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## OR 709 Dynamic Programming 3. Prerequisite: MA 405, ST 421.

Introduction to theory and computational aspects of dynamic programming and its application to sequential decision problems.

## ISE 709 Dynamic Programming 3. Prerequisite: MA 405, ST 421.

Introduction to theory and computational aspects of dynamic programming and its application to sequential decision problems.

## ISE 361 Deterministic Models in Industrial Engineering 3. Prerequisite: (MA 303 or MA 341 or MA 405 )and C or better in ISE 110.

Introduction to mathematical modeling, analysis techniques, and solution procedures applicable to decision-making problems in a deterministic environment. Linear programming models and algorithms and associated computer codes are emphasized.

## OR 506 Algorithmic Methods in Nonlinear Programming 3. Prerequisite: MA 301, MA 405, knowledge of computer language, such as FORTRAN or PL1.

Introduction to methods for obtaining approximate solutions to unconstrained and constrained minimization problems of moderate size. Emphasis on geometrical interpretation and actual coordinate descent, steepest descent, Newton and quasi-Newton methods, conjugate gradient search, gradient projection and penalty function methods for constrained problems. Specialized problems and algorithms treated as time permits.

## EMS 490 School Mathematics from an Advanced Perspective 3. Prerequisite: MA 403 or MA 407, MA 308 or MA 408, MA 205 or MA 305 or MA 405.

This course will serve as a culminating experience for all students majoring in mathematics education and intending to become high school mathematics teachers. Course content includes functions in both secondary and collegiate mathematics, development of euclidean geometry from euclid's elements, and historical overview of albebra, and other mathematics subject matter, a trigonometry review from both triangle basis and funtion basis, connections between linear algebra and the high school presentation of matrices, and other topics. For Math Education majors only.

## ECG 766 Computational Methods in Economics and Finance 3. Prerequisite: (MA 305 or MA 405) and MA 341 and EC 301 and EC 302 and (CSC 112 or 114) or equivalents..

Fundamental methods for forumlating and solving economic models numerically will be developed. Emphasis on defining the mathematical structure of problems and practical computer methods for obtaining model solutions. Major topics include solution of systems of equations, complementarity relationships and optimization. Finite and infinite dimensional problems will be addressed, the latter through the use of finite dimensional approximation techniques. Particular emphasis placed on solving dynamic asset pricing, optimization and equilibrium problems. MS in Financial Mathematics Program required.

## OR 708 Integer Programming 3. Prerequisite: MA 405, OR (MA,IE) 505, Corequisite: Some familiarity with computers (e.g., CSC 112).

General integer programming problems and principal methods of solving them. Emphasis on intuitive presentation of ideas underlying various algorithms rather than detailed description of computer codes. Students have some "hands on" computing experience that should enable them to adapt ideas presented in course to integer programming problems they may encounter.

## ISE 708 Integer Programming 3. Prerequisite: MA 405, OR (MA,IE) 505, Corequisite: Some familiarity with computers (e.g., CSC 112).

General integer programming problems and principal methods of solving them. Emphasis on intuitive presentation of ideas underlying various algorithms rather than detailed description of computer codes. Students have some "hands on" computing experience that should enable them to adapt ideas presented in course to integer programming problems they may encounter.

## MA 405 Introduction to Linear Algebra 3. Prerequisite: MA 241 (MA 225 recommended); Corequisite: MA 341 is recommended.

This course offiers a rigorous treatment of linear algebra, including systems of linear equations, matrices, determinants, abstract vector spaces, bases, linear independence, spanning sets, linear transformations, eigenvalues and eigenvectors, similarity, inner product spaces, orthogonality and orthogonal bases, factorization of matrices. Compared with MA 305 Introductory Linear Algebra, more emphasis is placed on theory and proofs. MA 225 is recommended as a prerequisite. Credit is not allowed for both MA 305 and MA 405.

## MA 407 Introduction to Modern Algebra for Mathematics Majors 3. Prerequisite: MA 225 and MA 405.

## MA 426 Mathematical Analysis II 3. Prerequisite: MA 425 and 405.

## MA 430 Mathematical Models in the Physical Sciences 3. Prerequisite: MA 341 and MA 405.

Application of mathematical techniques to topics in the physical sciences. Problems from such areas as conservative and dissipative dynamics, calculus of variations, control theory, and crystallography.

## MA 432 Mathematical Models in Life and Social Sciences 3. Prerequisite: MA 341, (MA 305 or MA 405), and programming proficiency; Corequisite: (MA 421 or ST 371).

Topics from differential and difference equations, probability, and matrix algebra applied to formulation and analysis of mathematical models in biological and social science (e.g., population growth).

## MA 437 Applications of Algebra 3. Prerequisite: MA 403 or 407, MA 405.

Error correcting codes, cryptography, crystallography, enumeration techniques, exact solutions of linear equations, and block designs.

## MA 504 Introduction to Mathematical Programming 3. Prerequisite: MA 242, MA 405.

Basic concepts of linear, nonlinear and dynamic programming theory. Not for majors in OR at Ph.D. level.

## OR 504 Introduction to Mathematical Programming 3. Prerequisite: MA 242, MA 405.

Basic concepts of linear, nonlinear and dynamic programming theory. Not for majors in OR at Ph.D. level.

## MA 520 Linear Algebra 3. Prerequisite: MA 405.

Vector spaces. Bases and dimension. Changes of basis. Linear transformations and their matrices. Linear functionals. Simultaneous triangularization and diagonalization. Rational and Jordan canonical forms. Bilinear forms.

## MA 521 Abstract Algebra I 3. Prerequisite: MA 405 and MA 407.

Groups, normal subgroups, quotient groups, Cayley's theorem, Sylow's theorem. Rings, ideals and quotient rings, polynomial rings. Elements of field theory.

## MA 522 Computer Algebra 3. Prerequisite: MA 407 or MA 521 and MA 405 or MA 520.

Basic techniques and algorithms of computer algebra. Integer arithmetic, primality tests and factorization of integers, polynomial arithmetic, polynomial factorization, Groebner bases, integration in finite terms.

## MA 523 Linear Transformations and Matrix Theory 3. Prerequisite: MA 405.

Vector spaces, linear transformations and matrices, orthogonality, orthogonal transformations with emphasis on rotations and reflections, matrix norms, projectors, least squares, generalized inverses, definite matrices, singular values.

## MA 524 Combinatorics I 3. Prerequisite: MA 405 and MA 407.

Enumerative combinatorics, including placements of balls in bins, the twelvefold way, inclusion/exclusion, sign-reversing involutions and lattice path enumeration. Partically ordered sets, lattices, distributive lattices, Moebius functions, and rational generating functions.

## MA 532 Ordinary Differential Equations I 3. Prerequisite: MA 341, 405, 425 or 511, Corequisite: MA 426 or 512.

Existence and uniqueness theorems, systems of linear equations, fundamental matrices, matrix exponential, nonlinear systems, plane autonomous systems, stability theory.

## MA 537 Nonlinear Dynamics and Chaos 3. Prerequisite: MA 341 and MA 405.

Usage of computer experiments for demonstration of nonlinear dynamics and chaos and motivation of mathematical definitions and concepts. Examples from finance and ecology as well as traditional science and engineering. Difference equations and iteration of functions as nonlinear dynamical systems. Fixed points, periodic points and general orbits. Bifurcations and transition to chaos. Symbolic dynamics, chaos, Sarkovskii's Theorem, Schwarzian derivative, Newton's method and fractals.

## MA 555 Introduction to Manifold Theory 3. Prerequisite: MA 405 and MA 426.

An introduction to smooth manifolds. Topics include: topological and smooth manifolds, smooth maps and differentials, vector fields and flows, Lie derivatives, vector bundles, tensors, differential forms, exterior calculus, and integration on manifolds.

## MA 719 Vector Space Methods in System Optimization 3. Prerequisite: MA 405, 511.

Introduction to algebraic and function-analytic concepts used in system modeling and optimization: vector space, linear mappings, spectral decomposition, adjoints, orthogonal projection, quality, fixed points and differentials. Emphasis on geometricinsight. Topics include least square optimization of linear systems, minimum norm problems in Banach space, linearization in Hilbert space, iterative solution of system equations and optimization problems. Broad range of applications in operations research and system engineering including control theory, mathematical programming, econometrics, statistical estimation, circuit theory and numerical analysis.

## OR 719 Vector Space Methods in System Optimization 3. Prerequisite: MA 405, 511.

Introduction to algebraic and function-analytic concepts used in system modeling and optimization: vector space, linear mappings, spectral decomposition, adjoints, orthogonal projection, quality, fixed points and differentials. Emphasis on geometricinsight. Topics include least square optimization of linear systems, minimum norm problems in Banach space, linearization in Hilbert space, iterative solution of system equations and optimization problems. Broad range of applications in operations research and system engineering including control theory, mathematical programming, econometrics, statistical estimation, circuit theory and numerical analysis.

## MA 746 Introduction To Stochastic Processes 3. Prerequisite: MA 405 and MA(ST) 546 or ST 521.

Markov chains and Markov processes, Poisson process, birth and death processes, queuing theory, renewal theory, stationary processes, Brownian motion.

## ST 746 Introduction To Stochastic Processes 3. Prerequisite: MA 405 and MA(ST) 546 or ST 521.

Markov chains and Markov processes, Poisson process, birth and death processes, queuing theory, renewal theory, stationary processes, Brownian motion.

## MA 788 Numerical Nonlinear Partial Differential Equations 3. Prerequisite: MA 405 or 520 and MA 501 or 534; knowledge of a high level programming language.

Nonlinear discrete equations; Newton and monotone methods for nonlinear equations; computational algorithms and applications; finite difference method-convergence, stability and error estimates; multiplicity of solutions and bifurcation; asymptotic behavior of solutions; and coupled systems of equations.

## CSC 428 Introduction to Numerical Analysis II 3. Prerequisite: MA 405 and programming language proficiency; MA (CSC) 427 is not a prerequisite.

Computational procedures including direct and iterative solution of linear and nonlinear equations, matrices and eigenvalue calculations, function approximation by least squares, smoothing functions, and minimax approximations.

## MA 428 Introduction to Numerical Analysis II 3. Prerequisite: MA 405 or MA 305 and programming language proficiency..

Computational procedures including direct and iterative solution of linear and nonlinear equations, matrices and eigenvalue calculations, function approximation by least squares, smoothing functions, and minimax approximations.

## CSC 580 Numerical Analysis I 3. Prerequisite: MA 405; MA 425 or MA 511; high-level computer language.

Algorithm behavior and applicability. Effect of roundoff errors, systems of linear equations and direct methods, least squares via Givens and Householder transformations, stationary and Krylov iterative methods, the conjugate gradient and GMRES methods, convergence of method.

## MA 580 Numerical Analysis I 3. Prerequisite: MA 405; MA 425 or MA 511; high-level computer language.

Algorithm behavior and applicability. Effect of roundoff errors, systems of linear equations and direct methods, least squares via Givens and Householder transformations, stationary and Krylov iterative methods, the conjugate gradient and GMRES methods, convergence of method.

## E 531 Dynamic Systems and Multivariable Control I 3. Prerequisite: MA 341, MA 405.

Introduction to modeling, analysis and control of linear discrete-time and continuous-time dynamical systems. State space representations and transfer methods. Controllability and observability. Realization. Applications to biological, chemical, economic, electrical, mechanical and sociological systems.

## OR 531 Dynamic Systems and Multivariable Control I 3. Prerequisite: MA 341, MA 405.

Introduction to modeling, analysis and control of linear discrete-time and continuous-time dynamical systems. State space representations and transfer methods. Controllability and observability. Realization. Applications to biological, chemical, economic, electrical, mechanical and sociological systems.

## MA 531 Dynamic Systems and Multivariable Control I 3. Prerequisite: MA 341, MA 405.

Introduction to modeling, analysis and control of linear discrete-time and continuous-time dynamical systems. State space representations and transfer methods. Controllability and observability. Realization. Applications to biological, chemical, economic, electrical, mechanical and sociological systems.

## MA 505 Linear Programming 3. Prerequisite: MA 405.

Introduction including: applications to economics and engineering; the simplex and interior-point methods; parametric programming and post-optimality analysis; duality matrix games, linear systems solvability theory and linear systems duality theory; polyhedral sets and cones, including their convexity and separation properties and dual representations; equilibrium prices, Lagrange multipliers, subgradients and sensitivity analysis.

## OR 505 Linear Programming 3. Prerequisite: MA 405.

Introduction including: applications to economics and engineering; the simplex and interior-point methods; parametric programming and post-optimality analysis; duality matrix games, linear systems solvability theory and linear systems duality theory; polyhedral sets and cones, including their convexity and separation properties and dual representations; equilibrium prices, Lagrange multipliers, subgradients and sensitivity analysis.

## ISE 505 Linear Programming 3. Prerequisite: MA 405.

Introduction including: applications to economics and engineering; the simplex and interior-point methods; parametric programming and post-optimality analysis; duality matrix games, linear systems solvability theory and linear systems duality theory; polyhedral sets and cones, including their convexity and separation properties and dual representations; equilibrium prices, Lagrange multipliers, subgradients and sensitivity analysis.

## MA 708 Integer Programming 3. Prerequisite: MA 405, OR (MA,IE) 505, Corequisite: Some familiarity with computers (e.g., CSC 112).

General integer programming problems and principal methods of solving them. Emphasis on intuitive presentation of ideas underlying various algorithms rather than detailed description of computer codes. Students have some "hands on" computing experience that should enable them to adapt ideas presented in course to integer programming problems they may encounter.

## ST 430 Introduction to Regression Analysis 3. Prerequisites: (ST 305 or ST 312 or ST 372) and ST 307 and (MA 305 or MA 405).

Regression analysis as a flexible statistical problem solving methodology. Matrix review; variable selection; prediction; multicolinearity; model diagnostics; dummy variables; logistic and non-linear regression. Emphasizes use of computer.

## TT 405 Advanced Nonwovens Processing 3. Prerequisite: MA 241, PY 208, TT 305.

Mechanisms used in the production of nonwoven materials. Design and operation of these mechanisms. Process flow, optimization of process parameters, influence of process parameters on product properties.

## MA 518 Geometry of Curves and Surfaces 3. Prerequisite: MA 242 and MA 405.

Geometry of curves and surfaces in space; Arclength, torsion, and curvature of curves; Tangent spaces, shape operators, and curvatures of surfaces; metrics, covariant derivatives, geodesics, and holonomy. Applications in the physical sciences and/or projects using computer algebra.

## ISE 362 Stochastic Models in Industrial Engineering 3. Prerequisite: C or better in ISE/TE 110 and (MA 303 or MA 341 or MA 405) and C- or better in ST 371 or ST 370.

Introduction to mathematical modeling, analysis, and solution procedures applicable to uncertain (stochastic) production systems. Methodologies covered include probability theory and stochastic processes. Applications relate to design and analysisof problems, capacity planning, inventory control, waiting lines, and system reliability and maintainability.

## OR 565 Graph Theory 3. Prerequisite: MA 231 or MA 405.

Basic concepts of graph theory. Trees and forests. Vector spaces associated with a graph. Representation of graphs by binary matrices and list structures. Traversability. Connectivity. Matchings and assignment problems. Planar graphs. Colorability. Directed graphs. Applications of graph theory with emphasis on organizing problems in a form suitable for computer solution.

## PY 509 General Relativity 3. P: MA 401 and MA 405 and PY 412 and PY 415; R: Graduate Standing.

This course provides in-depth knowledge of general relativity covering: Einstein's equation, Schwarzschild metric, Kerr metric, Friedman-Robertson-Walker metric, Christoffel symbols, Killing vectors, Riemann curvature,and Ricci tensors. Theoretical computations are compared with experimental data including the precession rate of the perihelion for Mercury and the deflection in the solar eclipse, the geodelic affect and the frame dragging effect measured in Gravity Probe B experiment.

## FIM 549 Financial Risk Analysis 3. Prerequisites: MA 405 and (MA 421 or ST 421) and (MA/ST 412 or MA/ST 413).

This course focuses on mathematical methods to analyze and manage risks associated with financial derivatives. Topics covered include aggregate loss distributions, extreme value theory, default probabilities, Value-at-Risk and expected shortfall, coherent risk measures, correlation and copula, applications of principle component analysis and Monte Carlo simulations in financial risk management, how to use stochastic differential equations to price financial risk derivatives, and how to back-test and stress-test models.

## MA 402 Mathematics of Scientific Computing 3. P: (MA 341 or MA 405) and programming proficiency (MATLAB, C++, Java, Fortran, or other language).

This course will provide an overview of methods to solve quantitative problems and analyze data. The tools to be introduced are mathematical in nature and have links to Algebra, Analysis, Geometry, Graph Theory, Probability and Topology. Students will acquire an appreciation of (I) the fundamental role played by mathematics in countless applications and (II) the exciting challenges in mathematical research that lie ahead in the analysis of large data and uncertainties. Students will work on a project for each unit. While this is not a programming class, the students will do some programming through their projects.

## ECG 528 Options and Derivatives Pricing 3. Prerequisites: MA 341 and MA 405 and MA 421.

The course covers (i) structure and operation of derivative markets, (ii) valuation of derivatives, (iii) hedging of derivatives, and (iv) applications of derivatives in areas of risk management and financial engineering. Models and pricing techniques include Black-Scholes model, binomial trees, Monte-Carlo simulation. Specific topics include simple no-arbitrage pricing relations for futures/forward contracts; put-call parity relationship; delta, gamma, and vega hedging; implied volatility and statistical properties; dynamic hedging strategies; interest-rate risk, pricing of fixed-income product; credit risk, pricing of defaultable securities.

## MBA 528 Options and Derivatives Pricing 3. Prerequisites: MA 341 and MA 405 and MA 421.

The course covers (i) structure and operation of derivative markets, (ii) valuation of derivatives, (iii) hedging of derivatives, and (iv) applications of derivatives in areas of risk management and financial engineering. Models and pricing techniques include Black-Scholes model, binomial trees, Monte-Carlo simulation. Specific topics include simple no-arbitrage pricing relations for futures/forward contracts; put-call parity relationship; delta, gamma, and vega hedging; implied volatility and statistical properties; dynamic hedging strategies; interest-rate risk, pricing of fixed-income product; credit risk, pricing of defaultable securities.

## FIM 528 Options and Derivatives Pricing 3. Prerequisites: MA 341 and MA 405 and MA 421.

The course covers (i) structure and operation of derivative markets, (ii) valuation of derivatives, (iii) hedging of derivatives, and (iv) applications of derivatives in areas of risk management and financial engineering. Models and pricing techniques include Black-Scholes model, binomial trees, Monte-Carlo simulation. Specific topics include simple no-arbitrage pricing relations for futures/forward contracts; put-call parity relationship; delta, gamma, and vega hedging; implied volatility and statistical properties; dynamic hedging strategies; interest-rate risk, pricing of fixed-income product; credit risk, pricing of defaultable securities.

## MA 528 Options and Derivatives Pricing 3. Prerequisites: MA 341 and MA 405 and MA 421.

## MA 548 Monte Carlo Methods for Financial Math 3. Prerequisites: (MA 421 or ST 421), MA 341, and MA 405.

Monte Carlo (MC) methods for accurate option pricing, hedging and risk management. Modeling using stochastic asset models (e.g. geometric Brownian motion) and parameter estimation. Stochastic models, including use of random number generators, random paths and discretization methods (e.g. Euler-Maruyama method), and variance reduction. Implementation using Matlab. Incorporation of the latest developments regarding MC methods and their uses in Finance.

## ST 442 Introduction to Data Science 3. P: (MA 305 or MA 405) and (ST 305 or ST 312 or ST 370 or ST 372) and (CSC 111 or CSC 112 or CSC 113 or CSC 116 or ST 114 or ST 445).

Overview of data structures, data lifecycle, statistical inference. Data management, queries, data cleaning, data wrangling. Classification and prediction methods to include linear regression, logistic regression, k-nearest neighbors, classification and regression trees. Association analysis. Clustering methods. Emphasis on analyzing data, use and development of software tools, and comparing methods.

## CSC 442 Introduction to Data Science 3. P: (MA 305 or MA 405) and (ST 305 or ST 312 or ST 370 or ST 372) and (CSC 111 or CSC 112 or CSC 113 or CSC 116 or ST 114 or ST 445).

Overview of data structures, data lifecycle, statistical inference. Data management, queries, data cleaning, data wrangling. Classification and prediction methods to include linear regression, logistic regression, k-nearest neighbors, classification and regression trees. Association analysis. Clustering methods. Emphasis on analyzing data, use and development of software tools, and comparing methods.

## MA 549 Financial Risk Analysis 3. Prerequisites: MA 405 and (MA 421 or ST 421) and (MA/ST 412 or MA/ST 413).

This course focuses on mathematical methods to analyze and manage risks associated with financial derivatives. Topics covered include aggregate loss distributions, extreme value theory, default probabilities, Value-at-Risk and expected shortfall, coherent risk measures, correlation and copula, applications of principle component analysis and Monte Carlo simulations in financial risk management, how to use stochastic differential equations to price financial risk derivatives, and how to back-test and stress-test models.

## FIM 548 Monte Carlo Methods for Financial Math 3. Prerequisites: (MA 421 or ST 421), MA 341, and MA 405.

Monte Carlo (MC) methods for accurate option pricing, hedging and risk management. Modeling using stochastic asset models (e.g. geometric Brownian motion) and parameter estimation. Stochastic models, including use of random number generators, random paths and discretization methods (e.g. Euler-Maruyama method), and variance reduction. Implementation using Matlab. Incorporation of the latest developments regarding MC methods and their uses in Finance.

## ECE 765 Probabilistic Graphical Models for Signal Processing and Computer Vision 3. Prerequisites: Programming experience (MATLAB, C++ or other object oriented language such as Python), linear algebra (MA 405 or equivalent), and probability (ECE 514, equivalent or instructor permission).

Techniques for machine learning using probabilistic graphical models. Emphasis on Bayesian and Markov networks with applications to signal processing and computer vision.