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Course Criteria
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3.00 - 4.00 Credits
Equilibria and the solution of linear systems and linear least squares problems. Dynamical systems and the eigenvalue problem with the Jordan form and Laplace transform via complex integration. Optional 1-credit laboratory motivates concepts from the course via physical experiments involving circuits, spring networks, and vibrating mechanical systems.
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3.00 Credits
Classical and numerical solution techniques for ordinary and partial differential equations. Fourier series and the finite element method for initial and boundary value problems arising in diffusion and wave propagation phenomena.
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3.00 Credits
An introductory course in numerical analysis with computer applications. Topics include floating point arithmetic; algorithms for the solution of linear systems, linear least square problems, and nonlinear equations; interpolation; Fourier transform; numerical integration; numerical solution of ordinary differential equations. Computer programming in Matlab is required.
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3.00 Credits
Continuation of Analysis I. Vector spaces of functions, sequences and series, convergence. Continuity and differentiability of functions of several variables, the derivative as a linear map, the contraction mapping principle, inverse and implicit function theorems, fundamental theorems on differential equations, multivariable integration, Stoke's theorem and relatives. Credit may not be received for both CAAM 402 and CAAM 502.
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3.00 Credits
The course will introduce methods to model the neurobiological basis of learning and memory, sensory coding, and the relation between neural activity and behavior.
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3.00 Credits
Existence and uniqueness for solutions of ordinary differential equations and difference equations, linear systems, nonlinear systems, stability, periodic solutions, bifurcation theory. Theory and theoretical examples are complemented by computational, model driven examples from biological and physical sciences.
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3.00 Credits
Derivation and properties of solutions of the partial differential equations of continuum physics. Basic concepts of continuum mechanics, ideal fluids, Navier-Stokes equations, linear elasticity, acoustics, basic principles of thermodynamics, Newtonian heat flow, porous flow, Maxwell's equations, electrical circuits.
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3.00 Credits
This course covers various numerical methods for solving partial differential equations: aspects of finite difference methods, finite element methods, finite volume methods, mixed methods, discontinuous Galerkin methods, and Meshless methods. Both theoretical convergence and practical implementation of the methods are studied for hyperbolic, elliptic, and parabolic problems.
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3.00 Credits
Iterative methods for linear systems of equations including Krylov subspace methods; Newton and Newton-like methods for nonlinear systems of equations; Gradient and Newton-like methods for unconstrained optimization and nonlinear least squares problems; techniques for improving the global convergence of these algorithms; linear programming duality and primal-dual interior-point methods. Credit may not be received for both CAAM 454 and CAAM 554.
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3.00 Credits
Study of the structure and properties of graphs, together with a variety of applications. Includes paths, cycles, trees, connectivity, matchings, colorings, planarity, directed graphs, and algorithms. Some knowledge of linear algebra is recommended.
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