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Course Criteria
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0.00 - 3.00 Credits
Cr. 3. Alt. S., offered 2008. Prereq : 501 or 515. Topics selected from: Geometry of curves and surfaces. Manifolds, coordinate systems. Tensors, differential forms, Riemannian metrics. Connections, covariant differentiation, curvature tensors.
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0.00 - 3.00 Credits
Cr. 3. Alt. F., offered 2007. Prereq : Permission of instructor. Fundamental theory of normed linear spaces and algebras emphasizing aspects that provide a framework for the study of boundary-value problems, eigenvalue problems, harmonic analysis, analytic function theory, and modern operator theory.
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0.00 - 3.00 Credits
Cr. 3. Alt. S., offered 2008. Prereq : 633. Continuation of Math 633.
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0.00 - 4.00 Credits
(4-0) Cr. 4. F. Prereq : Stat 542. Measure spaces, extension theorem and construction of Lebesgue-Stieljes measures on Euclidean spaces, Lebesgue integration and the basic convergence theorems, Lp-spaces, absolute continuity of measures and the Radon Nikodym theorem, absolute continuity of functions on R and the fundamental theorem of Lebesgue integration, product spaces and Fubini- Tonelli Theorems, convolutions. Fourier series and transforms, probability spaces; Kolmogorov's existence theorem for stochastic processes; expectation; Jensen's inequality and applications, independence, Borel-Cantelli lemmas; weak and strong laws of large numbers and applications, renewal theory.
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0.00 - 3.00 Credits
(3-0) Cr. 3. S. Prereq: Permission of instructor. Weak convergence. Random walks and Brownian motion. Martingales. Stochastic integration and Ito's Formula. Stochastic differential equations and applications.
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0.00 - 3.00 Credits
Cr. 3. S. Prereq: Permission of instructor. Modeling of the dynamics of complex systems on multiple scales: Classical and dissipative molecular dynamics, stochastic modeling and Monte- Carlo simulation; macroscale non-linear dynamics and pattern formation.
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0.00 - 3.00 Credits
Cr. 3. F. Prereq : 515 or 519. First order equations and systems, conservation laws, general theory of linear partial differential equations of elliptic, parabolic and hyperbolic types, maximum principles, fundamental solutions, Sobolev spaces, variational and Hilbert space methods.
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0.00 - 3.00 Credits
Cr. 3. S. Prereq: 655. Continuation of Math 655.
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0.00 - 3.00 Credits
Cr. 3. Alt. S., offered 2009. Prereq : 501 or 515. Smooth mappings and fl ows. Fixed points, stable, unstable and center manifolds, normal forms. Structural stability, bifurcations. Horseshoe maps, introduction to chaotic behavior.
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0.00 - 3.00 Credits
Cr. 3. S. Prereq: 503 or 516 or 520 or 656. Elements of functional analysis; Sobolev spaces; variational principles and weak formulations; approximation theory in fi nite element spaces; analysis of fi nite element methods; implementation issues; applications.
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