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ACMS 40900: Topics in Applied and Computational Mathematics and Methods
3.00 Credits
University of Notre Dame
This course will include the study of topics related to the instructor's research interests in applied and computational mathematics and methods.
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ACMS 40900 - Topics in Applied and Computational Mathematics and Methods
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ACMS 46800: Directed Readings
0.00 - 10.00 Credits
University of Notre Dame
Readings not covered in the curriculum which relate to the student's area of interest.
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ACMS 46800 - Directed Readings
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ACMS 48498: Undergraduate Research
1.00 - 3.00 Credits
University of Notre Dame
Research in collaboration with members of the faculty. Evaluation of performance will be accomplished through regular discussions with the faculty member in charge of the course.
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ACMS 48498 - Undergraduate Research
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ACMS 50051: Numerical PDE Techniques for Scientists and Engineers
3.00 Credits
University of Notre Dame
Partial Differential Equations (PDEs) are ubiquitous in science and engineering and are usually discussed in classes as analytic solutions for specialized cases. This course will teach the students the basic methods for their numerical solution. The course starts with an overview of PDEs, then moves on to discuss finite difference approximations. Hyperbolic systems are introduced by the scalar advection and scalar non-linear conservation laws, followed by the Riemann problem for hyperbolic systems and approximate Riemann solvers. Multidimensional schemes for non-linear hyperbolic systems are then presented. Elliptic and parabolic systems and their solution methodologies are then discussed including Krylov subspace methods and Multigrid methods. The course explains the theory underlying the numerical solution of PDEs and also provides hands-on experience with computer codes. A recommended prerequisite for this course is programming courses or a programming background.
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ACMS 50051 - Numerical PDE Techniques for Scientists and Engineers
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ACMS 50052: Numerical PDE Techniques for Scientists and Engineers II
3.00 Credits
University of Notre Dame
Partial Differential Equations (PDEs) are ubiquitous in science and engineering and are usually discussed in courses as analytic solutions for specialized cases. In PHYS 50051, students saw an overview of PDEs and were taught basic methods for their solution. Emphasis was on the theory underlying the numerical solution of PDEs and providing hands-on experience with computer codes. For background, students were expected to have some computer literacy at the level of familiarity with the Linux operating system and Fortran. The text was the first half of a book in development, written by the instructor. This second course will cover the second half of the text, with topics ranging from stiff source terms in hyperbolic PDEs to multigrid methods and Krylov subspace methods for elliptic and parabolic PDEs and adaptive mesh refinement. Interested advanced undergraduates and graduate students from applied mathematics, engineering, and the sciences may take the second course without having taken the first one.
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ACMS 50052 - Numerical PDE Techniques for Scientists and Engineers II
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ACMS 50730: Mathematical and Computational Modeling in Biology and Physics
3.00 Credits
University of Notre Dame
Introductory course on applied mathematics and computational modeling with emphasis on modeling of biological problems in terms of differential equations and stochastic dynamical systems. Students will be working in groups on several projects and will present them in class in the end of the course.
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ACMS 50730 - Mathematical and Computational Modeling in Biology and Physics
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ACMS 60590: Finite Elements in Engineering
3.00 Credits
University of Notre Dame
Fundamental aspects of the finite-element method are developed and applied to the solution of PDEs encountered in science and engineering. Solution strategies for parabolic, elliptic, and hyperbolic equations are explored. Spring.
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ACMS 60590 - Finite Elements in Engineering
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ACMS 60610: Discrete Mathematics
3.00 Credits
University of Notre Dame
The course will provide an introduction into different subjects of discrete mathematics. Topics include (1) Graph Theory: Trees and graphs, Eulerian and Hamiltonian graphs; tournaments; graph coloring and Ramsey's theorem. Applications to electrical networks. (2) Enumerative Combinatorics: Inclusion-exclusion principle, Generating functions, Catalan numbers, tableaux, linear recurrences and rational generating functions, and Polya theory. (3) Partially Ordered Sets: Distributive lattices, Dilworth's theorem, Zeta polynomials, Eulerian posets. (4) Projective and combinatorial geometries, designs and matroids.
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ACMS 60610 - Discrete Mathematics
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ACMS 60620: Optimization
3.00 Credits
University of Notre Dame
Vector spaces and convex sets; convex Hull; theorems of Caratheodory and Radon; Helly's Theorem; convex sets in Euclidean space; the Krein-Milman theorem in Euclidean space; extreme points of polyhedra; applications; the moment curve and the cyclic polytope; the cone of nonnegative polynomials; the cone of positive semidefinite matrices; the idea of semidefinite relaxation; semidefinite programming; cliques and the chromatic number of a graph; the Schur-Horn theorem; and the Toeplitz-Hausdorff theorem.
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ACMS 60620 - Optimization
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ACMS 60630: Nonlinear Dynamical Systems
3.00 Credits
University of Notre Dame
An introduction to the theory of nonlinear dynamical systems. Topics include: geometry of the phase space, symplectic structures, variational methods, nonlinear Hamiltonian systems, bifurcation theory, perturbation theory and transition to chaos, discrete dynamical systems, lattice based models, theory of pattern formation with examples from physics and biology.
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ACMS 60630 - Nonlinear Dynamical Systems
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