# AM - Applied Mathematics

## AM3 Precalculus for the Social Sciences

Introduces mathematical functions and their uses for modeling real-life problems in the social sciences. Includes inequalities, linear and quadratic equations, functions (linear, quadratic, polynomial, rational, power, exponential, logarithmic, trigonometric), inverses, and the composition of functions. Students cannot receive credit for both this course and Mathematics 3. Mathematics 3 can substitute for this course. (Formerly Applied Mathematics and Statistics 3.)

### Credits

5

## AM6 Precalculus for Statistics

Reviews and introduces mathematical methods useful in the elementary study of statistics, including logic, real numbers, inequalities, linear and quadratic equations, functions, graphs, exponential and logarithmic functions, and summation notation. (Formerly AMS 6.)

### Credits

5

## AM10 Mathematical Methods for Engineers I

Applications-oriented course on complex numbers and linear algebra integrating Matlab as a computational support tool. Introduction to complex algebra. Vectors, bases and transformations, matrix algebra, solutions of linear systems, inverses and determinants, eigenvalues and eigenvectors, and geometric transformations. Students cannot receive credit for this course and for courses 10A or Mathematics 21. (Formerly AMS 10.)

### Credits

5

## AM11A Mathematical Methods for Economists I

Introduction to mathematical tools and reasoning, with applications to economics. Topics are drawn from differential calculus in one variable and include limits, continuity, differentiation, elasticity, Taylor polynomials, and optimization. Students cannot receive credit for both this course and MATH 11A or MATH 19A or AM 15A. (AM 11A formerly AMS 11A.)

### Credits

5

## AM11B Mathematical Methods for Economists II

Mathematical tools and reasoning, with applications to economics. Topics are drawn from multivariable differential calculus and single variable integral calculus, and include partial derivatives, linear and quadratic approximation, optimization with and without constraints, Lagrange multipliers, definite and indefinite integrals, and elementary differential equations. Students cannot receive credit for both this course and MATH 11B or MATH 19B or AM 15B. (AM 11B formerly AMS 11B.)

### Credits

5

## AM15A Case-Study Calculus I

Case-study-based, first-quarter introduction to single-variable calculus, with computing labs/discussion sections featuring contemporary symbolic, numerical, and graphical computing tools. Case studies drawn from biology, environmental sciences, health sciences, and psychology. Includes functions, mathematical modeling, limits, continuity, tangents, velocity, derivatives, the chain rule, implicit differentiation, higher derivatives, exponential and logarithmic functions and their derivatives, differentiating inverse functions, the mean value theorem, concavity, inflection points, function optimization, and curve-sketching. Students cannot receive credit for this course and course 11A or Economics 11A or Mathematics 11A or 19A. (Formerly AMS 15A.)

### Credits

5

## AM15B Case-Study Calculus II

Case-study based, second-quarter introduction to single-variable calculus, with computing labs/discussion sections featuring symbolic numerical, and graphical computing tools. Case studies are drawn from biology, environmental science, health science, and psychology. Includes indefinite and definite integrals of functions of a single variable; the fundamental theorem of calculus; integration by parts and other techniques for evaluating integrals; infinite series; Taylor series, polynomial approximations. Students cannot receive credit for this course and course 11B or Economics 11B or Mathematics 11B or 19B. (Formerly AMS 15B.)

### Credits

5

## AM20 Mathematical Methods for Engineers II

Applications-oriented class on ordinary differential equations (ODEs) and systems of ODEs using Matlab as a computational support tool. Covers linear ODEs and systems of linear ODEs; nonlinear ODEs using substitution and Laplace transforms; phase-plane analysis; introduction to numerical methods. Students cannot receive credit for this course and for courses 20A or Mathematics 24. (Formerly AMS 20.)

### Credits

5

## AM30 Multivariate Calculus for Engineers

Advanced multivariate calculus for engineering majors. Coordinate systems, parametric curves, and surfaces; partial derivatives, gradient, Taylor expansion, stationary points, constrained optimization; integrals in multiple dimensions; integrals over curves and surfaces. Applications to engineering form an integral part of the course.

### Credits

5

## AM100 Mathematical Methods for Engineers III

Applications-oriented course on complex analysis and partial differential equations using Maple as symbolic math software support. In addition, introduces Fourier analysis, special functions, and asymptotic methods. Students cannot receive credit for this course and Physics 116B or Physics 116C.

### Credits

5

## AM107 Introduction to Fluid Dynamics

Covers fundamental topics in fluid dynamics: Euler and Lagrange descriptions of continuum dynamics; conservation laws for inviscid and viscous flows; potential flows; exact solutions of the Navier-Stokes equation; boundary layer theory; gravity waves. Students cannot receive credit for this course and AM 217. (AM 107 formerly AMS 107.)

### Credits

5

## AM114 Introduction to Dynamical Systems

Introduces continuous and discrete dynamical systems. Topics include: fixed points; stability; limit cycles; bifurcations; transition to and characterization of chaos; fractals. Examples are drawn from sciences and engineering. Students cannot receive credit for this course and course 214. (Formerly AMS 114.)

### Credits

5

## AM115 Stochastic Modeling in Biology

Application of differential equations, probability, and stochastic processes to problems in cell, organismal, and population biology. Topics include life-history theory, behavioral ecology, and population biology. Students may not receive credit for this course and course 215.

### Credits

5

## AM129 Foundations of Scientific Computing for Scientists and Engineers

Covers fundamental aspects of scientific computing for research. Students are introduced to algorithmic development, programming (including the use of compilers, libraries, debugging, optimization, code publication), computational infrastructure, and data analysis tools, gaining hands-on experience through practical assignments. Basic programming experience is assumed. (Formerly AMS 129.)

### Credits

5

## AM147 Computational Methods and Applications

Applications of computational methods to solving mathematical problems using Matlab. Topics include solution of nonlinear equations, linear systems, differential equations, sparse matrix solver, and eigenvalue problems. (Formerly AMS 147.)

### Credits

5

## AM148 GPU Programming for Scientific Computations

This second course in scientific computing focuses on the use of parallel processing on GPUs with CUDA. Basic topics covered include the idea of parallelism and parallel architectures. The course then presents key parallel algorithms on GPUs such as scan, reduce, histogram and stencil, and compound algorithms. Applications to scientific computing are drawn from problems in linear algebra, curve fitting, FFTs, systems of ODEs and PDEs, and image processing. Finally, the course presents optimization strategies specific to GPUs. Basic knowledge of Unix, and C is assumed. (Formerly AMS 148.)

### Credits

5

## AM198 Independent Study or Research

Students submit petition to sponsoring agency.

### Credits

5

## AM198F Independent Study or Research

Students submit petition to sponsoring agency.

### Credits

2

## AM200 Research and Teaching in Applied Mathematics

Basic teaching techniques for teaching assistants, including responsibilities and rights; resource materials; computer skills; leading discussions or lab sessions; presentation techniques; maintaining class records; and grading. Examines research and professional training, including use of library; technical writing; giving talks in seminars and conferences; and ethical issues in science and engineering. (Formerly AMS 200.)

### Credits

3

## AM209 Foundations of Scientific Computing

Covers the fundamental aspects of scientific computing for research. Introduces algorithmic development; programming (including the use of compilers, libraries, debugging, optimization, and code publication); computational infrastructure; and data-analysis tools. Students gain hands-on experience through practical assignments. (Formerly AMS 209.) Basic programming experience will be assumed. May be taught in conjunction with AM 129 some quarters.

### Credits

5

## AM211 Foundations of Applied Mathematics

Accelerated class reviewing fundamental applied mathematical methods for all sciences. Topics include: multivariate calculus, linear algebra, Fourier series and integral transform methods, complex analysis, and ordinary differential equations. (Formerly AMS 211.)

### Credits

5

## AM212A Applied Mathematical Methods I

Focuses on analytical methods for partial differential equations (PDEs), including: the method of characteristics for first-order PDEs; canonical forms of linear second-order PDEs; separation of variables; Sturm-Liouville theory; Green's functions. Illustrates each method using applications taken from examples in physics. Course 211 or equivalent is strongly recommended as preparation. (Formerly AMS 211.)

### Credits

5

## AM212B Applied Mathematical Methods II

Covers perturbation methods: asymptotic series, stationary phase and expansion of integrals, matched asymptotic expansions, multiple scales and the WKB method, Padé approximants and improvements of series. (Formerly AMS 212B.)

### Credits

5

## AM213A Numerical Linear Algebra

Focuses on numerical solutions to classic problems of linear algebra. Topics include: LU, Cholesky, and QR factorizations; iterative methods for linear equations; least square, power methods, and QR algorithms for eigenvalue problems; and conditioning and stability of numerical algorithms. Provides hands-on experience in implementing numerical algorithms for solving engineering and scientific problems. Basic knowledge of mathematical linear algebra is assumed. (Formerly AMS 213A.)

### Credits

5

## AM213B Numerical Methods for the Solution of Differential Equations

Introduces the numerical solutions of ordinary and partial differential equations (ODEs and PDEs). Focuses on the derivation of discrete solution methods for a variety of differential equations, and their stability and convergence. Also provides hands-on experience in implementing such numerical algorithms for the solution of engineering and scientific problems using MATLAB software. The class consists of lectures and hands-on programming sections. Basic mathematical knowledge of ODEs and PDEs is assumed, and a basic working knowledge of programming in MATLAB is expected. (Formerly AMS 213B.)

### Credits

5

## AM214 Applied Dynamical Systems

Introduces continuous and discrete dynamical systems. Topics include: fixed points; stability; limit cycles; bifurcations; transition to and characterization of chaos; and fractals. Examples drawn from sciences and engineering; founding papers of the subject are studied. Students cannot receive credit for this course and course 114. (Formerly AMS 214.)

### Credits

5

## AM215 Stochastic Modeling in Biology

Application of differential equations and probability and stochastic processes to problems in cell, organismal, and population biology. Topics include: life-history theory, behavioral ecology, and population biology. Students may not receive credit for this course and course 115. (Formerly AMS 215.)

### Credits

5

## AM216 Stochastic Differential Equations

Introduction to stochastic differential equations and diffusion processes with applications to biology, biomolecular engineering, and chemical kinetics. Topics include Brownian motion and white noise, gambler's ruin, backward and forward equations, and the theory of boundary conditions. (Formerly AMS 216.)

### Credits

5

## AM217 Introduction to Fluid Dynamics

Covers fundamental topics in fluid dynamics at the graduate level: Euler and Lagrange descriptions of continuum dynamics; conservation laws for inviscid and viscous flows; potential flows; exact solutions of the Navier-Stokes equation; boundary layer theory; gravity waves. Students cannot receive credit for this course and AM 107. (Formerly AMS 217.)

### Credits

5

## AM227 Waves and Instabilities in Fluids

Advanced fluid dynamics course introducing various types of small-amplitude waves and instabilities that commonly arise in geophysical and astrophysical systems. Topics covered include, but are not limited to: pressure waves, gravity waves, Rossby waves, interfacial instabilities, double-diffusive instabilities, and centrifugal instabilities. Advanced mathematical methods are used to study each topic. Undergraduates are encouraged to take this course with permission of the instructor. (Formerly AMS 227.)

### Credits

5

## AM229 Convex Optimization

Focuses on recognizing, formulating, analyzing, and solving convex optimization problems encountered across science and engineering. Topics include: convex sets; convex functions; convex optimization problems; duality; subgradient calculus; algorithms for smooth and non-smooth convex optimization; applications to signal and image processing, machine learning, statistics, control, robotics and economics. Students are required to have knowledge of calculus and linear algebra, and exposure to probability. (Formerly AMS 229.)

### Credits

5

## AM230 Numerical Optimization

Introduces numerical optimization tools widely used in engineering, science, and economics. Topics include: line-search and trust-region methods for unconstrained optimization, fundamental theory of constrained optimization, simplex and interior-point methods for linear programming, and computational algorithms for nonlinear programming. (Formerly AMS 230.)

### Credits

5

## AM231 Nonlinear Control Theory

Covers analysis and design of nonlinear control systems using Lyapunov theory and geometric methods. Includes properties of solutions of nonlinear systems, Lyapunov stability analysis, effects of perturbations, controllability, observability, feedback linearization, and nonlinear control design tools for stabilization. (Formerly AMS 231.)

### Credits

5

## AM232 Applied Optimal Control

Introduces optimal control theory and computational optimal control algorithms. Topics include: calculus of variations, minimum principle, dynamic programming, HJB equation, linear-quadratic regulator, direct and indirect computational methods, and engineering application of optimal control. (Formerly AMS 232.)

### Credits

5

## AM238 Fundamentals of Uncertainty Quantification in Computational Science and Engineering

Computing the statistical properties of nonlinear random system is of fundamental importance in many areas of science and engineering. Introduces students to state-of-the-art methods for uncertainty propagation and quantification in model-based computations, focusing on the computational and algorithmic features of these methods most useful in dealing with systems specified in terms of stochastic ordinary and partial differential equations. Topics include: polynomial chaos methods (gPC and ME-gPC), probabilistic collocation methods (PCM and ME-PCM), Monte-Carlo methods (MC, quasi-MC, multi-level MC), sparse grids (SG), probability density function methods, and techniques for dimensional reduction. Basic knowledge of probability theory and elementary numerical methods for ODEs and PDEs is recommended. (Formerly AMS 238.)

### Credits

5

## AM250 An Introduction to High Performance Computing

Designed for STEM students and others. Through hands-on practice, this course introduces high-performance parallel computing, including the concepts of multiprocessor machines and parallel computation, and the hardware and software tools associated with them. Students become familiar with parallel concepts and the use of MPI and OpenMP together with some insight into the use of heterogeneous architectures (CPU, CUDA, OpenCL), and some case-study problems. (Formerly AMS 250.)

### Credits

5

## AM260 Computational Fluid Dynamics

Introduces modern computational approaches to solving the differential equations that arise in fluid dynamics, particularly for problems involving discontinuities and shock waves. Examines the fundamentals of the mathematical foundations and computation methods to obtain solutions. Focuses on writing practical numerical codes and analyzing their results for a full understanding of fluid phenomena. (Formerly AMS 260.)

### Credits

5

## AM275 Magnetohydrodynamics

Studies the interaction of fluid motion and magnetic fields in electrically conducting fluids, with applications in many natural and man-made flows ranging from, for example, planetary physics and astrophysics to industrial metallurgic engineering. (Formerly AMS 275.)

### Credits

5

## AM280A Seminar in Mathematical and Computational Biology

Weekly seminar on mathematical and computational biology. Participants present research findings in organized and critical fashion, framed in context of current literature. Students present own research on a regular basis. (Formerly AMS 280A.)

### Credits

2

## AM280B Seminar in Applied Mathematical Modeling

Weekly seminar series covering topics of current research in applied mathematics and statistics. Permission of instructor required. Enrollment is restricted to graduate students. (Formerly AMS 280B.)

### Credits

2

## AM280C Seminar in Geophysical and Astrophysical Fluid Dynamics

Weekly seminar/discussion group on geophysical and astrophysical fluid dynamics covering both analytical and computational approaches. Participants present research progress and findings in semiformal discussions. Students must present their own research on a regular basis. (Formerly AMS 280C.)

### Credits

2

## AM296 Masters Project

Independent completion of a masters project under faculty supervision. Students submit petition to sponsoring agency. Enrollment is restricted to graduate students.

### Credits

2

## AM297A Independent Study or Research

Independent study or research under faculty supervision. Students submit petition to sponsoring agency. Enrollment is restricted to graduate students.

### Credits

5

## AM297B Independent Study or Research

Independent study or research under faculty supervision. Students submit petition to sponsoring agency. Enrollment is restricted to graduate students.

### Credits

10

## AM297C Independent Study or Research

Independent study or research under faculty supervision. Students submit petition to sponsoring agency. Enrollment is restricted to graduate students.

### Credits

15

## AM297F Independent Study

### Credits

2

## AM299A Thesis Research

Thesis research under faculty supervision. Students submit petition to sponsoring agency. Enrollment restricted to graduate students.

### Credits

5

## AM299B Thesis Research

Thesis research under faculty supervision. Students submit petition to sponsoring agency. Enrollment restricted to graduate students.

### Credits

10

## AM299C Thesis Research

Thesis research under faculty supervision. Students submit petition to sponsoring agency. Enrollment restricted to graduate students.

### Credits

15