Graduate Course Offerings

Current course information is available via the Graduate Bulletin .

DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS I AMS 501
Examples of initial and boundary value problems in which differential equations arise. Existence and uniqueness of solutions, systems of linear differential equations, and the fundamental solution matrix. Power series solutions. Sturm-Liouville theory and eigenfunction expansion. Green's functions.
Spring, 3 credits, Letter graded (A, A-, B+, etc.)

DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS II AMS 502
Analytic solution techniques for, and properties of solutions of, partial differential equations, with concentration on second order PDEs. Techniques covered include: method of characteristics, separation of variables, eigenfunction expansions, spherical means, Green's functions and fundamental solutions, and Fourier transforms. Solution properties include: energy conservation, dispersion, dissipation, existence and uniqueness, maximum and mean value principles.
Prerequisite: AMS 501
3 credits, Letter graded (A, A-, B+, etc.)

APPLICATIONS OF COMPLEX ANALYSIS AMS 503
A study of those concepts and techniques in complex function theory that are of interest for their applications. Pertinent material is selected from the following topics: harmonic functions, calculus of residues, conformal mapping, and the argument principle. Application is made to problems in heat conduction, potential theory, fluid dynamics, and feedback systems.
Spring, 3 credits, Letter graded (A, A-, B+, etc.)

FOUNDATIONS OF APPLIED MATHEMATICS AMS 504
An introductory course for the purpose of developing certain concepts and techniques that are fundamental in modern approaches to the solution of applied problems. An appropriate selection of topics is based on the concepts of metric spaces, compactness, sequences and convergence, continuity, differentiation and integration, function sequences, contraction mapping theorem. Strong emphasis on proofs.
Fall, 3 credits, Letter graded (A, A-, B+, etc.)

APPLIED LINEAR ALGEBRA AMS 505
Review of matrix operations. Elementary matrices and reduction of general matrices by elementary operations, canonical forms, and inverses. Applications to physical problems. Coscheduled as AMS 505 or HPH 695.
Fall, 3 credits, Letter graded (A, A-, B+, etc.)

FINITE STRUCTURES AMS 506
Problem solving in combinatorial analysis and graph theory using generating functions, recurrence relations, Polya's enumeration formula, graph coloring, and network flows.
3 credits, Letter graded (A, A-, B+, etc.)

INTRODUCTION TO PROBABILITY AMS 507
The topics include sample spaces, axioms of probability, conditional probability and independence, discrete and continuous random variables, jointly distributed random variables, characteristics of random variables, law of large numbers and central limit theorem, Markov chains.
Note: Crosslisted with HPH 696.
3 credits, Letter graded (A, A-, B+, etc.)

ANALYTICAL METHODS FOR APPLIED MATHEMATICS AND STATISTICS AMS 510
Review of techniques of multivariate calculus, convergence and limits, matrix analysis, vector space basics, and Lagrange multipliers.
Fall, 3 credits, Letter graded (A, A-, B+, etc.)

FOUNDATIONS OF QUANTITATIVE FINANCE AMS 511
Introduction to capital markets, securities pricing and modern portfolio theory, including the organization and operation of securities market, the Efficient Market Hypothesis and its implications, the Capital Asset Pricing Model, the Arbitrage Pricing Theory and more general factor models. Common stocks and their valuation, statistical analysis, and portfolio selection in a single-period, mean-variance context will be explored along with its solution as a quadratic program. Fixed income securities and their valuation, statistical analysis, and portfolio selection. Discussion of the development and use of financial derivatives. Introduction to risk neutral pricing, stochastic calculus and the Black-Scholes Formula. Whenever practical examples will use real market data. Numerical exercises and projects in a high-level programming environment will also be assigned.
Prerequisite: AMS 505 or AMS 510; AMS 507 3 credits, Letter graded (A, A-, B+, etc.)

CAPITAL MARKETS PORTFOLIO THEORY AMS 512
Development of capital markets and portfolio theory in both continuous time and multi-period settings. Utility theory and its application to the determination of optimal consumption and investment policies. Asymptotic growth under conditions of uncertainty. Applications to problems in strategic asset allocation over finite horizons and to problems in public finance. Whenever practical, examples will use real market data. Numerical exercises and projects in a high-level programming environment will also be assigned.
Prerequisite: AMS 511 3 credits, Letter graded (A, A-, B+, etc.)

FINANCIAL DERIVATIVES AND STOCHASTIC CALCULUS AMS 513
Further development of derivative pricing theory including the use of equivalent martingale measures, the Girsanov Theorem, the Radon-Nikodym Derivative, and a deeper, more general understanding of the Arbitrage Theorem. Numerical approaches to solving stochastic PDE's will be further developed. Applications involving interest rate sensitive securities and more complex options will be introduced. Whenever practical examples will use real markeet data. Numerical exercises and projects in a high-level programming environment will also be assigned. Prerequisite: AMS 5113 credits, Letter graded (A, A-, B+, etc.)

COMPUTATIONAL FINANCE AMS 514
Review of foundations: stochastic calculus, martingales, pricing, and arbitrage. Basic principles of Monte Carlo and the efficiency and effectiveness of simulation estimators. Generation of pseudo- and quasi-random numbers with sampling methods and distributions. Variance reduction techniques such as control variates, antithetic variates, stratified and Latin hypercube sampling, and importance sampling. Discretization methods including first and second order methods, trees, jumps, and barrier crossings. Applications in pricing American options, interest rate sensitive derivatives, mortgage-backed securities and risk management. Whenever practical examples will use real market data. Extensive numerical exercises and projects in a general programming environment will also be assigned.
Prerequisite: AMS 512 and AMS 513 3 credits, Letter graded (A, A-, B+, etc.)

CASE STUDIES IN COMPUTATIONAL FINANCE AMS 515
Actual applications of Quantitative Finance to problems of risk assessment, product design, portfolio management and securities pricing will be covered. Particular attention will be paid to data collection and analysis, the design and implementation of software, and, most importantly, to differences the occur between "theory and practice" in model application, and to the development of practical strategies for handling cases in which "model failure" makes the naive use of quantitative techniques dangerous. Extensive use of guest lecturers drawn from the industry will be made. Prerequisite: AMS 512 and AMS 513 3 credits, Letter graded (A, A-, B+, etc.)

MATHEMATICAL MODELING IN THE ANALYSIS OF PUBLIC SYSTEMS AMS 520
Review of models relating to the questions of the improvement in delivery of urban service systems (e.g., fire, police, health, sanitation, transit). Topics include optimal location and districting of public facilities, distribution networks, models of congestion and delay in municipal services, and optimal deployment of emergency vehicles.
3 credits, Letter graded (A, A-, B+, etc.)

NUMERICAL ANALYSIS I AMS 526
Direct and indirect methods for solving simultaneous linear equations and matrix inversion, conditioning, and round-off errors. Computation of eigenvalues and eigenvectors.
Corequisite: AMS 505 Fall, 3 credits, Letter graded (A, A-, B+, etc.)

NUMERICAL ANALYSIS II AMS 527
Numerical methods based upon functional approximation: polynomial interpolation and approximation; and numerical differentiation and integration. Solution methods for ordinary differential equations. AMS 527 may be taken whether or not the student has completed AMS 526. Spring, 3 credits, Letter graded (A, A-, B+, etc.)

NUMERICAL ANALYSIS III AMS 528
An introduction to scientific computation, this course considers the basic numerical techniques designed to solve problems of physical and engineering interest. Finite difference methods are covered for the three major classes of partial differential equations: parabolic, elliptic, and hyperbolic. Practical implementation will be discussed. The student is also introduced to the important packages of scientific software algorithms. AMS 528 may be taken whether or not the student has completed AMS 526 or AMS 527.
Spring, 3 credits, Letter graded (A, A-, B+, etc.)

PRINCIPLES IN PARALLEL COMPUTING AMS 530
This course is designed for both academic and industrial scientists interested in parallel computing and its applications to large-scale scientific and engineering problems. It focuses on the three main issues in parallel computing: analysis of parallel hardware and software systems, design and implementation of parallel algorithms, and applications of parallel computing to selected problems in physical science and engineering. The course emphasizes hands-on practice and understanding of algorithmic concepts of parallel computing.
Prerequisite: A course in basic computer science such as operating systems or architectures or some programming experience.
Spring, 3 credits, Letter graded (A, A-, B+, etc.)

INTRODUCTION TO COMPUTATIONAL STRUCTURAL BIOLOGY AND DRUG DESIGN AMS 535
This course will provide an introduction to Computational Structural Biology with application to Drug Design. Methods and applications that use computation to model biological systems involved in human disease will be emphasized. The course aims to foster collaborative learning and will consist of presentations by the instructor, guest lecturers, and by course participants with the goal of summarizing key methods, topics and papers relevant to Computational Structural Biology.
0 - 3 credits, Letter graded (A, A-, B+, etc.) May be repeated for credit.

MOLECULAR MODELING OF BIOLOGICAL MOLECULES AMS 536
This course is designed for students who wish to gain hands on experience modeling biological molecules at the atomic level. In conjunction with the individual interests, Molecular Mechanics, Molecular dynamics, Monte Carlo, Docking (virtual screening), or Quantum Mechanics software packages can be used to study relevant biological systems(s). Projects will include setup, execution, and analysis. Course participants will give literature presentations relevant to the simulations being performed and a final project report will be required. Familiarity with Unix (Linux) is desirable.
0 - 3 credits, Letter graded (A, A-, B+, etc.)

LINEAR PROGRAMMING AMS 540
Formulation of linear programming problems and solutions by simplex method. Duality, sensitivity analysis, dual simplex algorithm, decomposition. Applications to the transportation problem, two-person games, assignment problem, and introduction to integer and nonlinear programming. This course is offered as both MBA 540 and AMS 540.
Prerequisite: A course in linear algebra. 3 credits, Letter graded (A, A-, B+, etc.)

ANALYSIS OF ALGORITHMS AMS 542
Techniques for designing efficient algorithms, including choice of data structures, recursion, branch and bound, divide and conquer, and dynamic programming. Complexity analysis of searching, sorting, matrix multiplication, and graph algorithms. Standard NP-complete problems and polynomial transformation techniques. This course is offered as both AMS 542 and CSE 548.
Spring, 3 credits, Letter graded (A, A-, B+, etc.)

DISCRETE AND NONLINEAR OPTIMIZATION AMS 544
Theoretical and computational properties of discrete and nonlinear optimization problems: integer programming, including cutting plane and branch and bound algorithms, necessary and sufficient conditions for optimality of nonlinear programs, and performance of selected nonlinear programming algorithms. This course is offered as both MBA 544 and AMS 544.
Prerequisite: AMS 540 or MBA 540 3 credits, Letter graded (A, A-, B+, etc.)

COMPUTATIONAL GEOMETRY AMS 545
Study of the fundamental algorithmic problems associated with geometric computations, including convex hulls, Voronoi diagrams, triangulation, intersection, range queries, visibility, arrangements, and motion planning for robotics. Algorithmic methods include plane sweep, incremental insertion, randomization, divide-and-conquer, etc. This course is offered as both AMS 545 and CSE 555.
Spring, 3 credits, Letter graded (A, A-, B+, etc.)

NETWORK FLOWS AMS 546
Theory of flows in capacity-constrained networks. Topics include maximum flow, feasibility criteria, scheduling problems, matching and covering problems, minimum-length paths, minimum-cost flows, and associated combinatorial problems. This course is offered as both MBA 546 and AMS 546.
Spring, 3 credits, Letter graded (A, A-, B+, etc.)

DISCRETE MATHEMATICS AMS 547
This course introduces such mathematical tools as summations, number theory, binomial coefficients, generating functions, recurrence relations, discrete probability, asymptotics, combinatorics, and graph theory for use in algorithmic and combinatorial analysis. This course is offered as both CSE 547 and AMS 547.
Spring, 3 credits, Letter graded (A, A-, B+, etc.)

OPERATIONS RESEARCH: STOCHASTIC MODELS AMS 550
Includes Poisson processes, renewal theory, discrete-time and continuous-time Markov processes, Brownian motion, applications to queues, statistics, and other problems of engineering and social sciences. This course is offered as both MBA 550 and AMS 550.
Prerequisite: AMS 507 or equivalent 3 credits, Letter graded (A, A-, B+, etc.)

GAME THEORY I AMS 552
Elements of cooperative and noncooperative games. Matrix games, pure and mixed strategies, and equilibria. Solution concepts such as core, stable sets, and bargaining sets. Voting games, and the Shapley and Banzhaff power indices. This course is offered as both ECO 604 and AMS 552.
Prerequisite for ECO 604: Graduate standing in the Economics Department or permission of the Graduate Director. 0 - 3 credits, Letter graded (A, A-, B+, etc.)

SIMULATION AND MODELING AMS 553
A comprehensive course in formulation, implementation, and application of simulation models. Topics include data structures, simulation languages, statistical analysis, pseudo-random number generation, and design of simulation experiments. Students apply simulation modeling methods to problems of their own design. This course is offered as CSE 529, AMS 553 and MBA 553.
Prerequisite: CSE 214 or equivalent; AMS 310 or 507 or equivalent; or permission of instructor.
3 credits, Letter graded (A, A-, B+, etc.)

QUEUING THEORY AMS 554,
Introduction to the mathematical aspects of congestion. Birth and death processes. Queues with service priorities and bulk-service queues. Analysis of transient- and steady-state behavior. Estimation of parameters. Applications to engineering, economic, and other systems. This course is offered as both MBA 554 and AMS 554.
Fall, even years, 3 credits, Letter graded (A, A-, B+, etc.)

GAME THEORY II AMS 555
Refinements of strategic equilibrium, games with incomplete information, repeated games with and without complete information, and stochastic games. The Shapley value of games with many players, and NTU-values. This course is offered as both ECO 605 and AMS 555.
Spring, 0 - 3 credits, Letter graded (A, A-, B+, etc.)

DYNAMIC PROGRAMMING AMS 556
Stochastic and deterministic multistage optimization problems. Stochastic path problems. Principle of optimality. Recursive and functional equations. Method of successive approximations and policy iteration. Applications to finance, economics, inventory control, maintenance, inspection, and replacement problems. This course is offered as both MBA 556 and AMS 556.
Prerequisite: MBA/AMS 550 or MBA/AMS 558 3 credits, Letter graded (A, A-, B+, etc.)

NUMERICAL HYDROLOGY AMS 562
Hydrology Numerical solution methods for the equations of incompressible flow in porous media with special emphasis on groundwater flow. Finite difference and finite element methods for steady-state and transient flows-boundary conditions, range of validity and stability of the numerical schemes, and numerical artifacts. The approach is hands on, with example problems being computed. This course is offered as both GEO 564 and AMS 562.
Fall, alternate years, 3 credits, Letter graded (A, A-, B+, etc.)

WAVE PROPAGATION AMS 565
Theory of propagation of vector and scalar waves in bounded and unbounded regions. Development of methods of geometrical optics. Propagation in homogeneous and anisotropic media.
3 credits, Letter graded (A, A-, B+, etc.)

COMPRESSIBLE FLUID DYNAMICS AMS 566
Physical, mathematical, and computational description in compressible fluid flows. Integral and differential forms of the conservation equations, one-dimensional flow, shocks and expansion waves in two and three dimensions, quasi-one-dimensional flow, transient flow, numerical methods for steady supersonic flow, numerical methods for transient flow.
Spring, 3 credits, Letter graded (A, A-, B+, etc.)

PROBABILITY THEORY I AMS 569
Probability spaces and sigma-algebras. Random variables as measurable mappings. Borel-Cantelli lemmas. Expectation using simple functions. Monotone and dominated convergence theorems. Inequalities. Stochastic convergence. Characteristic functions. Laws of large numbers and the central limit theorem. This course is offered as both AMS 569 and MBA 569.
Prerequisite: AMS 504 or equivalent 3 credits, Letter graded (A, A-, B+, etc.)

INTRODUCTION TO MATHEMATICAL STATISTICS AMS 570
Probability and distributions; multivariate distributions; distributions of functions of random variables; sampling distributions; limiting distributions; point estimation; confidence intervals; sufficient statistics; Bayesian estimation; maximum likelihood estimation; statistical tests.
Prerequisite: AMS 312 or equivalent 3 credits, Letter graded (A, A-, B+, etc.)

MATHEMATICAL STATISTICS AMS 571
Sampling distribution; convergence concepts; classes of statistical models; sufficient statistics; likelihood principle; point estimation; Bayes estimators; consistency; Neyman-Pearson Lemma; UMP tests; UMPU tests; Likelihood ratio tests; large sample theory.
Prerequisite: AMS 312; AMS 570 is preferred, but not required .3 credits, Letter graded (A, A-, B+, etc.)

DATA ANALYSIS I AMS 572
Introduction to basic statistical procedures. Survey of elementary statistical procedures such as the t-test and chi-square test. Procedures to verify that assumptions are satisfied. Extensions of simple procedures to more complex situations and introduction to one-way analysis of variance. Basic exploratory data analysis procedures (stem and leaf plots, straightening regression lines, and techniques to establish equal variance). Coscheduled as AMS 572 or HPH 698.
Fall, 3 credits, Letter graded (A, A-, B+, etc.)

DESIGN AND ANALYSIS OF CATEGORICAL DATA AMS 573
Measuring the strength of association between pairs of categorical variables. Methods for evaluating classification procedures and inter-rater agreement. Analysis of the associations among three or more categorical variables using log linear models. Logistic regression.
Spring, 3 credits, Letter graded (A, A-, B+, etc.)

INTERNSHIP IN STATISTICAL CONSULTING AMS 575
Directed quantitative research problem in conjunction with currently existing research programs outside the department. Students specializing in a particular area work on a problem from that area; others work on problems related to their interests, if possible. Efficient and effective use of computers. Each student gives at least one informal lecture to his or her colleagues on a research problem and its statistical aspects.
Prerequisite: Permission of instructor 3 - 4 credits, Letter graded (A, A-, B+, etc.)

STATISTICAL METHODS FOR SOCIAL SCIENTISTS AMS 576
This course is an introduction to statistical thinking in the social sciences. The course covers statistical variability, standard scores, regression correlation, sampling notions, estimation, confidence intervals, significance testing, conditional probability, and Bayesian manipulations. This course is offered as both CET 555 and AMS 576.
Prerequisite: AMS 310 or permission of instructor 3 credits, Letter graded (A, A-, B+, etc.)

MULTIVARIATE ANALYSIS AMS 577
The multivariate distribution. Estimation of the mean vector and covariance matrix of the multivariate normal. Discriminant analysis. Canonical correlation. Principal components. Factor analysis. Cluster analysis.
Prerequisites: AMS 572 and AMS 578 3 credits, Letter graded (A, A-, B+, etc.)

REGRESSION THEORY AMS 578
Classical least-squares theory for regression including the Gauss-Markov theorem and classical normal statistical theory. An introduction to stepwise regression, procedures, and exploratory data analysis techniques. Analysis of variance problems as a subject of regression. Brief discussions of robustness of estimation and robustness of design.
Prerequisite: AMS 572 or equivalent 3 credits, Letter graded (A, A-, B+, etc.)

ANALYSIS OF VARIANCE AMS 581
Analysis of models with fixed effects. The Gauss-Markov theorem; construction of confidence ellipsoids and tests with Gaussian observations. Problems of multiple tests of hypotheses. One-way, two-way, and higher-way layouts. Analysis of incomplete designs such as Latin squares and incomplete blocks. Analysis of covariance problems. Prerequisite: AMS 570 or equivalent.
3 credits, Letter graded (A, A-, B+, etc.)

DESIGN OF EXPERIMENTS AMS 582
Discussion of the accuracy of experiments, partitioning sums of squares, randomized designs, factorial experiments, Latin squares, confounding and fractional replication, response surface experiments, and incomplete block designs. Coscheduled as AMS 582 or HPH 699.
Prerequisite: AMS 572 or equivalent 3 credits, Letter graded (A, A-, B+, etc.)

TIME SERIES AMS 586
Analysis in the frequency domain. Periodograms, approximate tests, relation to regression theory. Pre-whitening and digital fibers. Common data windows. Fast Fourier transforms. Complex demodulation, Gibbs' phenomenon issues. Time-domain analysis.
Prerequisites: AMS 507 and AMS 570 3 credits, Letter graded (A, A-, B+, etc.)

NONPARAMETRIC STATISTICS AMS 587
This course covers the applied nonparametric statistical procedures: one-sample Wilcoxon tests, two-sample Wilcoxon tests, runs test, Kruskal-Wallis test, Kendall 's tau, Spearman's rho, Hodges-Lehman estimation, Friedman analysis of variance on ranks. The course gives the theoretical underpinnings to these procedures, showing how existing techniques may be extended and new techniques developed. An excursion into the new problems of multivariate nonparametric inference is made.
Fall, 3 credits, Letter graded (A, A-, B+, etc.)

BIOSTATISTICS AMS 588
Statistical techniques for planning and analyzing medical studies. Planning and conducting clinical trials and retrospective and prospective epidemiological studies. Analysis of survival times including singly censored and doubly censored data. Quantitative and quantal bioassays, two-stage assays, routine bioassays. Quality control for medical studies.
Fall, 3 credits, Letter graded (A, A-, B+, etc.)

QUANTITATIVE GENETICS AMS 589
Definition of relevant terminology. Statistical and genetic models for inheritance of quantitative traits. Estimation of effects of selection, dominance polygenes, epistatis, and environment. Linkage studies and threshold characteristics.
Spring, odd years, 3 credits, Letter graded (A, A-, B+, etc.)

TOPICS FOR M.S. STUDENTS AMS 591
Various topics of current interest in applied mathematics will be offered if sufficient interest is shown. Several topics may be taught concurrently in different sections.
Prerequisite: Permission of instructor. 3 credits, Letter graded (A, A-, B+, etc.) .May be repeated for credit.

MATHEMATICAL METHODS FOR FINANCE AND INVESTMENTS AMS 592
A broad-based course in mathematical modeling and quantitative analysis of financial transactions and investment management issues such as debt and equity, measures of risk and returns, efficient markets and efficient set mathematics, asset pricing, one-factor and multiple-factor models, portfolio selection, futures and options.
Fall, 3 credits, Letter graded (A, A-, B+, etc.)

MATHEMATICAL THEORY OF INTEREST AND PORTFOLIO PRICING AMS 593
Calculation of simple and compound interest poses elementary arithmetic or algebraic problems. Considers how variable interest rates (including indexing), inflation, changes in the exchange rates of foreign currency, and changes in the laws, such as income tax, create investment risks. The course is intended to develop problem-solving skills and adopts both deterministic and stochastic approaches. The perspectives of the consumer and the investor are taken into account. The material helps students prepare for the actuarial examinations. Topics are selected from the following: simple and compound interest, fixed-rate loans and mortgages, annuities and capital budgeting of pension plans, variable interest rates, bonds, prepayment and default scenarios, and currency baskets. Fall, 3 credits, Letter graded (A, A-, B+, etc.)

MATHEMATICAL METHODS FOR FINANCE AND INVESTMENTS II AMS 594
This course employs the techniques of mathematical statistics and empirical finance, e.g., estimation theory, linear and nonlinear regression, time series analysis, modeling and simulation to examine critically various models of prediction for asset-pricing, pricing of derivative products and term-structure of interest rates assuming stochastic volatility. Statistics necessary for analysis is incorporated in the course.
Fall, 3 credits, Letter graded (A, A-, B+, etc.)

FUNDAMENTALS OF COMPUTING AMS 595
Introduction to Unix operating system, C language, graphics, and parallel supercomputing.
Fall, 1 credit, Letter graded (A, A-, B+, etc.)

FUNDAMENTALS OF LARGE-SCALE COMPUTING AMS 596
Overview of the design and maintenance of large-scale computer projects in applied mathematics, including basic programming techniques for massively parallel supercomputers.
Spring, 1 credit, Letter graded (A, A-, B+, etc.)

STATISTICAL COMPUTING AMS 597
Introduction to statistical computing using SAS and S plus.
Fall, 1 credit, Letter graded (A, A-, B+, etc.)

RESEARCH AMS 599
Variable and repetitive credit

PROBABILITY THEORY II AMS 605
Advanced probability. Conditional sigma-fields, stochastic processes, Brownian motion, Markov property, weak convergence, infinitely divisible distributions, martingales, stochastic integrals, stochastic differential equations, and stochastic approximation.
Prerequisite: AMS 569 or permission of instructor. 3 credits, Letter graded (A, A-, B+, etc.)

ADVANCED STOCHASTIC PROCESSES I AMS 607
Theory and application of continuous time stochastic processes, continuous time martingales, square-integrable martingales, Brownian motion, stochastic integrals and Ito's formula, stochastic differential equations, and applications to financial mathematics.
Spring, 3 credits, Letter graded (A, A-, B+, etc.)

ADVANCED STOCHASTIC PROCESSES II AMS 615
Existence, uniqueness, and continuity theorems. Approximate solutions by method of iteration. Study of autonomous systems. Phase-plane analysis, periodic solutions. Singular points, cycles, and limit cycles. Theory of bifurcation. Stability theory and Liapunov functions. Analytical and geometrical investigations of second-order equations such as van der Pol's and Lienard's equations. Prerequisite: AMS 501 3 credits, Letter graded (A, A-, B+, etc.)

THEORY AND APPLICATIONS OF LARGE-SCALE NETWORKS AMS 620
A rigorous treatment of mathematical techniques used to answer many practical questions arising in the study and design of large-scale networks. Emphasis on the development of algorithms. Several lectures devoted to specific applications to computer networks to be used throughout the course. Prerequisite: AMS 540 or equivalent . 3 credits, Letter graded (A, A-, B+, etc.)

FINITE ELEMENT METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS AMS 621
Variational form of the problem, Ritz-Galerkin, collocation, and mixed methods; triangular, rectangular (2-D), and tetrahedral (3-D) elements; accuracy, convergence, and stability; solutions of linear, nonlinear steady-state, and dynamic problems; implicit and explicit time integration; equivalence of finite-element and finite-difference methods.
Prerequisite: AMS 502 or equivalent 3 credits, Letter graded (A, A-, B+, etc.)

TOPICS IN SYSTEMS AND CONTROL THEORY AMS 623
This course is designed for second- and third-year graduate students who wish to pursue research in the area of systems and control theory. The students are expected to have a strong research background in linear algebra and differential equations and basic knowledge in systems and control theory.
Spring, 3 credits, Letter graded (A, A-, B+, etc.) . May be repeated for credit.

THEORY OF INTEGRAL EQUATIONS AND THEIR APPLICATIONS AMS 627
Integral equations with degenerate kernels, equations of the second kind, iterative solutions, contraction mapping principle, Fredholm theory, and spectral theory for symmetric kernels. Volterra equations of the first and second kind, equations with weakly singular kernels, simultaneous systems, and applications.
Prerequisites: AMS 504 and AMS 505 . 3 credits, Letter graded (A, A-, B+, etc.)

APPLICATIONS OF FUNCTIONAL ANALYSIS AMS 628
Introduction to such topics as unbounded operators and the closed-graph theorem, convexity, weak convergence in Hilbert space, and degree theory. Applications to monotone operators and the stability of nonlinear systems, to Schwartz distributions and passive linear systems, and to the solution of nonlinear equations.
Prerequisite: AMS 504 or equivalent . 3 credits, Letter graded (A, A-, B+, etc.)

SPECIAL TOPICS IN MATHEMATICAL PROGRAMMING AMS 641
The course is designed for second- and third-year graduate students with a strong foundation in linear algebra and analysis who wish to pursue research in applied mathematics. Varying topics from nonlinear programming and optimization to applied graph theory and applied combinatorics may be offered concurrently.
Prerequisites: AMS 540 and permission of instructor. 3 credits, Letter graded (A, A-, B+, etc.) . May be repeated for credit.

SPECIAL TOPICS IN APPLIED PROBABILITY AMS 644
The course is designed for second- and third-year graduate students with a background in probability and stochastic modeling who wish to pursue research in applications of the probability theory. Several topics may be taught concurrently in different sections.
Fall, 3 credits, Letter graded (A, A-, B+, etc.). May be repeated for credit.

NONLINEAR ANALYSIS AND OPTIMIZATION AMS 651
Iterative methods for solving nonlinear operator equations. Frechet differentials. The Newton-Raphson method in function space and nonlinear boundary value problems. The Courant penalty concept and constrained optimization. General multiplier rules. Variable metric gradient projection for nonlinear least-square methods, with applications.
3 credits, Letter graded (A, A-, B+, etc.)

SPECIAL TOPICS IN GAME THEORY AMS 652
The course is designed for second- and third-year graduate students who wish to specialize in the mathematical theory of games.
Prerequisites: AMS 552 and permission of instructor. 3 credits, Letter graded (A, A-, B+, etc.)

SPECIAL TOPICS IN PROBABILITY AND MATHEMATICAL STATISTICS AMS 670
The course is designed for second- and third-year graduate students with a strong foundation in analysis and statistics who wish to pursue research in mathematical statistics. Several topics may be taught concurrently in different sections.
Prerequisites: AMS 569, AMS 570. 3 credits, Letter graded (A, A-, B+, etc.). May be repeated for credit.

SPECIAL TOPICS IN APPLIED STATISTICS AMS 675
The course is designed for second- and third-year students with a strong foundation in statistical analysis who wish to pursue research in applied statistics.
Prerequisites: AMS 507, AMS 572. 3 credits, Letter graded (A, A-, B+, etc.). May be repeated for credit.

INTERNSHIP IN APPLIED MATHEMATICS AMS 676
Directed research and/or practical experience in industry, financial and consulting firms, and research institutions. Students are required to have a department faculty adviser who coordinates and supervises the internship. Submission of the final report is required. 0 - 9 credits, S/U grading

BIOLOGICAL PHYSICS & BIOPHYSICAL CHEMISTRY : THEORETICAL PERSPECTIVES AMS 683
This course will survey a selected number of topics in biological physics and biophysical chemistry. The emphasis is on the understanding of physical organization principles and fundamental mechanisms involved in the biological process. The potential topics include: Protein Folding, Protein Dynamics, Biomolecular Interactions and Recognition, Electron and Proton Transfer, Motors, Membranes, Single Molecules and Single Cells, Cellular Networks, Development and Differentiation, Brains and Neural Systems, Evolution. There will be no homework or exams. The grades will be based on the performance of the term projects. Crosslisted with PHY 680 and CHE 683.
0 - 3 credits, Letter graded (A, A-, B+, etc.)

SPECIAL TOPICS IN DIFFERENTIAL EQUATIONS AND APPLIED ANALYSIS AMS 690
The course is designed for second- and third-year graduate students with a strong foundation in analysis who wish to pursue research in applied mathematics. Several topics may be taught concurrently in different sections.
Prerequisites: AMS 501, AMS 504. 3 credits, Letter graded (A, A-, B+, etc.) . May be repeated for credit

TOPICS IN APPLIED MATHEMATICS AMS 691
Varying topics selected from the list below if sufficient interest is shown. Several topics may be taught concurrently in different sections: Advanced Operational Methods in Applied Mathematics Approximate Methods in Boundary Value Problems in Applied Mathematics Control Theory and Optimization Foundations of Passive Systems Theory Game Theory Mixed Boundary Value Problems in Elasticity Partial Differential Equations Quantitative Genetics Stochastic Modeling.
3 credits, Letter graded (A, A-, B+, etc.). May be repeated for credit.

SPECIAL TOPICS IN NUMERICAL ANALYSIS AND SCIENTIFIC COMPUTING AMS 695
Analysis and Scientific Computing. The course is designed for second- and third-year graduate students with a strong foundation in applied linear algebra and numerical analysis who wish to pursue research in applied mathematics. Several topics may be taught concurrently in different sections. Prerequisites: AMS 505, AMS 526 .
3 credits, Letter graded (A, A-, B+, etc.) . May be repeated for credit.

PRACTICUM IN TEACHING AMS 698
May be repeated for credit

DISSERTATION RESEARCH ON CAMPUS AMS 699
Prerequisite: Must be advanced to candidacy (G5). Major portion of research must take place on SBU campus, at Cold Spring Harbor , or at the Brookhaven National Lab. Fall, Spring, and Summer, 1 - 12 credits, S/U grading. May be repeated for credit

DISSERTATION RESEARCH OFF CAMPUS - DOMESTIC AMS 700
Prerequisite: Must be advanced to candidacy (G5). Major portion of research will take place off-campus, but in the United States and/or U.S. provinces. Please note, Brookhaven National Labs and the Cold Spring Harbor Lab are considered on-campus. All international students must enroll in one of the graduate student insurance plans and should be advised by an International Advisor.
Fall, Spring, Summer. Prerequisite: G5 Standing. 1 - 9 credits, S/U grading. May be repeated for credit.

DISSERTATION RESEARCH OFF CAMPUS - INTERNATIONAL AMS 701
Prerequisite: Must be advanced to candidacy (G5). Major portion of research will take place outside of the United States and/or U.S. provinces. Domestic students have the option of the health plan and may also enroll in MEDEX. International students who are in their home country are not covered by mandatory health plan and must contact the Insurance Office for the insurance charge to be removed. International students who are not in their home country are charged for the mandatory health insurance. If they are to be covered by another insurance plan they must file a waiver be second week of classes. The charge will only be removed if other plan is deemed comparable.
All international students must received clearance from an International Advisor. Fall, Spring, Summer . Prerequisite: G5 Standing . 1 - 9 credits, S/U grading . May be repeated for credit.

SUMMER RESEARCH AMS 800
May be repeated for credit.

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