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Repeatable for credit Individual investigation in mathematics. Repeatable for credit Research in mathematics. Prerequisite: Graduate standing.
Prerequisite: Applied mathematics or pure mathematics major; and graduate standing. Topics included are: canonical forms of matrices, diagonalizability criteria. Topics include the distinction between syntactic, object-level proofs and semantic, meta-level proofs, the distinction between axiomatic systems and natural deduction systems of object-level proofs, various systems of modal logic and some non-classical logics. Topics include transversability, colorability, networks, inclusion and exclusion, matching and designs. Prerequisite: Graduate standing; and special approval.
Mathematical methods from optimization, dynamical systems and probability are developed and applied.
Written and oral reports required. Linear systems, least-square data fitting, nonlinear equations and systems and optimization problems. Repeatable for credit Studies in special topics in pure and applied mathematics. Sampling distributions. Decision spaces and loss functions. Sufficiency and completeness.
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Estimation theory. Rao Blackwell and the Cramer Rao theorems. Neyman Pearson Lemma. Exponential families and invariance. Sequential tests. Non-parametric procedures. Repeatable for credit Seminar on current research in statistics and probability. Repeatable for credit Studies of special topics in mathematics. Not acceptable for credit toward a graduate degree in mathematics without approval of the student's adviser.
Repeatable for credit Techniques and problems in the teaching of college-level mathematics. Student presentations of mathematical papers and colloquia will be included. Includes one hour problem session each week. Repeatable for credit Seminar on current research in algebra.
Dimensional analysis Buckingham Pi Theorem. Perturbation methods singular perturbations, matched asymptotic expansions, WKB approximation. Variational methods Euler-Lagrange equations. Integral equations and Green's functions Fredholm alternative, compact operators, distributions, weak solutions. Wave phenomena dispersion, KdV equation. Stability and bifurcation linearized stability analysis, turning points, Hopf bifurcation. Included are basic topics in functional analysis and Hilbert space theory.
Numerical quadrature Newton-Cotes, Gauss ,extrapolation, Romberg integration. Least squares, orthogonalization methods. Algebraic eigenvalue problems, QR algorithm, singular value decomposition.