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Course Criteria
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3.00 Credits
Theoretical concepts and analysis of wave problems in science and engineering with examples chosen from elasticity, acoustics, geophysics, hydrodynamics, blood flow, nondestructive evaluation, and other applications. Progressive waves, group velocity and dispersion, energy density and transport. Reflection, refraction and transmission of plane waves by an interface. Mode conversion in elastic waves. Rayleigh waves. Waves due to a moving load. Scattering by a two-dimensional obstacle. Reciprocity theorems. Parabolic approximation. Waves on the sea surface. Capillary-gravity waves. Wave resistance. Radiation of surface waves. Internal waves in stratified fluids. Waves in rotating media. Waves in random media.
Prerequisite:
Prereq: 2.003, 18.075
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3.00 Credits
A unified treatment of nonlinear oscillations and wave phenomena with applications to mechanical, optical, geophysical, fluid, electrical and flow-structure interaction problems. Nonlinear free and forced vibrations; nonlinear resonances; self-excited oscillations; lock-in phenomena. Nonlinear dispersive and nondispersive waves; resonant wave interactions; propagation of wave pulses and nonlinear Schrodinger equation. Nonlinear long waves and breaking; theory of characteristics; the Korteweg-de Vries equation; solitons and solitary wave interactions. Stability of shear flows. Some topics and applications may vary from year to year.
Prerequisite:
Prereq: Permission of instructor
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3.00 Credits
The applied mathematics of continuous media and classical physics. Reading and presentation of papers from recent applied mathematics and physics literature. Topics and papers include fluid mechanics, solid mechanics, and biophysics. Instruction and practice in written and oral communication provided. Enrollment limited.
Prerequisite:
Prereq: 18.311, 18.354, or permission of instructor
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3.00 Credits
Introduction to the theory of nonlinear dynamical systems with applications from science and engineering. Local and global existence of solutions, dependence on initial data and parameters. Elementary bifurcations, normal forms. Phase plane, limit cycles, relaxation oscillations, Poincare-Bendixson theory. Floquet theory. Poincare maps. Averaging. Near-equilibrium dynamics. Synchronization. Introduction to chaos. Universality. Strange attractors. Lorenz and Rossler systems. Hamiltonian dynamics and KAM theory. Uses MATLAB computing environment.
Prerequisite:
Prereq: 18.03 or 18.034
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3.00 Credits
Advanced subject on the modern theory of nonlinear dynamical systems with an emphasis on applications in science and engineering. Invariant manifolds, homoclinic orbits, global bifurcations. Hamiltonian systems, completely integrable systems, KAM theory. Different mechanisms for chaotic dynamics, Shilnikov-type orbits, attractors, horseshoes, symbolic dynamics. Geometric singular perturbation theory. Physical applications.
Prerequisite:
Prereq: 18.385 or permission of instructor
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3.00 Credits
Selection of topics from the theory of finite groups, Lie groups, and group representations, motivated by quantum mechanics and particle physics. 8.322 and 8.323 helpful.
Prerequisite:
Prereq: 8.321
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3.00 Credits
Topics selected from the following: SUSY algebras and their particle representations; Weyl and Majorana spinors; Lagrangians of basic four-dimensional SUSY theories, both rigid SUSY and supergravity; supermultiplets of fields and superspace methods; renormalization properties, and the non-renormalization theorem; sponteneous breakdown of SUSY; and phenomenological SUSY theories. Some prior knowledge of Noether's theorem, derivation and use of Feynman rules, l-loop renormalization, and gauge theories is essential.
Prerequisite:
Prereq: Permission of instructor
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3.00 Credits
For students who want to have a clear understanding of quantum field theories. Appropriate for students who have not taken such a subject as well as students who have but are not entirely comfortable with the basic concepts and techniques. The topics begin with classical mechanics and end with gauge field theories and the renormalization of the standard model.
Prerequisite:
Prereq: Permission of instructor
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4.00 Credits
Provides an introduction to some of the central ideas of theoretical computer science, including circuits and decision trees, finite automata, Turing machines and computability, efficient algorithms and reducibility, the P versus NP problem, NP-completeness, the power of randomness, cryptography and one-way functions, computational learning theory, and quantum computing. Examines the classes of problems that can and cannot be solved in various computational models.
Prerequisite:
Prereq: 6.042
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4.00 Credits
A more extensive and theoretical treatment of the material in 6.045J/18.400J, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems.
Prerequisite:
Prereq: 18.310 or 18.062J
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