APAM E3105x. Programming methods for scientists and engineers. 3 pts. Lect: 2.5 Lab: 1 Not offered in 2009-2010.

Introduction to modern techniques of computer programming for the numerical solutions to familiarity with basic and advanced concepts of modern numerical programming and acquire practical experience solving representative problems in math and physics.

APMA E4001y. Principles of Applied Mathematics. Lect: 3. 3 pts.

Prerequisites: Introductory Linear Algebra required. Ordinary Differential Equations recommended. Review of finite-dimensional vector spaces and elementary matrix theory. Linear transformations, change of basis, eigenspaces. Matrix representation of linear operators and diagonalization. Applications to difference equations, Markov processes, ordinary differential equations, and stability of nonlinear dynamical systems. Inner product spaces, projection operators, orthogonal bases, Gram-Schmidt orthogonalization. Least squares method, pseudo-inverses, singular value decomposition. Adjoint operators, Hermitian and unitary operators, Fredholm Alternative Theorem. Fourier series and eigenfunction expansions. Introduction to the theory of distributions and the Fourier Integral Transform. Green's functions. Application to Partial Differential Equations.

APMA E4101x. Introduction to Dynamical Systems. 3 pts. Lect: 3. Prerequisites:APMA E2101 (or MATH V1210) and APMA E3101 or their equivalents, or permission of the instructor.

An introduction to the analytic and geometric theory of dynamical systems; basic existence, uniqueness and parameter dependence of solutions to ordinary differential equations; constant coefficient and parametrically forced systems; Fundamental solutions; resonance; limit points, limit cycles and classification of flows in the plane (Poincare-Bendixson Theorem); conservative and dissipative systems; linear and nonlinear stability analysis of equilibria and periodic solutions; stable and unstable manifolds; bifurcations, e.g. Andronov-Hopf; sensitive dependence and chaotic dynamics; selected applications.

APPH E4130y. Physics of solar energy. 3 pts. Lect: 3. Prerequisites: general physics (PHYS C1403 or C1602) and mathematics including ordinary differential equations and complex numbers (such as MATH V1202 or E1210) or permission of the instructor.

The physics of solar energy including solar radiation, the analemma, atmospheric efforts, thermodynamics of solar energy, physics of solar cells, energy storage and transmission, and physics and economics in the solar era.

APMA E4150x. Applied functional analysis. 3 pts. Prerequisites: Advanced calculus and a course in basic analysis, or instructor's approval.

Introduction to modern tools in functional analysis that are used in the analysis of deterministic and stochastic partial differential equations and in the analysis of numerical methods: metric and normed spaces, Banach space of continuous functions, measurable spaces, the contraction mapping theorem, Banach and Hilbert spaces bounded linear operators on Hilbert spaces and their spectral decomposition, and time permitting distributions and Fourier transforms.

APMA E4200x. Partial Differential Equations. 3 pts. Lect: 3.

Prerequisite: a course in ordinary differential equations. Techniques of solution of partial differential equations. Separation of the variables. Orthogonality and characteristic functions, nonhomogeneous boundary value problems. Solutions in orthogonal curvilinear coordinate systems. Applications of Fourier integrals, Fourier and Laplace transforms. Problems from the fields of vibrations, heat conduction, electricity, fluid dynamics, and wave propagation are considered.

APMA E4204x. Functions of a Complex Variable. 3 pts. Lect. 3. Prerequisite: MATH V1202 or the equivalent. Complex numbers, functions of a complex variable, differentiation and integration in the complex plane. Analytic functions, Cauchy integral theorem and formula, Taylor and Laurent series, poles and residues, branch points, evaluation of contour integrals. Conformal mapping. Schwarz-Christoffel transformation. Applications to physical problems.

APMA E4300y. Introduction to Numerical Methods. 3 pts. Lect: 3. Prerequisites:MATH V1201, MATH E1210, and APMA E3101 or their equivalents. Some programming experience and Matlab will be extremely useful.

Introduction to fundamental algorithms and analysis of numerical methods commonly used by scientists, mathematicians and engineers. This course is designed to give a fundamental understanding of the building blocks of scientific computing that will be used in more advanced courses in scientific computing and numerical methods for PDE's. Topics include numerical solutions of algebraic systems, linear least-squares, eigenvalue problems, solution of non-linear systems, optimization, interpolation, numerical integration and differentiation, initial value problems and boundary value problems for systems of ODE's. All programming exercises will be in Matlab.

APMA E4301x. Numerical Methods for Partial Differential Equations. 3 pts. Lect: 3. Prerequisites:APMA E4300 and APMA E3102 or APMA E4200 or equivalents.

Numerical solution of partial differential equations (PDE) arising in various physical fields of application. Finite difference, finite element, and spectral methods. Elementary finite volume methods for conservation laws. Time stepping, method of lines, and simultaneous space-time discretization. Direct and iterative methods for boundary-value problems. Applied numerical analysis of PDE, including sources of numerical error and notions of convergence and stability, to an extent necessary for successful numerical modeling of physical phenomena. Applications will include the Poisson equation, heat equation, wave equation, and nonlinear equations of fluid, solid, and gas dynamics. Homework assignments will involve substantial programming.

AMCS E4302x. Parallel Scientific Computing. 3 pts. Lect: 3. Not offered in 2009-2010. Prerequisites:APMA E3101, APMA E3102, and APMA E4300, or their equivalents. Corequisites:APMA E4301, and programming ability in C/C++ or FORTRAN/F90.

An introduction to the concepts, the hardware and software environments, and selected algorithms and applications of parallel scientific computing, with an emphasis on tightly coupled computations that are capable of scaling to thousands of processors. Includes high-level descriptions of motivating applications and low-level details of implementation, in order to expose the algorithmic kernels and the shifting balances of computation and communication between them. Students run demonstration codes provided on a Linux cluster. Modest programming assignments using MPI and PETSc culminate in an independent project leading to an in-class report.

APMA E4400y. Introduction To Biophysical Modeling. 3 pts. Lect: 3. Prerequisites:PHYS W1401 or equivalent, and APMA E2101 or MATH E1210 or equivalent.

Introduction to physical and mathematical models of cellular and molecular biology. Physics at the cellular scale (viscosity, heat, diffusion, statistical mechanics). RNA transcription and regulation of genetic expression. Genetic and biochemical networks. Bioinformatics as applied to reverse-engineering of naturally-occurring networks and to forward-engineering of synthetic biological networks. Mathematical and physical aspects of functional genomics.

APMA E6209x. Approximation Theory. 3 pts. Lect: 2. Not offered in 2009-2010. Prerequisite: MATH W4061 or some knowledge of modern analysis. Theory and application of approximate methods of analysis from the viewpoint of functional analysis. Approximate numerical and analytical treatment of linear and nonlinear algebraic, differential, and integral equations. Topics include function spaces, operators in normed and metric spaces, fixed point theorems and their applications.

APMA E6301y. Analytic Methods for Partial Differential Equations. 3 pts. Lect: 2. Prerequisites: Advanced calculus, basic concepts in analysis, APMA E3101 and E4200 or their equivalents, or permission of the instructor. Introduction to analytic theory of PDEs of fundamental and applied science; wave (hyperbolic), Laplace and Poisson equations (elliptic), heat (parabolic) and Schroedinger (dispersive) equations; fundamental solutions, Green's functions, weak/distribution solutions, maximum principle, energy estimates, variational methods, method of characteristics; elementary functional analysis and applications to PDEs; introduction to nonlinear PDEs, shocks; selected applications.

APMA E6302x. Numerical Analysis of Partial Differential Equations. 3 pts. Lect: 2.

Prerequisite: APMA E3102 or E4200. Numerical analysis of initial and boundary value problems for partial differential equations. Convergence and stability of the finite difference method, the spectral method, the finite element method and applications to elliptic, parabolic, and hyperbolic equations.

APMA E6304y. Integral Transforms. 3 pts. Lect: 2. Not offered in 2009-2010.

Prerequisites: APMA E4204 and MATH E1210, or their equivalents. Laplace, Fourier, Hankel, and Mellin transforms. Selection of suitable transform for a given partial differential equation boundary value problem. Operational properties of transforms. Inversion theorems. Approximate evaluation of inversion integrals for small and large values of parameter. Application to the solution of integral equations.

APMA E6901x and y. Special Topics In Applied Mathematics. 3 pts. Lect: 3.

Prerequisites: advanced calculus and junior year applied mathematics, or their equivalents. This course may be repeated for credit. Topics and instructors from the Applied Mathematics Committee and the staff change from year to year. For students in engineering, physical sciences, biological sciences, and other fields.

APMA E8308y. Asymptotic Methods In Applied Mathematics. 3 pts. Lect: 2. Prerequisites:APMA E4204 or the equivalent.

Asymptotic treatment of ordinary and partial differential equations in problems arising in applied mathematics. Asymptotic series. Asymptotic evaluation of integrals. Expansion of solutions of ordinary differential equations: connection problem and turning points. Stoke's phenomenon. Differential equations with a parameter: "boundary layer" phenomenon. Application to partial differential equations: problems from fluid dynamics, wave propagation theory, electromagnetic theory.

APPH E4010x. Introduction To Nuclear Science. 3 pts.

Prerequisites: MATH V1202 and E1210 and PHYS C1403 or their equivalents. This introductory course is for individuals with an interest in medical physics and other branches of radiation science. Topics covered include: basic concepts, nuclear models, semi-empirical mass formula, interaction of radiation with matter, nuclear detectors, nuclear structure and instability, radioactive decay process and radiation, particle accelerators, and fission and fusion processes and technologies.

APPH E4018y. Applied Physics Laboratory. 2 pts. Lab: 4.

Prerequisite: ELEN E3401 or the equivalent. Typical experiments are in the areas of plasma physics, microwaves, laser applications, optical spectroscopy physics, and superconductivity.

APPH E4100x. Quantum Physics of Matter. 3 pts. Lect: 3. Prerequisites:APPH E3100. Corequisites:APMA E3102 or equivalent.

Basic theory of quantum mechanics, well and barrier problems, the harmonic oscillator, angular momentum identical particles, quantum statistics, perturbation theory and applications to the quantum physics of atoms, molecules, and solids.

APPH E4112y. Laser Physics. 3 pts. Not offered in 2009-2010. Prerequisites: Recommended but not required: APPH E3100 and APPH E3300 or their equivalents.

Optical resonators, interaction of radiation and atomic systems, theory of laser oscillation, specific laser systems, rate processes, modulation, detection, harmonic generation, and applications.

APPH E4200x. Physics of Fluids. 3 pts. Prerequisite: APMA E3102 or equivalent; PHYS 1401 or 1601 or equivalent. An introduction to the physical behavior of fluids for science and engineering students. Derivation of basic equations of fluid dynamics: conservation of mass, momentum, and energy. Dimensional analysis. Vorticity. Laminar boundary layers. Potential flow. Effects of compressibility, stratification, and rotation. Waves on a free surface; shallow water equations. Turbulence.

APPH E4210y. Geophysical Fluid Dynamics. 3 pts. Lect. 3. Prerequisites:APMA E3101, APMA E3102 (or equivalents) and APPH E4200 (or equivalent), or permission from instructor. Fundamental concepts in the dynamics of rotating, stratified flows. Geostrophic and hydrostatic balances, potential vorticity, f and beta plane approxima-tions, gravity and Rossby waves, geostrophic adjustment and quasigeostrophy, baroclinic and barotropic instabilities, Sverdrup balance, boundary currents, Ekman layers.

APPH E4300x. Applied Electrodynamics. 3 pts. Lect: 3. Prerequisites:APPH E3300. Overview of properties and interactions of static electric and magnetic fields. Study of phenomena of time dependent electric and magnetic fields including induction, waves, and radiation as well as special relativity. Applications are emphasized.

APPH E4301y. Introduction To Plasma Physics. 3 pts. Lect: 3. Prerequisites:PHYS W3008 or APPH E3300.

Definition of a plasma. Plasmas in laboratories and nature, plasma production. Motion of charged particles in electric and magnetic fields, adiabatic invariants. Heuristic treatment of collisions, diffusion, transport, and resistivity. Plasma as a conducting fluid. Electrostatic and magnetostatic equilibria of plasmas. Waves in cold plasmas. Demonstration of laboratory plasma behavior, measurement of plasma properties. Illustrative problems in fusion, space, and nonneutral or beam plasmas.

APPH E4990x and y. Special topics in Applied Physics. 1-3 pts. Prerequisites: Permission of the instructor.

This course may be repeated for credit. Topics and instructors change from year to year. For advanced undergraduate students and graduate students in engineering, physical sciences, and other fields.

APPH E6081x. Solid State Physics, I. 3 pts. Lect: 3. Prerequisites:APPH E3100 or the equivalent. Knowledge of statistical physics on the level of MSAE E3111 or PHYS G4023 strongly recommended.

Crystal structure, reciprocal lattices, classification of solids, lattice dynamics, anharmonic effects in crystals, classical electron models of metals, electron band structure, and low-dimensional electron structures.

APPH E6082y. Solid State Physics, II. 3 pts. Lect: 3.

Prerequisite: APPH E6081 or the instructor's permission. Semiclassical and quantum mechanical electron dynamics and conduction, dielectric properties of insulators, semiconductors, defects, magnetism, superconductivity, low-dimensional structures, and soft matter.

APPH E6101x. Plasma Physics, I. 3 pts. Lect: 3. Prerequisites:APPH E4300.

Debye screening. Motion of charged particles in space- and time-varying electromagnetic fields. Two-fluid description of plasmas. Linear electrostatic and electromagnetic waves in unmagnetized and magnetized plasmas. The magnetohydrodynamic (MHD) model, including MHD equilibrium, stability, and MHD waves in simple geometries. Fluid theory of transport.

APPH E6102y. Plasma Physics, II. 3 pts. Lect: 3. Prerequisites:APPH E6101.

Magnetic coordinates. Equilibrium, stability, and transport of torodial plasmas. Ballooning and tearing instabilities. Kinetic theory, including Vlasov equation, Fokker-Planck equation, Landau damping, kinetic transport theory. Drift instabilities.

APPH E6110x. Laser Interactions With Matter. 3 pts. Lect: 3. Not offered in 2009-2010. Prerequisites: APPH E4112 or the equivalent, and quantum mechanics. Principles and applications of laser-matter coupling, non-linear optics, three- and four-wave mixing, harmonic generation, laser processing of surfaces, laser probing of materials, spontaneous and stimulated light scattering, saturation spectroscopy, multiphoton excitation, laser isotope separation, transient optical effects.

APPH E9142x-E9143y. Applied Physics Seminar. 3 pts. Sem: 3. These courses may be repeated for credit. Selected topics in applied physics.