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Course Criteria
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1.00 - 3.00 Credits
Specialized study in a selected area of Applied Mathematics.
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3.00 Credits
Necessity and sufficiency conditions for constrained optimization problems are derived. The derived conditions are used to help answer questions concerning whether a given optimization problem has a solution, whether a solution is unique and how a solution can be found.
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3.00 Credits
Selected methods for unconstrained and constrained optimization problems with applications.
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3.00 Credits
An in-depth study of computer arithmetic, the solution of non-linear equations, the solution of systems of linear equations, eigenvalue problems and interpolation. Algorithms and methods are developed and then implemented on a computer.
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3.00 Credits
An in-depth study of orthogonal polynomials, numerical integration, and numerical solutions of ordinary and partial differential equations. Development and computer implementation of algorithms and methods.
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3.00 Credits
State-space techniques from modern control system theory. Topics include realization theory for MIMO systems, state-space techniques for feedback control, closed loop observer design, and state-space techniques in optimal control. The study of linear maps on finite dimensional vector spaces. Topics include: diagonalization (direct sums, invariant subspaces and Cayley-Hamilton theorem for linear operators), inner product spaces (self-adjoint, orthogonal operators, orthogonal projections and the spectral theorem, bilinear and quadratic forms), canonical forms (Jordan and rational forms, minimal polynomials), special matrices (non-negative matrices), and the exponential of a linear operator.
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3.00 Credits
A study of topics from the classical analytic theory of numbers. Topics will be chosen from arithmetic functions, the distribution of primes, congruences, the Riemann-zeta functions, the prime number theorem, Eisenstein series, quadratic resides, Dirichlet series, Euler products, the Dedekind eta function, the Jacobi theta functions, integer partitions, and modular forms.
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3.00 Credits
The theory of ordinary differential equations and dynamical systems. Topics include: Sturm-Liouville boundary value problems, eigenfunction expansions, Lyapunov stability, limit cycles, Poincare Bendixson theorem, Floquet's theory and Invariance theorems.
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3.00 Credits
State-space techniques from modern control system theory. Topics include realization theory for MIMO systems, state-space techniques for feedback control, closed loop observer design, and state-space techniques in optimal control.
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3.00 Credits
The study of normed linear spaces and linear operators. Topics include: Hilbert spaces (projection theorem, Riesz representation, Parseval relation); Banach spaces (convexity, duality, bounded and compact operators, theorems of Hahn-Banach, Banach-Steinhaus, open mapping, closed graph, Fredholm alternative); Stone-Weierstrass and Banach fixed point theorems.
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