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Course Criteria
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1.00 Credits
An introduction to the use of quantitative modeling techniques in solving problems in biology. Each year one major biological area is explored in detail from a modeling perspective. The particular topic will vary from year to year. Mathematical techniques will be discussed as they arise in the context of biological problems. Prerequisites: introductory level biology, APMA 0330, 0340 or 0350, 0360, or written permission. Offered in alternate years.
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1.00 Credits
Sequencing of genomes (human and other) has generated a massive quantity of fundamental data that is revolutionizing the life sciences. The focus of this course is on drawing traditional and Bayesian statistical inferences from these data, including: alignment of biopolymer sequences; prediction of their structures, regulatory signals, and compositional characteristics; significances in database searches; phylogeny; and functional genomics. Emphasis is on inferences of the discrete high dimensional objects that are common in this field. Statistical topics: parameter estimation, hypothesis testing and false discovery rates, statistical decision theory, and Bayesian posterior inference. Prerequisite: APMA 1650 or MATH 1610 or equivalent; BIOL 0200 or equivalent; and experience with Matlab or another programming language.
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1.00 Credits
Focuses on fundamental algorithms in computational linear algebra with relevance to all science concentrators. Basic linear algebra and matrix decompositions (Cholesky, LU, QR, etc.), round-off errors and numerical analysis of errors and convergence. Iterative methods and conjugate gradient techniques. Computation of eigenvalues and eigenvectors, and an introduction to least squares methods. A brief introduction to Matlab is given. Prerequisites: MATH 0520 is recommended, not required.
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1.00 Credits
Fundamental numerical techniques for solving ordinary and partial differential equations. Overview of techniques for approximation and integration of functions. Development of multistep and multistage methods, error analysis, step-size control for ordinary differential equations. Solution of two-point boundary value problems, introduction to methods for solving linear partial differential equations. Introduction to Matlab is given but some programming experience is expected. Prerequisites: APMA 0330, 0340 or 0350, 0360. APMA 1170 is recommended.
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1.00 Credits
Basic probabilistic problems and methods in operations research and management science. Methods of problem formulation and solution. Markov chains, birth-death processes, stochastic service and queueing systems, the theory of sequential decisions under uncertainty, dynamic programming. Applications. Prerequisite: APMA 1650 or MATH 1610, or equivalent.
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1.00 Credits
An introduction to the basic mathematical ideas and computational methods of optimizing allocation of effort or resources, with or without constraints. Linear programming, network models, dynamic programming, and integer programming.
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1.00 Credits
An introduction to the dynamics of fluid flow and deforming elastic solids for students in the physical or mathematical sciences. Topics in fluid mechanics include statics, simple viscous flows, inviscid flows, potential flow, linear water waves, and acoustics. Topics in solid mechanics include elastic/plastic deformation, strain and stress, simple elastostatics, and elastic waves with reference to seismology. Offered in alternate years.
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1.00 Credits
Mathematical methods based on functions of a complex variable. Fournier series and its applications to the solution of one-dimensional heat conduction equations and vibrating strings. Series solution and special functions. Vibrating membrance. Sturm-Liouville problem and eigenfunction expansions. Fournier transform and wave propagations.
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1.00 Credits
See Methods Of Applied Mathematics III, IV (AM0133) for course description.
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1.00 Credits
Overview and introduction to dynamical systems. Local and global theory of maps. Attractors and limit sets. Lyapunov exponents and dimensions. Fractals: definition and examples. Lorentz attractor, Hamiltonian systems, homoclinic orbits and Smale horseshoe orbits. Chaos in finite dimensions and in PDEs. Can be used to fulfill the senior seminar requirement in applied mathematics. Prerequisites: differential equations and linear algebra.
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