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Research
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Prelim Courses - Applied Mathematics
This is not necessarily the official description for the courses. For the official descriptions, consult the 2012 - 2013 graduate catalog.
Description: Abstract integration: Lebesgue integration theory, outer measures and Caratheodory's theorem, Fatou's lemma, monotone and dominated convergence theorems. Measure theory: positive Borel measures, Riesz representation theorem for positive linear functionals on C(K), complex measures, Hahn-Jordan and Lebesgue decompositions, Radon-Nykodim theorem and differentiation of measures, Riemann-Stieltjes integral, the Lebesgue measure on Rd. Lp spaces: Cauchy-Bunyakovsky-Schwarz, Hoelder, Minkowski and Jensen inequalities, L2 and Lp spaces as Hilbert and Banach spaces, Riesz representation theorem for bounded functionals on Lp. Integration on product spaces, Fubini's and Tonelli's theorems, Fourier transform and Plancherel theorem. For prelim preparation, see the prelim study guide.
Prerequisites: MATH 5110
Offered: Spring
Credits: 3
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Description: Complex plane, Riemann sphere, Euler's formula, complex differentiable functions and Cauchy-Riemann equations, conformal maps, linear fractional transformations. Integration along simple rectifiable curves, Cauchy-Goursat and Morera's theorems, Cauchy integral formula, Cauchy estimates and Schwarz lemma. Power series and the disk of convergence, Taylor and Laurent series, classification of singularities. The argument principle, winding numbers and Rouche's theorem. Cauchy's residue theorem and its use in evaluating real-valued integrals. Maximum modulus, Liouville and Picard theorems, the Fundamental Theorem of Algebra, Schwarz reflection principle. Harmonic functions and harmonic conjugates. Normal families and Montel's theorem. The Riemann mapping theorem. A practical purpose of the class is to prepare students to take the qualifying exams.
For prelim preparation, see the prelim study guide.
Prerequisites: MATH 5110
Credits: 3
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Description: Banach spaces, linear operator theory and application to differential equations, nonlinear operators, compact sets on Banach spaces, the adjoint operator on Hilbert space, linear compact operators, Fredholm alternative, fixed point theorems and application to differential equations, spectral theory, distributions.
Credits: 3
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Description: The study of convergence, numerical stability, roundoff error, and discretization error arising from the approximation of differential and integral operators.
Prerequisites: MATH 5110, which may be taken concurrently.
Offered: Fall
Credits: 3
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