, "A Survey of Numerical Mathematics," Vol. II, Dover Publications, 1988. (Offered Fall 2002, U# 57170, MWF 9-10, RLM 12.166)

M391C - Nonlinear Functional Analysis (Professor J. Bona) This course will be concerned with the developments from the mid-20th century of nonlinear analysis set in infinite dimensional spaces. The course will assume familiarity with the rudiments of linear functional analysis. The level will correspond to that of the first-year graduate course, Applied Mathematics I & II. The course will commence with calculus in an infinite-dimensional setting, including the implicit and inverse-function theorems. It will continue with topics selected from the following ambitious list:
1. Fixed-point theory
2. Degree theory
3. Bifurcation analysis
4. k-set mappings
5. Monotone operators
6. Nonlinear semigroups and applications to evolution equations
(Next offering to be determined)

M 391C - Functional Analysis II (Professor C. Radin). This is a second year graduate mathematics course, given on a regular basis - every year or two. (It was offered for the first time in the Spring of 1999.) The text for the course is Essential Results of Functional Analysis by Robert Zimmer, an inexpensive paperback. Consent of instructor will not be required. (Next offering to be determined)

M 391C - Topics in Analysis- Theory of Wavelets (Professor Gilbert) This graduate level course will deal with topics of current research interest in the theory of wavelets and their applications to signal analysis, partial differential equations, and mathematical physics. The course will begin with a rapid discussion of relevant ideas from Fourier Analysis of Euclidean Space and Functional Analysis, followed by a development of wavelet theory. A major part of the course will be devoted to applications including: numerical solution of boundary value problems in partial differential equations; speech and image processing; time-frequency methods. Graduate students with applied interests are encouraged to attend, and student participation will be encouraged. (Next offering to be determined)

M 393C - Dynamical Systems (Professor de la LLave). The modern theory of dynamical systems uses a large array of methods )topology, geometry, analysis, computation) to study the possible trajectories of a system whose law of evolution is known. A perennial example is the Newton's laws of gravitation, which, even if known for over 300years still contain surprises. The course aims to introduce students to the variety of tools employed in this rapidly developing field and to develop a taste for problem driven research. No previous experience will be assumed and we will try to accommodate different backgrounds and possible future interests'. The subject lends itself very well to students undertaking research projects ( possibly but not necessarily computational). The instructor will encourage these projects. No prerequisite. Recommended Textbooks: Introduction to the modern theory of dynamical systems by A. Katok & B. Hasselblatt, Cambridge Univ. Press and An Introduction to dynamical systems by a. Arrowsmith & C. M. Place, Cambridge Univ. Press. (Offered Fall 2002, U# 57255, TTH 9:30-11P, RLM 9.166)

M 393C - Introduction to Partial Differential Equations (Professor M. Vishik). The emphasis of this course is on basic examples: the Laplace equation, the wave equation, the heat equation, etc. Topics covered will include: 1) The general first-order nonlinear equation. 2) The 1D wave equation: systems of first-order equations. 3) Cauchy-Kovalevskaya theorem and Homgren uniqueness theorem. 4) The Laplace equation: maximum principle, properties of harmonic functions, Poisson's formula, Dirichlet problem using Hilbert space methods. 5) nD wave equation: the fundamental solution, mixed problems, symmetric hyperbolic systems (if time allows). 6) The heat equation: initial value problem, maximum principle, uniqueness and regularity. (Next offering to be announced.)

M 393C - Navier-Stokes Equations (Professor M. Vishik). Navier-Stokes equations are a basic mathematical model to describe motion of a viscous incompressible fluid. In the early 1930s, Jean Leray proved existence of a weak solution defined globally in time. Uniqueness of weak solutions remains an open question. At the beginning we will cover the fundamentals of the theory including results of Leray and contributions of the later authors such as E. Hopf, O. Ladyzhenskaya, J.-L. Lions, G. Prodi, and others. In the remaining time, we will concentrate on some of the striking recent advances.

1. Variational formulation of the Navier-Stokes equations. Weak solutions.
2. Uniqueness in dimension 2.
3. Wavelets and a divergence-free wavelet basis.
4. The L3 theory of T. Kato.
5. Littlewood-Paley decomposition and paraproducts.
6. Function spaces (Besov, Morrey-Campanato, Lorentz...)
7. Uniqueness of mild solutions of T. Kato.

If time permits, we will also discuss recent results on existence (and nonexistence) of self-similar solutions.

PREREQUISITE: Functional analysis as in the applied Math prelim course or equivalent. PDEs as in the Introduction to PDEs course.

TEXTBOOK(S):
1. J. Leray, OEuvres scientifiques, vol. 2, Springer and Soc. Math. France, 1998.
2. O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Breach, 1969.
3. J.-L. Lions, Quelques méthods de résolution de problèmes aux limites nonlinéaires, Dunod, 1969.
4. R. Temam, Navier-Stokes equations, North Holland, 1984.
5. M. Cannone, Ondelettes, paraproduits et Navier-Stokes, Diderot, 1995.
6. Y. Meyer, Wavelets, paraproducts and Navier-Stokes equations.
7. P.-L. Lions, Mathematical topics in fluid mechanics, vol. 1, Oxford University Press, 1996.

Consent of instructor is not needed. (Next course offering to be determined)

M 393C (same as CAM 393C) - Topics in Partial Differential Equations (Professor L. Caffarelli). This course covers fully nonlinear equations, the monge Ampere and related equations and optimal allocation. The theory of fully nonlinear elliptic equations arises in geometry, probability, and control theory. In the 80's a powerful theory was devised (the Crandall Lions viscosity method). The Monge Ampere equation is a fully nonlinear equation that does not exactly fit the theory due to the large family of invariants, and it is related to affine geometry, and to several problems of optimal allocations that have found applications in image enhancing, frontonogenesis, and assymptotic behavior of classical nonlinear evolution equations. Text: Caffareli Cabre: Fully Nonlinear Elliptic Equations and several articles (Next Offering to be determined)

M 393C - Partial Differential Equations (Professor Souganidis). We shall introduce various classes of initial- boundary-value problems for partial differential equations of systems as models for classical problems of continuum mechanics. These include the deformation or vibration of elastic materials, the flow of fluids, and diffusion through porous media. Extensions to visco-elastic or plastic materials, free-boundary problems, and diffusion through deformable media may be presented. The theory of evolution equations with monotone operators in Banach space will be developed and applied to some of these initial-boundary value problems. Related topics will include variational methods in Sobolev spaces, convex functions and elliptic problems, Cauchy problem and parabolic equations, wave equations, and related systems. The discussion of applications will be self contained, and appropriate notes will be available. The remaining topics are discussed in the text: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Prerequiste: Some familiarity with Lp spaces. M383 or CAM 385 (Methods of Applied Math) would be adequate. No previous PDE course is assumed. (Offered by Professor Cabre, Fall 2002, U# 57265, MWF 12-1P, RLM 11.176)

M 393C (same as CAM 393C) - Rational Mechanics (Professor de la Llave). The goal of this course is to introduce the students to models used to describe mechanical systems and their mathematical treatment (variational methods, qualitative theory of ODE and PDE). No background in Physics will be assumed. We hope that the course will be helpful for mathematicians to develop intuition about ODE's and PDE's and for Physicists and Engineers to get a taste of the Mathematical analysis of models. We will try to accomodate the different backgrounds of the students. Roughly a little less than half of the course will be devoted to discrete systems and the rest to continuum models. Students will be encouraged to carry out a computer project (e.g. solving some ODE's and PDE's that are particularly significant, working out some asymptotic expansions, etc.) Recommended textbooks: Classical Dynamics A Contemporary Approach by Jose and Saletan (this recent book fits exactly a large fraction of the course.) Also, Mathematics Applied to Deterministic Problems in the Natural Sciences (SIAM) by Lin and Segal and A Course in Mathematical Physics Vol I, II, by W. Thirring. Consent of instructor is not required. (Next offering to be determined)

M 393C (same as CAM 393C) - Introduction to Dynamics (Professor de la Llave). This course aims to be an introduction to the general theory of dynamical systems, which is the theory that aims to describe the long term behavior of systems whose law of evolution is known. The point of view adopted is that the way of understanding chaotic behavior is to find landmarks that organize the long term behavior. For the landmarks we discuss, we will present both a discussion of the phenomenology and mathematically rigorous proofs. In most of the cases, the proofs are quite constructive and lend themselves to algorithms. Whether the emphasis lies on the proofs or on the computations depends on the student. To be covered: Consent of instructor is not required. (Next offering to be determined.)

M 393C - Statistical Mechanics (Professor C. Radin) The basic aim of the course is a careful introduction to the qualitative behavior of systems with many degrees of freedom - more specifically, systems of interacting particles in thermodynamic equilibrium. The lectures will concentrate on ``rigorous results'' for models of condensation, melting and other phase transitions. General results, valid beyond exactly solvable models, will be emphasized. We will start with a brief summary of thermodynamics, then discuss the thermodynamic limit, various existence theorems for phase transitions (van der Waals, Lee-Yang, Peierls etc.) and end with results on correlation functions and order parameters. The lectures will be aimed at a diverse audience of graduate students in mathematics, physics and other sciences, and will attempt to accommodate a diversity of backgrounds. There will not be a textbook for the course, though we will be getting much of the material from "Statistical Physics" by David Ruelle. Consent of instructor will not be required. (Next offering to be determined)

M 393C - Topics in Applied Mathematics--Nonlinear Waves (Professor J. Bona). The following topics will be covered: 1) Derivation and analysis of model equations for the two-way propagation of nonlinear dispersive waves in shallow water (Boussinesq equations). 2) Existence and stability of solitary-wave solutions. 3) Smoothing properties and singularity formation of dispersive evolution equations. 4) Models for wave-bottom interaction. (This topic considers wave motion over a sandy bed and is concerned with the evolution of the two free boundaries - the water and the sand. Of particular interest is the formation of stable, long-term structures such as sand bars, sand ridges and the like). 5) Derivation and analysis of model equations for internal-wave propagation in two- and three-fluid systems and in continuously stratified flows. The course assumes the student has some familiarity with mechanics, real variables, Fourier analysis, partial differential equations and functional analysis, but when prior knowledge is used, it is usually developed rapidly within the course context itself (Next offering to be determined).

M 393C (same as CAM 393C) - Kinetic Theory (Professor Gamba). (Offered Spring 2002, U# 55745, TTH 11-12:30p, RLM 11.176).

M 393C (same as CAM 393C) - TPC Nonlin Partial Diff Equatn (Professor Souganidis). (Offered Spring 2002, U# 55750, MF 9:30-11, RLM 10.176).

M 393C (same as CAM 393C) - Computational Modeling (Professor Gonzalez). (Offered Spring 2002, U# 55730, TTH 2-3:30p, RLM 12.166).

M 393D (same as CAM 393D and C S 393D) - Approximation Theory (Professor W. Cheney). Some classical approximation theory; polynomials, rational functions, trigonometric polynomials, spline functions. Emphasis is, however, on modern developments, including radial basis functions for interpolation of multivariate functions, approximation by ridge functions, wavelets, quasi-interpolation, and neural networks. The material is taken from the book, A Course in Approximation Theory by Cheney and Light (Brooks/Cole Publishing Co., 1999). Prospective students can look at the book in the Physics-Mathematics-Astronomy Library. (Offered Fall 2002, U# 57270, MWF 2-3P, RLM 11.176).


M 393N (also C S 393N and CAM 393M) - Numerical Solutions of Elliptic Partial Differential Equations (Professor D. Young). The course is concerned with the numerical solution of elliptic partial differential equations and the solution of large systems of linear algebraic equations with sparse matrices by iterative methods. Particular emphasis will be given to methods which are suitable for use on supercomputers. Included are: a review of relevant topics in linear algebra and matrix theory; basic iterative methods including the Jacobi, Gauss-Seidel, successive overrelaxation (SOR) and symmetric SOR methods. Methods derived from approximate factorizations of the given matrix will also be considered; the successive overrelaxation method for consistently ordered matrices and for L-matrices; acceleration of basic iterative methods by Chebyshev acceleration and by conjugate gradient acceleration; adaptive iterative procedures; special procedures for nonsymmetric systems; and multigrid methods. Additional topics: time dependent problems, fast direct methods, singular systems and nearly singular systems such as often arise in problems with Neumann boundary conditions. The course will cover both the theory and actual implementation of a number of methods. Homework assignments will include some problems requiring the use of the computer. A number of software packages, developed at the University of Texas and elsewhere will be used. Prerequisites: M 386K/CS 386K, "Numerical Treatment of Differential Equations" or equivalent, is recommended but not required. A review of the relevant material will be given. Students should have background in matrix algebra, at least at the level of M340L. The instructor should be contacted in case of questions about prerequisites. Textbooks (1) L. A. Hageman and D. M. Young, Applied Iterative Methods, Academic Press, 1981. (A limited number of loan copies will be available; (2) D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, 1971. (A limited number of loan copies will be available); (3) A workbook will be used which can be purchased at a local copying service. (Next offering to be determined)

M 427K - Avanced Calculus for Application I (Professor Bichteler). Infinite series, ordinary and partial differential equations. Five class hours a week for one semester. Prerequisite: Mathematics 408D with a grade of at least a C. Computer component will be featured and class projects will be provided for interested students. (Offered Fall 2002, U# 56775, MW 12-1P, GEA 105 & TTH 12:30-2P, CPG 2.214)

Mechanical Engineering

M E 380Q (Topic 3) - Perturbation Methods (Professor D. Wilson). Introduction to perturbation methods; regular expansions and sources of nonuniformities; methods of strained coordinates and multiple scales; a method of matched asymptotic and composite expansions. The course places strong emphasis on the relationship between the physical and the mathematical basis and on the crucial role of nondimensionalization in problem solving (Next course offering to be determined).

M E 381P (Topic 1) - Fundamentals of Incompressible Flow (Professor R. Panton). Course content: (Offered Fall 2002, U# 17490, MWF 10-11, ETC 7.146)

M E 381P (Topic 3) - Dynamics of Turbulent Flow (Professor D. Bogard). Turbulence fundamentals including scales of turbulence and Reynolds averaged momentum and turbulence kinetic energy equations. Statistical analysis and brief overview of experimental techniques. Homogeneous, isotropic turbulence including spectral dynamics. Stability and transition. Detailed study of turbulent wall flows. Brief overviews of turbulent free shear flows and turbulence modeling. (Next offering to be determined)

M E 381P (Topic 5) - Incompressible Flow II: Applications (Professor R. Panton). Course content: Evaluation: Homework 50%, midterm 25%, final project 25%. (Next offering to be determined)

M E 381P (Topic 6) - Modeling of Turbulent Flows (Professor Bogard). Objective:develop a physical understanding of the various models of turbulence in the open literature and how they are used in numerical simulations. Note: previous course in turbulence not required Scope:intro to the physics of turbulence for homogeneous, wall, and free shear flows turbulent transport equations for low-speed, constant property flows mixing-length (momentum) and turbulent Prandtl number (energy) turbulence models advanced turbulence models: k-l, k-e, Reynolds stress, turbulent heat flux testing of various models using the TEXSTAN finite-difference boundary layer code (Next offering to be determined)

M E 381R (Topic 2) - Convection Heat Transfer (Professor M. E. Crawford). Laminar and turbulent transport in boundary layers and inside tubes, with equal emphasis on momentum and energy transport; compressible and property effects, numerical simulation of convective transport. (Next offering to be determined)

M E 382N (Topic 1) - Intro. Computational Fluid Dynamics (Professor K. S. Ball). Applied numerical analysis, including solution of linear algebraic equations and ordinary and partial differential equations; modeling of physical processes, including fluid flow and heat and mass transfer, use of general purpose computer codes, including commercial computational fluid dynamics software packages. The CFD software packages include FIDAP and FLUENT which are two of the most widely used codes in industry. In this project-oriented class, practical problems like those encountered in industry will be considered (for example, turbulent flow through a heat exchanger with conjugate heat transfer). Students in this class will gain valuable experience with realistic engineering problems that are well beyond the scope of formal (textbook) analysis. This course is intended to serve as an introduction to the field of CFD and its applications, and no prior CFD experience is required (although a basic understanding of fluid mechanics and heat transfer is desirable) (Offered Fall 2002, U# 17510, MWF 11-12p, ETC 7.146).

M E 382N (Topic 2) - Spectral Methods in Fluid Dynamics (Professor K. S. Ball). The use of spectral approximation theory to solve partial differential equations will be examined in detail. The focus of the course will be on problems in fluid dynamics and heat transfer, including transition and turbulence, but will also be of general interest to a wide cross-section of all engineering students, as well as students in physics and other disciplines, who are interested in obtaining highly accurate solutions to partial differential equations. The course will begin with a review of the theory of spectral approximation, e .g., the use of Fourier series and Chebyshev polynomials in the representation of functions followed by ways of computing these functions, including the use of FFT's. In class actual working codes will be developed. Relatively simple programming assignments will be given, which will be selected to cover as broad a range of representative problems as possible. Ways of extending the codes to more general or complex flows will also be covered. Students will leave the course with working codes to use in their own research and will learn about the many other uses of spectral approximation, e.g., in time series analysis, power spectrum estimation, calculations of correlations and pdf's, etc. Prerequisites: No special background is required beyond calculus and differential equations. A basic understanding of fluid mechanics and heat transfer is desirable. Familiarity with other numerical techniques helpful, but not essential. Must be competent in a scientific programming language such as C or FORTRAN (Next course Offering to be determined)

M E 397 - same as PHY 392T (see for description).

Physics

PHY 382M - Fluid Mechanics (Professor Swift). Fundamental conservation laws and hydrodynamic equations; flow kinematics, streamlines, vortices; ideal fluid flow, potential flow, two-dimensional flow, complex potential and velocity, three-dimensional flow; viscous flow, Couette flow, Poiseuille flow; boundary layers, thickness of boundary layer, boundary layer equations; hydrodynamic instabilities: Rayleigh-Taylor, Kelvin-Helmholtz, Rayleigh-B=E9nard, Taylor-Couette, Orr-Sommerfeld Equation; turbulence, Kolmogorov Scaling, beta-model of intermittency.
Texts: Fundamental Mechanics of Fluids, Second Edition by I.G. Currie (McGraw-Hill, (1993)), Hydrodynamic Stability by P.G. Drazin and W.H. Reid (Cambridge U. Press (1981)) neither required
Grading: Approximately four problems sets = 70% of the grade. Final, in-class exam = 30% of the grade.
(Offered Fall 2002, U# 58890, MWF 9-10, RLM 6.114)

PHY 382N - Nonlinear Dynamics (Professor J. Swift). Topics will be: 1) Basic concepts of evolution and stability: deterministic evolution, different concepts of stability (asymptotic, global, linear, etc.), phase portraits, attractors, hyperbolicity, bifurcations; introduction of basic concepts - stability, sensitive dependence on initial conditions, etc. by means of the logistic map. 2) Examples of instabilities: Rayleigh-Benard convection, Taylor-Couette instability, reaction-diffusion systems. 3) Low dimensional dynamical systems: center manifold theorem and normal forms, dynamics and bifurcations in one and two dimensions. 4) Chaos: Poincare maps, Lorenz model, subharmonic route to chaos, intermittency, quasi-periodicity (circle map). 5) Characterization of temporal chaos: Lyapunov exponents, entropy, dimensions (fractals), implementations with data from experiment; shadowing lemma; noise reduction. 6) Pattern formation: basics of pattern formation, envelope formalism, phase diffusion, nonlinear wavelength selection, Turing patterns. 7) Dynamics and turbulence: spatio-temporal intermittency, hydrodynamics and turbulence. 8) Hamiltonian systems: integrability. Text: Dissipative Structures and Weak Turbulence, P. Manneville (Academic Press (1990)) Grading: Homework assignments (50%) and a comprehensive, in-class, final exam (50%) will form the basis for the grade in the course. PHY 392K is not a prerequisite. This course is only offered every two years. (Next offering to be determined)

PHY 382S - Nonlinear Dynamics Seminar (Professor H. Swinney). (Offered Fall 2002, U# 58895, MWF 1-2 pm, RLM 11.204).

PHY 385T - Irreversible Processes and Dynamical Systems (Professor I. Prigogine). Lecture series (Next offering to be determined).

PHY 391M - Nonlinear Plasma Theory (Professor W. Horton).
Course objective is to develop an understanding of the basic phenomena of nonlinear waves and coherent structures in ionized gases and to develop an understanding of the standard plasma physics paradigms.
1. Properties of Nonlinear Oscillations
2. Hamiltonian Dynamics, Poincare Surface of Section, KAM theory
3. Chaotic and Regular Orbits, Lyapunov exponents
4. The Standard map, whisker map, phase space structures
5. The diffusion approximations
6. Examples of orbits from geostrophic (EXB) flows and orbits in magnetic confinement systems
7. Solitons in plasmas and fluids
8. Vortex Structures in plasmas and fluids
9. Computer simulations for the nonlinear pde's
10. Renormalized turbulence theory

Examples will be drawn from:
The course will have homework and one individually assigned research project selected by the student from course topics relevant to his own research. There is no final exam. The course instruction will have lectures and self-paced individual instructions. Students will be given an account on the IFS workstation Orion with access to 3D visualization graphics and a help desk on how to prepare graphics files for publication. Accounts on the University super-computers will also be available for those designing a simulation project. Graduate level plasma physics is NOT a prerequisite. Text: Chaos and Structures in Nonlinear Plasma by Horton and Ichikawa, World Scientific, 1996. For more information please visit Dr. Horton's website. (Offered Fall 2002, U# 58985, TTH 2-3:30P, RLM 5.126)

PHY 392K - Solid-State Physics (Professor Qian Niu). (Next course offering to be determined).

PHY 392K - Solid-State Physics (Professor J.B. Shih). Topics: Drude theory of metals, Sommerfeld theory of metals, crystal lattices, reciprocal lattice, x-ray diffraction, band theory, electron transport, electron-electron interaction, phonons in insulators and metals, and electronphonon interaction as time permits. Prerequisites: PHY 389K and PHY 375S or the equivalent. Text (required): Solid State Physics by Ashcroft and Mermin. Grading policy: homework=50% and final exam=50%. (Next offering to be determined)

PHY 392T - Biological Physics (Professor Josef Käs). Biological physics emphasis novel biologically inspired condensed matter physics and innovative physical techniques which are applicable to biology. A good example of a physical instrument that benefits biology is atomic force microscopy. Conversely, recent fluorescent microscopy studies of DNA and filamentous proteins of cells have greatly advanced polymer physics, a subfield of condensed matter physics. The biophysics course offered in the spring will illustrate this productive interplay between biology and physics, pointing towards future directions like combined approaches of molecular biology and polymer physics. In particular, the course will focus on connections between cell biology and soft condensed matter physics. Content: Note: The course is also suitable for undergraduates, biologists (e.g. molecular biologists) and engineers (e.g. bioengineers, chemical engineers). (Next course offering to be determined.)

PHY 392T - Polymer Physics (Professor Josef Käs). Polymers such as ploystyrene make up a large number of the synthetic materials used in everyday life. Further, biological macromolecules - proteins - are nothing other than complex polymer chains. For example, shape and motility of eukaryotic cells is determined by the cytoskeleton, a network of protein filaments underlying the cell membrane. Given the complexity and chemical diversity of polymers it is surprising that universal theories of polymer physics can be derived. This is possible because the molecules themselves are very large and their behavior is dominated by the large scale properties of the molecules. Over the past twenty years polymer physics has undergone a dramatic evolution. However, these subjects are not covered in the classical curriculum of condensed matter physics. This course tries to fill in this gap by introducing selected topics in polymer physics -- an important subdomain of soft condensed matter physics with interdisciplinary ties to biology, materials science and chemical engineering. The first part of the course will establish the significant parameters that characterize a polymer chain and show how these parameters can be determined experimentally. Based on these static properties of a polymer, the dynamics of a single polymer chain will be derived. Then we will focus on the macroscopic properties of polymeric materials and how they relate to the properties of its constituent filaments. Among various dynamical properties of polymeric liquids, an important property is their viscoelastic response to mechanical forces, which can be characterized by rheological measurements. For example if one stretches chewing gum and releases it quickly, then the response is elastic, yet chewing gum is a liquid and can fill a container of any shape. A molecular theory of this behavior will be derived. Experiments will be described - in particular recently developed optical techniques - which allow one to relate viscoelasticity to the dynamics of single chains of the polymeric liquid. (Next offering to be determined)

PHY 392T (also M E 397) - Shock Compression of Condensed Matter (Professor Stephan Bless). The course will cover fundamental and applied aspects of the response of solids to impulse loading, with an emphasis on shock response. Topics will include propagation of shock waves in liquids and solids, high pressure behavior of solids, strength effects at high pressures and high strain rates, experimental techniques, and engineering applications. The course will be helpful to students interested in the behavior of matter at high energy densities, the nature of flow and failure of solids, impact physics, penetration mechanics, acoustics, spacecraft shielding from meteorites, geophysical cratering, shock metamorphosis, crash mechanics, impact erosion, fracture mechanics, etc. (Next offering to be determined)

PHY 395K - Nonlinear Optics and Lasers (Professor M.C. Downer). Topics:
Prerequisites: PHY 387K and 389K (or equivalent, or consent of instructor). Texts: A. Yariv, Quantum Electronics (Wiley & Sons, 1975), required; R.W. Boyd, Nonlinear Optics (Academic Press, 1992); Y.R. Shen, The Principles of Nonlinear Optics (Wiley & Sons, 1984); A.E. Siegman, Lasers (University Science Books, 1986). Yariv will be the core required text which you should buy or have readily available, but I will also use material from the other books and from recent research literature. All four books are on reserve in the PMA library along with a notebook containing other material relevant to the course. Requirements: several homeworks; final presentation, paper, or take-home final exam. Note: This course is strongly recommended for students whose research involves lasers, spectroscopy, and/or optics directly. However, it is not a special topics course. It will also be appropriate for students in unrelated research areas who would like a broad survey of the principles of nonlinear optics and lasers to satisfy an advanced course distribution requirement. (Next course offering to be determined)
Comments

9 August 2002