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Course Criteria
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3.00 Credits
Concepts and techniques for partial differential equations, especially nonlinear. Diffusion, dispersion and other phenomena. Initial and boundary value problems. Normal mode analysis, Green's functions, and transforms. Conservation laws, kinematic waves, hyperbolic equations, characteristics shocks, simple waves. Geometrical optics, caustics. Free-boundary problems. Dimensional analysis. Singular perturbation, boundary layers, homogenization. Variational methods. Solitons. Applications from fluid dynamics, materials science, optics, traffic flow, etc.
Prerequisite:
Prereq: 18.03 or 18.034; 18.04, 18.075, or 18.112
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3.00 Credits
Study of illustrative topics in discrete applied mathematics, including sorting algorithms, probability theory, information theory, coding theory, secret codes, generating functions, and linear programming. Instruction and practice in written communication provided. Enrollment limited.
Prerequisite:
Prereq: Calculus II (GIR)
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3.00 Credits
Covers fundamental concepts in continuous applied mathematics. Applications from traffic flow, fluids, elasticity, granular flows, etc. Also covers continuum limit; conservation laws, quasi-equillibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion and group velocity. Uses MATLAB computing environment.
Prerequisite:
Prereq: Calculus II (GIR); 18.03 or 18.034
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3.00 Credits
Applications of algebra to combinatorics. Topics include walks in graphs, the Radon transform, groups acting on posets, Young tableaux, electrical networks.
Prerequisite:
Prereq: 18.701 or 18.703
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3.00 Credits
Combinatorial problems and methods for their solution. Enumeration, generating functions, recurrence relations, construction of bijections. Introduction to graph theory. Prior experience with abstraction and proofs is helpful.
Prerequisite:
Prereq: Calculus II (GIR); 18.06, 18.700, or 18.701
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3.00 Credits
Content varies from year to year.
Prerequisite:
Prereq: Permission of instructor
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3.00 Credits
Content varies from year to year. Readings from current research papers in combinatorics. Topics to be chosen and presented by the class.
Prerequisite:
Prereq: Permission of instructor
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3.00 Credits
Topics vary from year to year.
Prerequisite:
Prereq: Permission of instructor
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3.00 Credits
Connections between combinatorics and geometry (and algebra). Discussion of combinatorial problems that arise in algebraic geometry, convex geometry, and algebraic topology. Topics include toric varieties, polytopes and fans, hyperplane arrangements, triangulations and tilings, matroids, topological combinatorics, Schubert calculus.
Prerequisite:
Prereq: Permission of instructor
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3.00 Credits
Topics vary from year to year.
Prerequisite:
Prereq: Permission of instructor
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