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Mathematical Sciences - Masters Programs

The Department of Mathematical Sciences offers the degree Master of Science in three areas; Mathematics, Applied Mathematics, and Stochastic Systems & Optimization.  Each program requires 30 credits of coursework (ten courses) to graduate.  Further program requirements are detailed in the Graduate Student Handbook, available from the Office of Graduate Admissions. 

Master of Science, Mathematics

A master’s degree in mathematics requires 30 credits of courses, including the following core courses:

Core Courses

MA 552 Axiomatic Linear Algebra Close Axiomatic Linear Algebra Fields and vector spaces; subspaces and quotient spaces; basis and dimension; linear transformations and matrices; determinants; and the theory of a single linear transformation.

MA 605 Foundations of Algebra I Close Foundations of Algebra I Topics covered in the sequence MA 605-606 include: elementary number theory, basic group theory, Lagrange’s theorem, isomorphism theorems, solvability, direct products, Jordan-Holder theorem, Sylow theorems, basic properties of rings, quotient rings, field of quotients of an integral domain, polynomial rings, factorization, elementary properties of fields, field extensions, and Galois theory.

MA 611 Probability Close Probability Foundations of probability, random variables and their distributions, discrete and continuous random variables, independence, expectation and conditioning, generating functions, multivariate distributions, convergence of random variables, and classical limit theorems.

MA 635 Real Variables I Close Real Variables I The real number system. Introduction to metric spaces and their applications. Lebesque measure and integral from a classical and/or modern approach.

MA 651 Topology I Close Topology I Metric spaces and topological spaces, bases and sub-bases, connectivity, local (path) connectivity, separation axioms, compactness and local compactness, concepts of convergence, Tychonoff’s theorem, Urysohn’s lemma, Tietze extension theorem, and selected topics as time permits.

MA 681 Functions of a Complex Variable I Close Functions of a Complex Variable I Complex numbers; elementary functions; Möbius transformations; analytic functions; power series; integration; Cauchy-Goursat theorems; Cauchy integral formula; Taylor and Laurent series; singularities; residue theory; and meromorphic and entire functions.

 

Master of Science, Applied Mathematics

MA 547 Advanced Calculus I Close Advanced Calculus I Elementary topology of Euclidean spaces; differential calculus of functions of several variables; inverse and implicit function theorems; integration; differential forms; and theorems of Gauss, Green, and Stokes.

or

MA 635 Real Variables I Close Real Variables I The real number system. Introduction to metric spaces and their applications. Lebesque measure and integral from a classical and/or modern approach.

MA 552 Axiomatic Linear Algebra Close Axiomatic Linear Algebra Fields and vector spaces; subspaces and quotient spaces; basis and dimension; linear transformations and matrices; determinants; and the theory of a single linear transformation.

MA 611 Probability Close Probability Foundations of probability, random variables and their distributions, discrete and continuous random variables, independence, expectation and conditioning, generating functions, multivariate distributions, convergence of random variables, and classical limit theorems.

MA 615 Numerical Analysis I Close Numerical Analysis I The MA 615-616 sequence covers topics in numerical analysis and numerical methods including: errors and accuracy; polynomial approximation; interpolation; numerical differentiation and integration; numerical solution of differential equations; least square and minimum-maximum error approximations; nonlinear equations; simultaneous linear equations; summing series, Fourier series, filter design, the frequency approach, design of numerical tools, and statistics of error analysis; eigenvalues and eigenvectors of matrices; and the orientation throughout is toward computers.

MA 649 Intermediate Differential Equations Close Intermediate Differential Equations Theory and application of ordinary differential equations (ODEs) with an emphasis on ODEs as continuous dynamical systems on a finite-dimensional phase space. Standard topics include existence and uniqueness theorems, general theory for linear equations, the exponential of linear map, stability of equilibrium points, hyperbolicity and structural stability, Lyapunov’s method, invariant manifolds, Floquet theory for periodic orbits, and Poincare-Bendixon theorem.

MA 681 Functions of a Complex Variable I Close Functions of a Complex Variable I Complex numbers; elementary functions; Möbius transformations; analytic functions; power series; integration; Cauchy-Goursat theorems; Cauchy integral formula; Taylor and Laurent series; singularities; residue theory; and meromorphic and entire functions.

MA 548 Advanced Calculus II Close Advanced Calculus II A continuation of MA 547, but with greater emphasis on mathematical rigor. Topics covered may include convergence of series, Riemann-Stieltjes integration, functions of bounded variation, metric spaces, introduction to measure theory, and functional analysis.

MA 627 Combinatorial Analysis Close Combinatorial Analysis Fundamental laws of counting, permutations, combinations, recurrence relations, Mšbius inversion, probleme des menages, probleme des recontres, partitions, trees, generating functions, Ramsey theory, transversal theory, and matroid theory.

MA 653 Numerical Solutions of Partial Differential Equations Close Numerical Solutions of Partial Differential Equations This course is an introduction to methods and theory in numerical solutions of partial differential equations. The finite difference and pseudo-spectral methods will be used as examples to solve partial differential equations, including parabolic, hyperbolic, and elliptic equations in one or higher dimensional space. The theory on consistency, convergence, and Von Neumann stability analysis of numerical schemes will be emphasized for a basic understanding about how to control numerical errors and to achieve higher order accuracy for numerical solutions. Students will also be assigned projects to obtain the first-hand experience in numerical computations.

CE 519 Advanced Structural Analysis Close Advanced Structural Analysis Analysis of structures using methods of work, slope deflection and moment distribution; force acceleration and energy methods; variable moments of inertia; continuous beams, trusses and frames; arch analysis; plasticity and limit design; slab and shell structures.

CE 601 Theory of Elasticity Close Theory of Elasticity Review of matrix algebra; the strain tensor, including higher order terms; the stress tensor; derivation of the linear form of Hooke's law and the higher order form of Hooke's law; equilibrium equations, boundary conditions and compatibility conditions; applications to the bending and torsion problems; variational and approximate methods of solving the Dirichlet type boundary value problems with particular application to the torsion problem. Fall semester.

CS 590 Algorithms Close Algorithms This is a course on more complex data structures, and algorithm design  and analysis, using the C++ language. Topics include: advanced and/or  balanced search trees; hashing; further asymptotic complexity analysis;  standard algorithm design techniques; graph algorithms; complex sort  algorithms; and other "classic" algorithms that serve as examples of  design techniques.

ME 674 Fluid Dynamics Close Fluid Dynamics Stress in a continuum; kinematics of fluid motion; rate of strain and vorticity; relation between stress and rate of strain; the Navier-Stokes equations; inviscid flow; stream function, velocity potential and circulation; Kelvin and Helmholtz theorems; two-dimensional incompressible flows; the Kuta-Joukowski theorem; introduction to compressible flows, boundary layers and drag-on bodies.

PEP 520 Computational Physics Close Computational Physics Numerical techniques. Numerical methods for integrating Newton’s laws, the diffusion equation, Poisson’s equation, and the wave equation are discussed. Topics also covered: discrete Fourier transform, stability theory,curve fitting , the diagonalization of matrices, and Monte Carlo methods. Spring semester. Typical text: Garcia, Computational Physics.

Master of Science, Stochastic Systems and Optimization

MA 547 Advanced Calculus I Close Advanced Calculus I Elementary topology of Euclidean spaces; differential calculus of functions of several variables; inverse and implicit function theorems; integration; differential forms; and theorems of Gauss, Green, and Stokes.

MA 611 Probability Close Probability Foundations of probability, random variables and their distributions, discrete and continuous random variables, independence, expectation and conditioning, generating functions, multivariate distributions, convergence of random variables, and classical limit theorems.

MA 612 Mathematical Statistics Close Mathematical Statistics Point estimation, method of moments, maximum likelihood, and properties of point estimators; confidence intervals and hypothesis testing; sufficiency; Neyman-Pearson theorem, uniformly most powerful tests, and likelihood ratio tests; and Fisher information and the Cramer-Rao inequality. Additional topics may include nonparametric statistics, decision theory, and linear models.

MA 623 Stochastic Processes Close Stochastic Processes Random walks and Markov chains; Brownian motions and Markov processes; and applications, stationary (wide sense) processes, infinite divisibility, and spectral decomposition.  By permission of instructor.

MA 629 Convex Analysis and Optimization Close Convex Analysis and Optimization The objective of this course is to introduce the students to the basic results of convex analysis and optimization. The properties of nonlinear non-smooth optimization models will be analyzed. Topics include: separation and representation of convex sets, properties of convex functions, subgradients, optimality conditions, saddle points, constraint qualifications, Fenchel and Lagrange duality, and sensitivity analysis.  Examples of optimization models from probability, statistics, and approximation theory will be discussed, as well as some basic models from management, finance, telecommunications, and other fields.

MA 661 Dynamic Programming & Stochastic Optimal Control Close Dynamic Programming & Stochastic Optimal Control The main purpose of this course is to present the foundations of the stochastic control theory, the corresponding numerical methods, and some applications. The focus will be on the idea of dynamic programming which will be developed starting from deterministic models, through finite-horizon stochastic problems, to infinite-horizon stochastic problems of various types. Applications to queuing systems, network design, and routing; supply-chain management and others will be discussed in detail. Topics to be covered: basic concepts of control theory for stochastic dynamic systems; controlled Markov chains; dynamic programming for finite horizon problems; infinite horizon discounted problems; numerical methods for infinite horizon problems; linear stochastic dynamic systems in discrete time; tracking and Kalman filtering; linear quadratic models; controlled Markov processes in continuous time; and elements of stochastic control theory in continuous time and state space.

MA 615 Numerical Analysis I Close Numerical Analysis I The MA 615-616 sequence covers topics in numerical analysis and numerical methods including: errors and accuracy; polynomial approximation; interpolation; numerical differentiation and integration; numerical solution of differential equations; least square and minimum-maximum error approximations; nonlinear equations; simultaneous linear equations; summing series, Fourier series, filter design, the frequency approach, design of numerical tools, and statistics of error analysis; eigenvalues and eigenvectors of matrices; and the orientation throughout is toward computers.

MA 627 Combinatorial Analysis Close Combinatorial Analysis Fundamental laws of counting, permutations, combinations, recurrence relations, Mšbius inversion, probleme des menages, probleme des recontres, partitions, trees, generating functions, Ramsey theory, transversal theory, and matroid theory.

MA 632 Theory of Games Close Theory of Games Strategic games and Nash equilibrium, strictly competitive (zero-sum) games and max-minimization, sStrategic games with imperfect information (Bayesian games), extensive games with perfect information (bargaining and repeated games), extensive games with imperfect information and signaling games, coalitional games (the core, stable sets, and bargaining sets), and auctions.

MA 641 Time Series Analysis I Close Time Series Analysis I Scope and applications of time series analysis: process control, financial data analysis and forecasting, and signal processing. Exploratory data analysis: graphical analysis, trend and seasonality detection and removal, and moving-average filtering. Review of basic statistical concepts related to the characterization of stationary processes. ARMA models and prediction of stationary processes. Estimation of ARMA models and model building and forecasting with ARMA models. Spectral analysis: periodogram testing for seasonality and periodicities and the maximum entropy and maximum-likelihood estimators. Asymptotic convergence. Selected topics, such as multivariate time series, nonlinear models, Kalman filtering, econometric forecasting, and long-memory processes. Selected applications, such as the unit-root problem in economics, forecasting and testing for market efficiency in financial time series, process control, and quality control.

MA 655 Optimal Control Theory Close Optimal Control Theory The main purpose of this course is to present the foundations of the optimal control theory, some applications, and their solutions. The students will be introduced to the core concepts and results of control and system theory. The foundational and basic results will be derived for discrete and continuous time scales, and state variables. Topics to be covered: proportional-derivative control; state-space and spectrum assignment; outputs and dynamic feedback; reachability; controllability; feedback and stability; Lyapunov theory; linearization principle of observability; dynamic programming algorithm; multipliers for unconstrained and constrained controls; and Pontryagin maximum principle.

MA 662 Stochastic Programming Close Stochastic Programming This course introduces students to modeling and numerical techniques for optimization under uncertainty and risk. Topics include: generalized concavity of measures, optimization problems with probabilistic constraints (convexity, differentiability, optimality, and duality), numerical methods for solving problems with probabilistic constraints, two-stage and multi-stage models (structure, optimality, duality), decomposition methods for two-stage and multi-stage models, risk averse optimization models,

MA 720 Advanced Statistics Close Advanced Statistics Selected topics may include: distribution theory; theory   of inference;  foundations of probability; spectral analysis;   multivariant analysis.

CS 535 Financial Computing Close Financial Computing This is a course in modeling the values of assets and financial derivatives and the software implementation of these models for pricing, simulations, and scenario analysis. The course includes an introduction to markets and financial derivatives, and a development of the necessary tools from the theories of stochastic processes and parabolic differential equations. An integral part of the course is the use of financial information sources and software packages available on the Internet for modeling and analysis.

EN 780 Nonlinear Correlation and System Identification Close Nonlinear Correlation and System Identification An investigation of tools to identify nonlinear processes and relationships. Mathematical tools covered include nonlinear regression, artificial neural networks, and multivariate polynomial regression. Applications include mass transfer correlations, prediction of drinking water quality, and modeling of wastewater treatment processes. Prerequisites: CE 679 or equivalent, and permission of instructor.

MGT 730 Design and Analysis of Experiments Close Design and Analysis of Experiments This course starts with the design and analysis of one factor analysis of variance. Methods of testing specific questions using planned comparisons are stressed. Models with two or more factors are considered with detailed instruction on the analysis of interactions. Repeated-measures designs are also covered, as well as designs with random and fixed factors.

Contact

Douglas Bauer
Professor
Kidde Building
Room 222
Phone: 201.216.5436
Fax: 201.216.8321
dbauer@stevens.edu

Ms. Linda Squier
Administrative Assistant
Kidde
Room 100
Phone: 201.216.5449
Fax: 201.216.8321
lsquier@stevens.edu

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