The course begins with the basics of compressible fluid dynamics, including governing equations, thermodynamic context and characteristic parameters. The next large block of lectures covers quasi-one-dimensional flow, followed by a discussion of disturbances and unsteady flows. The second half of the course comprises gas dynamic discontinuities, including shock waves and detonations, and concludes with another large block dealing with two-dimensional flows, both linear and non-linear.

Fundamental concepts and results for the compressible flow of gases. Topics include: appropriate conservation laws; propagation of disturbances; isentropic flows; normal shock wave relations, oblique shock waves, weak and strong shocks, and shock wave structure; compressible flows in ducts with area changes, friction, or heat addition; heat transfer to high speed flows; unsteady compressible flows, Riemann invariants, and piston and shock tube problems; steady 2D supersonic flow, Prandtl-Meyer function; and self-similar compressible flows. Emphasis on physical understanding of the phenomena and basic analytical techniques. 2.26 is a 6-unit Honors-level subject serving as the Mechanical Engineering department's sole course in compressible fluid dynamics. The prerequisites for this course are undergraduate courses in thermodynamics, fluid dynamics, and heat transfer. The goal of this course is to lay out the fundamental concepts and results for the compressible flow of gases. Topics to be covered include: appropriate conservation laws; propagation of disturbances; isentropic flows; normal shock wave relations, oblique shock waves, weak and strong shocks, and shock wave structure; compressible flows in ducts with area changes, friction, or heat addition; heat transfer to high speed flows; unsteady compressible flows, Riemann invariants, and piston and shock tube problems; steady 2D supersonic flow, Prandtl-Meyer function; and self-similar compressible flows. The emphasis will be on physical understanding of the phenomena and basic analytical techniques.

Theory for programmers. Introduction to programming and computability theory based on a term-rewriting, "substitution" model of computation by Scheme programs with side-effects. Computation as algebraic manipulation: provable and valid inequalities for multivariate polynomials. Scheme evaluation as algebraic manipulation and term rewriting theory. Paradoxes from self-application and introduction to formal programming semantics. Undecidability of the Halting Problem for Scheme. Properties of recursively enumerable sets, leading to Incompleteness Theorems for Scheme equivalences. Introduction to logic for program specification and verification. Hilbert's Tenth Problem. Alternate years. 6.844 is a graduate introduction to programming theory, logic of programming, and computability, with the programming language Scheme used to crystallize computability constructions and as an object of study itself. Topics covered include: programming and computability theory based on a term-rewriting, "substitution" model of computation by Scheme programs with side-effects; computation as algebraic manipulation: Scheme evaluation as algebraic manipulation and term rewriting theory; paradoxes from self-application and introduction to formal programming semantics; undecidability of the Halting Problem for Scheme; properties of recursively enumerable sets, leading to Incompleteness Theorems for Scheme equivalences; logic for program specification and verification; and Hilbert's Tenth Problem.

" 6.004 offers an introduction to the engineering of digital systems. Starting with MOS transistors, the course develops a series of building blocks ‰ŰÓ logic gates, combinational and sequential circuits, finite-state machines, computers and finally complete systems. Both hardware and software mechanisms are explored through a series of design examples. 6.004 is required material for any EECS undergraduate who wants to understand (and ultimately design) digital systems. A good grasp of the material is essential for later courses in digital design, computer architecture and systems. The problem sets and lab exercises are intended to give students "hands-on" experience in designing digital systems; each student completes a gate-level design for a reduced instruction set computer (RISC) processor during the semester."

This course covers the algorithmic and machine learning foundations of computational biology combining theory with practice. We cover both foundational topics in computational biology, and current research frontiers. We study fundamental techniques, recent advances in the field, and work directly with current large-scale biological datasets.

A computational camera attempts to digitally capture the essence of visual information by exploiting the synergistic combination of task-specific optics, illumination, sensors and processing. In this course we will study this emerging multi-disciplinary field at the intersection of signal processing, applied optics, computer graphics and vision, electronics, art, and online sharing through social networks. If novel cameras can be designed to sample light in radically new ways, then rich and useful forms of visual information may be recorded -- beyond those present in traditional photographs. Furthermore, if computational process can be made aware of these novel imaging models, them the scene can be analyzed in higher dimensions and novel aesthetic renderings of the visual information can be synthesized.We will discuss and play with thermal cameras, multi-spectral cameras, high-speed, and 3D range-sensing cameras and camera arrays. We will learn about opportunities in scientific and medical imaging, mobile-phone based photography, camera for HCI and sensors mimicking animal eyes. We will learn about the complete camera pipeline. In several hands-on projects we will build physical imaging prototypes and understand how each stage of the imaging process can be manipulated.

This course is an introduction to computational theories of human cognition. Drawing on formal models from classic and contemporary artificial intelligence, students will explore fundamental issues in human knowledge representation, inductive learning and reasoning. What are the forms that our knowledge of the world takes? What are the inductive principles that allow us to acquire new knowledge from the interaction of prior knowledge with observed data? What kinds of data must be available to human learners, and what kinds of innate knowledge (if any) must they have?

Introduces design as a computational enterprise in which rules are developed to compose and describe architectural and other designs. The class covers topics such as shapes, shape arithmetic, symmetry, spatial relations, shape computations, and shape grammars. It focuses on the application of shape grammars in creative design, and teaches shape grammar fundamentals through in-class, hands-on exercises with abstract shape grammars. The class discusses issues related to practical applications of shape grammars.

Why has it been easier to develop a vaccine to eliminate polio than to control influenza or AIDS? Has there been natural selection for a 'language gene'? Why are there no animals with wheels? When does 'maximizing fitness' lead to evolutionary extinction? How are sex and parasites related? Why don't snakes eat grass? Why don't we have eyes in the back of our heads? How does modern genomics illustrate and challenge the field? This course analyzes evolution from a computational, modeling, and engineering perspective. The course has extensive hands-on laboratory exercises in model-building and analyzing evolutionary data.

Study and discussion of computational approaches and algorithms for contemporary problems in functional genomics. Topics include DNA chip design, experimental data normalization, expression data representation standards, proteomics, gene clustering, self-organizing maps, Boolean networks, statistical graph models, Bayesian network models, continuous dynamic models, statistical metrics for model validation, model elaboration, experiment planning, and the computational complexity of functional genomics problems.

Topics in surface modeling: b-splines, non-uniform rational b-splines, physically based deformable surfaces, sweeps and generalized cylinders, offsets, blending and filleting surfaces. Non-linear solvers and intersection problems. Solid modeling: constructive solid geometry, boundary representation, non-manifold and mixed-dimension boundary representation models, octrees. Robustness of geometric computations. Interval methods. Finite and boundary element discretization methods for continuum mechanics problems. Scientific visualization. Variational geometry. Tolerances. Inspection methods. Feature representation and recognition. Shape interrogation for design, analysis, and manufacturing. Involves analytical and programming assignments.

16.225 is a graduate level course on Computational Mechanics of Materials. The primary focus of this course is on the teaching of state-of-the-art numerical methods for the analysis of the nonlinear continuum response of materials. The range of material behavior considered in this course will include: linear and finite deformation elasticity, inelasticity and dynamics. Numerical formulation and algorithms will include: Variational formulation and variational constitutive updates, finite element discretization, error estimation, constrained problems, time integration algorithms and convergence analysis. There will be a strong emphasis on the (parallel) computer implementation of algorithms in programming assignments. At the beginning of the course, the students will be given the source of a base code with all the elements of a finite element program which constitute overhead and do not contribute to the learning objectives of this course (assembly and equation-solving methods, etc.). Each assignment will consist of formulating and implementing on this basic platform, the increasingly complex algorithms resulting from the theory given in class, as well as in using the code to numerically solve specific problems. The application to real engineering applications and problems in engineering science will be stressed throughout.

This course introduces programming languages and techniques used by physical scientists: FORTRAN, C, C++, MATLAB, and Mathematica. Emphasis is placed on program design, algorithm development and verification, and comparative advantages and disadvantages of different languages.

This course is a graduate level introduction to automatic discourse processing. The emphasis will be on methods and models that have applicability to natural language and speech processing. The class will cover the following topics: discourse structure, models of coherence and cohesion, plan recognition algorithms, and text segmentation. We will study symbolic as well as machine learning methods for discourse analysis. We will also discuss the use of these methods in a variety of applications ranging from dialogue systems to automatic essay writing. This subject qualifies as an Artificial Intelligence and Applications concentration subject.

Wave equations for fluid and visco-elastic media. Wave-theory formulations of acoustic source radiation and seismo-acoustic propagation in stratified ocean waveguides. Wavenumber Integration and Normal Mode methods for propagation in plane-stratified media. Seismo-Acoustic modeling of seabeds and ice covers. Seismic interface and surface waves in a stratified seabed. Parabolic Equation and Coupled Mode approaches to propagation in range-dependent ocean waveguides. Numerical modeling of target scattering and reverberation clutter in ocean waveguides. Ocean ambient noise modeling. Students develop propagation models using all the numerical approaches relevant to state-of-the-art acoustic research.

The theoretical frameworks of Hartree-Fock theory and density functional theory are presented as approximate methods to solve the many-electron problem. A variety of ways to incorporate electron correlation are discussed. The application of these techniques to calculate the reactivity and spectroscopic properties of chemical systems, in addition to the thermodynamics and kinetics of chemical processes, is emphasized. This course also focuses on cutting edge methods to sample complex hypersurfaces, for reactions in liquids, catalysts and biological systems.

You develop procedural programming skills in a programming language designed for visual arts and visualization while exploring Earth science topics. In particular, you will learn and practice digital graphics capabilities in order to render Earth science concepts that are otherwise difficult to visualize due to complicated space and time scales. Both spatial and object visualization skills are key to success in the Earth sciences; you will build an awareness of these skills and practice them with an eye to being able to teach them to your own secondary school students.

In this course, you will interact with large, open, freely-available data sets by collecting, plotting, and analyzing them using a variety of computational methods. You will therefore be ready to teach your own secondary school students a range of Next Generation Science Standard skills involving data collecting, manipulation, analysis, and plotting.

You will also read and discuss current research regarding the teaching, learning, and evaluation of visualization skills, as well as multiple external representations of science concepts. For the course’s final project, you will apply your theoretical knowledge and practical skills by developing a teaching object for use with your own secondary science students.

This course covers the analytical, graphical, and numerical methods supporting the analysis and design of integrated biological systems. Topics include modularity and abstraction in biological systems, mathematical encoding of detailed physical problems, numerical methods for solving the dynamics of continuous and discrete chemical systems, statistics and probability in dynamic systems, applied local and global optimization, simple feedback and control analysis, statistics and probability in pattern recognition.

This course covers concepts of computation used in analysis of engineering systems. It includes the following topics: data structures, relational database representations of engineering data, algorithms for the solution and optimization of engineering system designs (greedy, dynamic programming, branch and bound, graph algorithms, nonlinear optimization), and introduction to complexity analysis. Object-oriented, efficient implementations of algorithms are emphasized.

Introduction to computer graphics hardware, algorithms, and software. Topics include: line generators, affine transformations, line and polygon clipping, splines, interactive techniques, perspective projection, solid modeling, hidden surface algorithms, lighting models, shading, and animation. Substantial programming experience required. 6.837 offers an introduction to computer graphics hardware, algorithms, and software. Topics include: line generators, affine transformations, line and polygon clipping, splines, interactive techniques, perspective projection, solid modeling, hidden surface algorithms, lighting models, shading, and animation. Substantial programming experience is required.