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Course Criteria
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3.00 Credits
Theory of nonlinear dynamical systems has applications to a wide variety of fields, from mathematics, physics, biology, and chemistry, to engineering, economics, and medicine. This is one of its most exciting aspects--that it brings researchers from many disciplines together with a common language. Dynamical systems are "deterministic" if there is a unique consequent to every state, and "stochastic" or "random" if there is more than one consequent chosen from some probability distribution. A dynamical system can have discrete or continuous time. The discrete case is defined by a map and the continuous case is defined by a "flow. Nonlinear dynamical systems have been shown to exhibit surprising and complex effects. Prominent examples of these include bifurcation and chaos. Applications to population dynamics, cancer growth and spread of infection will be discussed amongst others. This course will be self-contained.
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3.00 Credits
Laplace equations: Green's identity, fundamental solutions, maximum principles, Green's functions, Perron's methods. Parabolic equations: Heat equations, fundamental solutions, maximum principles, finite difference and convergence, Stefan Problems. First order equations: characteristic methods, Cauchy problems; vanishing of viscosity-viscosity solutions. Real analytic solutions: Cauchy-Kowalevski theorem, Holmgren theorem.
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3.00 Credits
A solid theoretical introduction to numerical analysis. Polynomial interpolation. Least squares and the basic theory of orthogonal functions. Numerical integration in one variable. Numerical linear algebra. Methods to solve systems of nonlinear equations. Numerical solution of ordinary differential equations. Solution of some simple partial differential equations by difference methods.
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3.00 Credits
A solid introduction to numerical partial differential equations with an emphasis on finite difference methods for time dependent equations and systems of equations. Interpolation. Stability and convergence of solutions in systems of PDE arising in science and engineering. High order accurate difference methods and Fourier methods. Well posed problems and general solutions for a variety of types of systems of equations with constant coefficients. Stability and convergence. Hyperbolic systems of equations.
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3.00 Credits
A thorough introduction to probability theory. Elements of measure and integration theory. Basic setup of probability theory (sample spaces, independence). Random variables, the law of large numbers. Discrete random variables (including random walks); continuous random variables, the basic distributions and sums of random variables. Generating functions, branching processes, basic theory of characteristic functions, central limit theorems. Markov chains. Various stochastic processes, including Brownian motion, queues and applications. Martingales. Other topics as time permits.
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3.00 Credits
This course gives an introduction to stochastic modeling and stochastic differential equations, with application to models from biology and finance. Some topics covered will be: stochastic versus deterministic models; Brownian motion and related processes, e.g., the Ornstein-Uhlenbeck Process; diffusion processes and stochastic differential equations; discrete and continuous Markov chain models with applications; the long run behavior of Markov chains; the Poisson processes with applications; and numerical methods for stochastic processes.
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1.00 Credits
This is a course in writing and oral presentation for graduate students in applied and computational mathematics and statistics. Communication with both technical and nontechnical audiences will be discussed.
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1.00 - 12.00 Credits
Readings not covered in the curriculum which relate to the student's area of interest.
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1.00 - 12.00 Credits
Research and dissertation for resident graduate students.
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3.00 Credits
A music appreciation course requiring no musical background and no prerequisites. General coverage of the history, various styles, and major performers of jazz, with an emphasis on current practice.
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