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Course Criteria
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3.00 Credits
A second course in matrix theory directed toward applications. Linear spaces, linear operators, equivalence and similarity, spectral theorem, canonical forms, congruence, inertia theorem, quadratic forms, singular value decomposition and other factorizations, generalized inverses. Applications to optimization, differential equations, stability. Prerequisites: Math 203, 208, or 302.
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1.00 - 3.00 Credits
Discussion of topics of current interest. Prerequisite: Graduate standing.
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3.00 Credits
Measure spaces, extensions of measures, probability spaces, measures and distributions in normed linear spaces, product measures, independence, integral and expectation, convergence theorems, Radon-Nikodyn theorem and applications. Lp spaces, selected topics. Prerequisite: Math 315.
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3.00 Credits
Abstract measures and integrals, the Daniell integration theory, integration on locally compact Hausdorff spaces, integration in function spaces, selected topics. Prerequisite: Must be preceded by Math 415.
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3.00 Credits
Linear transformations, Hahn-Banach theorem, open-mapping theorem, closed graph theorem, uniform boundedness theorem, self adjoint and normal operators, and related topics of Banach and Hilbert space theory. Prerequisites: Math 315 and (Math 308 or Math 385)
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3.00 Credits
Spectral analysis of linear operators, spectral theorems, selected applications, an introduction to the theory of topological linear spaces, and papers from the recent literature. Prerequisites: Math 415 and 417.
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3.00 Credits
Foundations of the abstract theory of linear operators in Hilbert spaces, Banach spaces, and topological linear spaces. Application of abstract theory in constructing computational techniques (method of Rayleigh-Ritz) in eigenvalue problems associated with linear differential and integral equations arising in physical applications. Introduction to theory of distributions and Green's functions. Prerequisite: Math 308.
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3.00 Credits
Continuation of Math 425. Theory of distributions (Dirac Delta function) and Green's functions. Applications in the solution of boundary value problems for linear partial differential equations arising in physical applications. Integral equations in several independent variables. Method of characteristics in solving partial differential equations. Prerequisite: Math 425.
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3.00 Credits
Stability theory, Liapunov's direct method, periodic solutions, Poincare-Bendixson theory, applications. Prerequisite: Math 302.
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3.00 Credits
Continuation of Math 430. Nonlinear oscillations, solutions near singular points, asymptotic methods, differential equations on manifolds, boundary-value problems. Prerequisite: Math 302.
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