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Course Criteria
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3.00 Credits
Credits: 3 Harmonic functions, generalizations of the maximum principle, entire and meromorphic functions, analytic continuation, and the Riemann mapping theorem. Prerequisites MATH 661. Hours of Lecture or Seminar per week 3 Hours of Lab or Studio per week 0
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3.00 Credits
Credits: 3 Covers the important results for analytic functions in several variables, including analyticity in several variables and the differences between the theory of one and the theory of several complex variables. Prerequisites MATH 661 and 762, or equivalent preparation in one complex variable. Hours of Lecture or Seminar per week 3 Hours of Lab or Studio per week 0
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3.00 Credits
Credits: 3 Cross-Listed with CSI 746 Study of the theory and computational aspects of wavelets and the wavelet transform. Emphasizes computational aspects of wavelets, defining the Fast Wavelet Transform in one and two dimensions. Developing the appropriate numerical algorithms. Includes developing the theory of wavelet bases on the real line, discussing multiresolution analysis, splines, time-frequency localization, and wavelet packets. Prerequisites MATH 315 or equivalent. Hours of Lecture or Seminar per week 3 Hours of Lab or Studio per week 0
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3.00 Credits
Credits: 3 Lebesque measure and integration. Theory of Lp spaces with p between one and infinity on the real line. Theory of linear operators on Banach spaces, including the Hahn-Banach theorem, open mapping theorem, closed graph theorem and the uniform boundedness principle. Prerequisites MATH 675. Hours of Lecture or Seminar per week 3 Hours of Lab or Studio per week 0
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3.00 Credits
Credits: 3 Bifurcation theory and perturbation methods for solutions in ordinary and partial differential equations. This course will develop and apply these mathematical tools in current scientific fields, such as biology, materials science, or financial mathematics. Prerequisites Permission of instructor. Different backgrounds may be appropriate, but generally, a student is expected to be an upper level graduate student who has already taken graduate courses including differential equations and dynamical systems. Hours of Lecture or Seminar per week 3 Hours of Lab or Studio per week 0
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3.00 Credits
Credits: 3 Techniques in nonlinear functional analysis with applications. Contraction mapping principle, Frechet and higher derivatives, the implicit function theorem, Lyapunov-Schmidt method, and bifurcation theory. Finite and infinite dimensional degree theory with applications in partial differential equations. Prerequisites Permission of instructor. Notes Different backgrounds may be appropriate, but generally, a student is expected to be an upper level graduate student who has already taken Linear Analysis. Since the applications given in the course are for differential equations, some familiarity with differential equations is extremely useful. Hours of Lecture or Seminar per week 3 Hours of Lab or Studio per week 0
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3.00 Credits
Credits: 3 Potential theory of Laplace's equation in Euclidean space. Harmonic functions, superharmonic functions, potentials, polar sets and capacity, the Dirichlet problem, the Martin boundary, boundary behavior of superharmonic functions using real variable techniques, and minimal fine limit techniques. Prerequisites Math 675 and 776 Hours of Lecture or Seminar per week 3 Hours of Lab or Studio per week 0
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1.00 Credits
Credits: 1 Mandatory for all PhD students. Weekly seminar graded on presentations and attendance. Faculty presentations on potential thesis topics and presentations by students. Prerequisites Admission to PhD program in mathematical sciences. Hours of Lecture or Seminar per week 1 Hours of Lab or Studio per week 0
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1.00 - 6.00 Credits
Credits: 1-6 Original or compilatory work evaluated by committee of three faculty members. Hours of Lecture or Seminar per week 0 Hours of Lab or Studio per week 0 Grading S/NC
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3.00 Credits
Credits: variable Program of studies designed by student's discipline director and approved by student's doctoral committee, which brings the student to participate in current research of discipline director and results in paper reporting the original contributions of student. Enrollment may be repeated.Prerequisites Admission to PhD in education program to study in mathematical sciences.
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