Department of Mathematical Sciences

**MATH 8500: Computational Algebraic Geometry, 3 cr. (3 and 0)**

Covers algebraic geometry and commutative algevra via Grobner bases. Includes ideals and varieties (affine and projective), Grobner bases, elimination theory, dimensions, solving plynomial systems via eigenvalues aand eigenvectors. Selected applications may include coding theory, computer vision, geometric theorem proving, integer programming, or statistics.

Prerequisite: MATH 3110, 4120.

**MATH 8510: Abstract Algebra I, 3 cr. (3 and 0)**

Basic algebraic structures: groups, rings and fields; permutation groups, Sylow theorems, finite abelian groups, polynomial domains, factorization theory and elementary field theory.**MATH 8520: Abstract Algebra II, 3 cr. (3 and 0)**

A continuation of MATH 8510 including selected topics from ring theory and field theory.**MATH 8530: Matrix Analysis, 3 cr. (3 and 0)**

Topics in matrix analysis that support an applied curriculum: similarity and eigenvalues; Hermitian and normal matrices; canonical forms; norms; eigenvalue localizations; singular value decompositions; definite matrices.

Prerequisite: MATH 3110, 4530 or 4630.**MATH 8540: Theory of Graphs, 3 cr. (3 and 0)**

Connectedness; path problems; trees; matching theorems; directed graphs; fundamental numbers of the theory of graphs; groups and graphs.

Prerequisite: permission of instructor.**MATH 8550: Combinatorial Analysis, 3 cr. (3 and 0)**

Combinations; permutations; permutations with restricted position; Polya's theorem; principle of inclusion and exclusion; partitions; recurrence relations; generating functions; Mobius inversion; enumeration techniques; Ramsey numbers; finite projective and affine geometries; Latin rectangles; orthogonal arrays; block designs; error detecting and error correcting codes.

Prerequisite: MATH 3110.

MATH 8560: Applicable Algebra, 3 cr. (3 and 0)

Applied algebraic ideas in lattice theory and Boolean Algebra; finite-state sequential machines; group theory as applied to network complexity and combinatorial enumeration; algebraic coding theory. Topics vary with background and interests of students.

Prerequisites: MATH 8510 and 8530 or permission of instructor.

**MATH 8570: Cryptography, 3 cr. (3 and 0)**

Classical and modern cryptography and their uses in modern communication systems are covered. Topics include entropy, Shannon's perfect secrecy theorem, Advanced Encryption Standard (AES), integer factorization, RSA cryptosystem, discrete logarithm problem, Diffie-Hellman key exchange, digital signatures, elliptic curve cryptosystems, hash functions, and identification schemes.

Prerequisite: MATH 3110, 4000 or 6000, 4120 or 8510.**MATH 9510: Algebraic Number Theory, 3 cr. (3 and 0)**

Covers arithmetic of number fields and number rings. Covers prime decomposition, ideal class groups, unit groups of number fields and distribution of prime ideals in number fields. Provides an overview of completions absolute values and valuation theory.

Prerequisite: MATH 8510.

**MATH 9540: Advanced Graph Theory, 3 cr. (3 and 0)**

Continuation of MATH 8540; topics not covered in 8540 including the four-color theorem, domination numbers, Ramsey theory, graph isomorphism, embeddings, algebraic graph theory and tournaments; research papers are also examined.

Prerequisite: MATH 8540 or permission of instructor.**MATH 9850: Selected Topics in Algebra and Combinatorics, 1-3 cr. (1-3 and 0)**

Advanced topics in algebra and combinatorics from current problems of interest. May be repeated for credit, but only if different topics are covered.

Sample Offerings:

*Introduction to Cryptography*

The purpose of this course is to acquaint the students with classical and modern methods of cryptography and their uses in modern communication systems. Main topics: Shannon's theory, conventional cryptosystems, DES, AES, finite fields and elementary number theory, RSA, Diffie-Hellman key exchange scheme, ElGamal cryptosystem, digital signature schemes, elliptic curves and elliptic curve cryptosystems, hash functions, pseudorandom numbers, identification schemes, and zero knowledge proofs.

This courses covers the basics of coding theory. Topics include cyclic codes, BCH codes, Reed-Solomon codes, and finite geometry.

This course covers basic finite field theory and applications.

Algebraic Curves

This course covers some basic results about algebraic curves that are useful in constructing error-correcting codes and in implementing public-key cryptosystems. Basic concepts in algebraic geometry and commutative algebra to be covered include varieties, polynomial and rational maps, divisors, (prime) ideals, function fields, valuations, local rings, Riemann-Roch Theorem, etc.

The course focuses heavily on the theory and applications of Grobner bases. Coding theory is emphasized as an area of application, including decoding of Reed-Solomon codes and Hermitian codes.

Fast Fourier transforms, fast multiplication of polynomials (integers), fast decoding of RS codes, sparse linear systems (from coding theory, cryptography and computer algebra), Krylov subspace methods (Lanczos and bi-orthogonal methods), Wiedemann's method a la Berlekamp-Massey, block algorithms (Coppersmith's and Montgomory's) and their analysis.

**MATH 9860: Selected Topics in Geometry, 1-3 cr. (1-3 and 0)**

Advanced topics in Geometry from current problems of interest. May be repeated for credit, but only if different topics are covered.

Course Substitution Policy

. . . back to Algebra and Discrete Mathematics page

**MATH 8210: Linear Analysis, 3 cr. (3 and 0)**

Metric spaces,completeness of a metric space and the completion of a metric space, infinite dimensional vectors spaces, Zorn's Lemma, normed spaces and compactness, Schauder Basis, linear operators bounded and unbounded, linear functionals, minimization results for normed spaces, inner product spaces, projection theorems and minimization, Hilbert spaces, Riesz - Fischer Theorem and self - adjoint operators, orthogonal systems.

Prerequisites: MATH 4540/6540 or 4530 and 8530**MATH 8220: Measure and Integration, 3 cr. (3 and 0)**

Riemann and Riemann - Stieljes integration, inner and outer measures, Cantor sets, measurability and additivity, abstract integration and Lebesgue integration, types of convergence and convergence interchange results, Lebesgue spaces; integration and differentiation, product measure, Fubini type results.

Prerequisite: MATH 4540/6540, MATH 8210**MATH 8230: Complex Analysis, 3 cr. (3 and 0)**

Topological concepts; complex integration; local and global properties of analytic functions; power series; analytic continuation; representation theorems; calculus of residues. Designed for nonengineering majors.

Prerequisite: MATH 4640/6640**MATH 8250: Introduction to Dynamical Systems Theory, 3 cr. (3 and 0)**

Techniques of analysis of dynamical systems; sensitivity analysis, linear systems, stability and control; theory of differential and difference equations.

Prerequisites: MATH 4540/6540 and 3110 or 4530 and 8530**MATH 8260: Partial Differential Equations, 3 cr. (3 and 0)**

First-order equations: elliptic, hyperbolic and parabolic; second-order equations: existence and uniqueness results, maximum principles, finite difference and Hilbert Space methods.

Prerequisite: MATH 8210 or permission of instructor

MATH 8270: Dynamical System Neural Networks, 3 cr. (3 and 0)

Scalar and planar maps with applications from biology; existence and uniqueness, bifurcations, periodic equations, stability of equilibria, conservative systems.

Prerequisites: MATH 4530, MATH 4540, MATH 8250 and MATH 8210**MATH 8310: Fourier Series, 3 cr. (3 and 0)**

Fourier series with applications to solution of boundary value problems in partial differential equations of physics and engineering; introduction to Bessel functions and Legendre polynominals.

Prerequisite: MATH 4640/6640

MATH 8370: Calculus of Variations and Optimal Control, 3 cr. (3 and 0)

Fundamental theory of the calculus of variations; variable end points; the parametric problem; the isoperimetric problem; constraint inequalities; introduction to the theory of optimal control; connections with the calculus of variations; geometric concepts.

Prerequisite: MATH 4530/6530 or 4630/6630**MATH 8410: Applied Mathematics I, 3 cr. (3 and 0)**

Derivation of equations from conservation laws, dimensional analysis, scaling and simplification; methods such as steepest descent, stationary phase, perturbation series, boundary layer theory, WKB theory, multiple-scale analysis and ray theory applied to problems in diffusion processes, wave propagation, fluid dynamics and mechanics.

Prerequisites: MATH 2080 and 4530/6530 or 4630/6630**MATH 8610: Advanced Numerical Analysis I, 3 cr. (3 and 0)**

Interpolation and approximation; numerical quadrature; numerical solution of functional differential equations; integral equations and overdetermined linear systems; eigenvalue problems; approximation using splines.

Prerequisites: MATH 4530 and 4600.**MATH 9270: Functional Analysis, 3 cr. (3 and 0)**

Topological Vector Spaces, Hahn - Banach Theorems, Closed and Open Mapping Theorems, Linear operators on specific spaces and spectral theory, distributions and Sobolev spaces.

Prerequisite: MATH 8210, MATH 8220**MATH 9740: Mathematical Models in Investment Science, 3 cr. (3 and 0)**

The course deals with a collection of concepts, constructs, and mathematical models that have been created to help deal with (in a rational manner) a portion of the myriad of problems that arise in the financial arena. There are two major themes in the course:

- How to decide the best course of action in an investment situation, e.g. how to devise the best portfolio, how to devise the optimal investment strategy for managing an investment, how to select a group of investment projects.
- How to determine the correct arbitrage-free, fair, or equilibrium value of an asset, e.g. the value of a firm, the value of a bond, the value of a derivative such as a put or call option.

Prerequisites: Individuals should have a technical background roughly equivalent to a bachelor's degree in engineering, mathematics, science, or economics; or have some familiarity with basic calculus, linear algebra, and probability theory. Most of the mathematics is at the level of undergraduate calculus.

**MATH 9820: Selected Topics in Analysis, 1-3 cr. (1-3 and 0)**

Advanced analysis topics from current problems of interest. May be repeated for credit if different topics are covered.

Sample Offerings:*Stochastic Calculus for Finance*

This special topics course is intended as an introduction to some basic ideas for modeling and simulation in finance. The course begins with a discussion of simple random walks and the analysis of certain gambling games. These topics are used to motivate the theory of martingales and continuous time stochastic processes. The course will then take up the Ito integral and enough of the theory of the diffusion equation to be able to solve the Black-Scholes PDE and prove the uniqueness of the solution. The foundations for the martingale theory of arbitrage pricing are then prefaced by a well-motivated development of the martingale representation theorems and Girsanov theory.

Prerequisites: Some analysis beyond calculus, an introduction to linear algebra, and basic methods from probability and statistics.

This special topics course is intended provide hands on familiarity with simulation of financial models. This will be a "soft" computing course, i.e., we will not prove convergence of the approximations used in the simulations. The course will make use of Maple and MatLab programs from the literature: e.g.,

- D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review 43 (2001), 525-546.
- D. J. Higham and P. E. Kloeden, Maple and MatLab for stochastic differential equations in finance, research report.
- D. J. Higham, Nine ways to implement the binomial method of option valuation in MatLab, research report.
- S. Cygnowski, L. Grüne, and P. E. Kloeden, Maple for stochastic differential equations, research report.

Prerequisites: Basic concepts from probability and stochastic processes. Familiarity with martingales, MatLab, and Maple would be helpful but not necessary.

Course Substitution Policy

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**MATH 8600: An Introduction to Scientific Computing, 3 cr. (3 and 0)**

Floating point models, conditioning and numerical stability, numerical linear algebra, integration, systems of ordinary differential equations and zero finding; emphasis is on the use of existing scientific software.

Prerequisites: MATH 2080, 3110 and CPSC 1100.**MATH 8610: Advanced Numerical Analysis I, 3 cr. (3 and 0)**

Interpolation and approximation; numerical quadrature; numerical solution of functional differential equations; integral equations and overdetermined linear systems; eigenvalue problems; approximation using splines.

Prerequisites: MATH 4530 and 4600.

**MATH 8630: Digital Models I, 3 cr. (3 and 0)**

Experimental mathematics; pseudostochastic processes; analytical and algebraic formulations of time-independent simulation; continuous-time simulation and discrete-time simulation;digital optimization; Fibonacci search; ravine search; gradient methods; current research in digital analysis. Offered Fall semester only.

Prerequisite: MATH 3110, 4530, and digital computer experience.**MATH 8650: Data Structures, 3 cr. (3 and 0)**

Representation and transformation of information; formal description of processes and data structures; tree and list structures; pushdown stacks; string and formula manipulation; hashing techniques; interrelation between data structure and program structure; storage allocation methods.

Prerequisites: Computational maturity and permission of instructor.

**MATH 8660: Finite Element Method, 3 cr. (3 and 0)**

Discusses the basic theory of the finite element method (FEM) for the numerical approximation of partial differential equations. Topics include Sobolev spaces, error estimation, and implementation of FEM in one and higher dimensions.

Prerequisite: MATH 8600 or consent of instructor.**MATH 9830: Selected Topics in Computational Mathematics, 1-3 cr. (1-3 and 0)**

Advanced topics in computational mathematics and numerical analysis from current problems of interest. May be repeated for credit if different topics are covered.

Sample Offerings:*Scientific Simulations in Java*

Because of its Object Orientation, its Platform Independence, and its tight specifications on arithmetic operations, Java is an appealing language for the development of Scientific Simulations. Since Java-based simulations can be distributed in the context of web-based documentation, such Java-based simulations can greatly enhance the dissemination of scientific knowledge and can substantially improve scientific training particularly in the area of understanding complex models. This course focuses on developing Scientific Simulations written in Java and distributed through the World Wide Web. The course is project oriented and involves the development of interactive web-based simulations of scientific topics chosen by the students.

This course, cross-listed as ChE 845, ME 893, and MATH 983, is team-taught by Math Sciences and Chemical Engineering faculty. The course presents a systems perspective of fiber and film processes using existing and new models developed by the Center for Advanced Engineering Fibers and Films. Constitutive equations are developed and applied to specific geometries and flow problems encountered in the production of fibers and films. Specific objectives are to develop the governing equations for polymeric fluids, derive various constitutive equations including those based on molecular models, explore analytical and numerical solution of the governing equations for special cases, develop an understanding for the strengths and weaknesses of the models to be discussed, and apply constitutive equations to fiber and film processing geometries.

Topics include classification of partial differential equations, the finite element method in one and higher dimensions, Sobolev spaces, interpolation theory, finite element spaces, and development of error estimates. Related topics are considered as time permits, including application of finite element methods to fluid flow problems which arise in science and engineering.

Course Substitution Policy

. . . back to Computational Mathematics page

**MATH 8000: Probability, 3 cr. (3 and 0)**

Basic probability theory with emphasis on results and techniques useful in operations research and statistics; axiomatic probability, advanced combinatorial probability, conditional informative expectation, functions of random variables, moment generating functions, distribution theory and limit theorems.

Prerequisite: MATH 2006

**MATH 8030: Stochastic Processes, 3 cr. (3 and 0)**

Theory and analysis of time series; recurrent events; Markov chains; random walks; renewal theory; application to communication theory; operations research.

Prerequisite: MATH 4000/6000 or MATH 8000**MATH 8100: Mathematical Programming, 3 cr. (3 and 0)**

Formulation and solution of linear programming models; mathematical development of the simplex method; revised simplex method; duality; sensitivity analysis; parametric programming, implementation, software packages.

Prerequisite: MATH 3110**MATH 8110: Nonlinear Programming, 3 cr. (3 and 0)**

Theoretical development of nonlinear optimization with applications; classical optimization; convex and concave functions; separable programming; quadratic programming; gradient methods.

Prerequisites: MATH 4400 and 4540

MATH 8120: Discrete Optimization, 3 cr. (3 and 0)

Principal methods used in integer programming and discrete optimization; branch and bound, implicit enumeration, cutting planes, group knapsack, Lagrangian relaxation, surrogate constraints, heuristics (performance analysis), separation/branching strategies and polynomial time algorithms for specific problems on special structures.

Prerequisite: MATH 8100 or equivalent**MATH 8130: Advanced Linear Programming, 3 cr. (3 and 0)**

Development of linear programming theory using inequality systems, convex cones, polyhedra and duality; solution algorithms and computational considerations for large scale and special structured problems using techniques of upper bounded variables, decomposition, partitioning and column generation; game theory; nonlinear representations and other methods such as ellipsoid and Karmarkar.

Prerequisite: MATH 4400/6400, 8100 or equivalent**MATH 8140: Network Flow Programming, 3 cr. (3 and 0)**

Max-flow/min-cut theorem; combinatorial applications; minimum cost flow problems (transportation, shortest path, transshipment); solution algorithms (including the out-of-kilter method); implementation and computational considerations.

Prerequisite: MATH 4400/6400, 8100 or equivalent

MATH 8160: Network Algorithms and Data Structures, 3 cr. (3 and 0)

Design, analysis and implementation of algorithms and data structures associated with the solution of problems formulated as networks and graphs; applications to graph theory, combinatorial optimization and network programming.

Corequisite: MATH 6400, 8100, 8540, 8630 or permission of instructor**MATH 8170: Stochastic Models in Operations Research I, 3 cr. (3 and 0)**

Stochastic control; structure of sequential decision processes; stochastic inventory models; recursive computation of optimal policies; discrete parameter finite Markov decision processes; various optimality criteria; computation by policy improvement and other methods; existence of optimal stationary policies; stopping-rule problems; examples from financial management, maintenance and reliability, search, queuing and shortest path.

Prerequisite: MATH 8030**MATH 8180: Stochastic Models in Operations Research II, 3 cr. (3 and 0)**

Introduction to queuing theory: Markovian queues, repairman problems, queues with an embedded Markov structure, the queue GI/G/1, queues with a large number of servers, decision making in queues; introduction to reliability theory; failure distributions; stochastic models for complex systems; maintenance and replacement policies; reliability properties of multicomponent structures.

Prerequisite: MATH 8170**MATH 8190: Multicriteria Optimization, 3 cr. (3 and 0)**

Theory and methodology of optimization problems with vector-valued objective functions; preference orders and domination structures; generating efficient solutions; solving mulitcriteria decision-making problems, noninteractive and interactive methods with applications.

Prerequisite: MATH 8100 or equivalent**MATH 8200: Complementarity Models, 3 cr. (3 and 0)**

Theory, algorithms and applications of linear and nonlinear complementarity; classes of matrices and functions and corresponding algorithms; applications to economics, mechanics and networks; generalizations to fixed-point problems and nonlinear systems of equations.

Prerequisite: MATH 8100

**MATH 9880: Selected Topics in Operations Research, 1-3 cr. (1-3 and 0)**

Advanced topics in operations research from current problems of interest. May be repeated for credit, but only if different topic covered.

Course Substitution Policy

. . . back to Operations Research page

**MATH 8010: General Linear Hypothesis I, 3 cr. (3 and 0)**

Least-square estimates; Gauss-Markov theorem; confidence ellipsoids and confidence intervals for estimable functions; tests of hypotheses; one-, two- and higher-way layouts; analysis of variance for other models.

Prerequisites: MATH 4030/6030and 3110**MATH 8020: General Linear Hypothesis II, 3 cr. (3 and 0)**

Continuation of MATH 8010.

Prerequisite: MATH 8010**MATH 8040: Statistical Inference, 3 cr. (3 and 0)**

Sampling distributions; maximum likelihood estimation and likelihood ratio tests; asymptotic confidence intervals for Binomial, Poisson and Exponential parameters; two sample methods; nonparametric tests; ANOVA; regression and model building.

Prerequisite: MATH 4000/6000 or equivalent or permission of instructor.

MATH 8050: Data Analysis, 3 cr. (3 and 0)

Methodology in analysis of statistical data emphasizing applications to real problems using computer-oriented techniques: computer plots, transformations, criteria for selecting variables, error analysis, multiple and stepwise regression, analysis of residuals, model building in time series and ANOVA problems, jackknife and random subsampling, multidimensional scaling, clustering.

Prerequisites: MATH 3010 and 4000/6000, or MATH 4010/6010 and 8000.

MATH 8060: Nonparametric Statistics, 3 cr. (3 and 0)

Order statistics; tolerance limits; rank-order statistics; Kolmogorov-Smirnov one-sample statistics; Chi-square goodness-of-fit test; two-sample problem; linear rank statistics; asymptotic relative efficiency.

Prerequisite: MATH 6000 or 8000.**MATH 8070: Applied Multivariate Analysis, 3 cr. (3 and 0)**

Applied multivariate analysis: computer plots of multivariate observations; multidimensional scaling; multivariate tests of means, covariances and equality of distributions; univariate and multivariate regressions and their comparisons; MANOVA; principle components analysis; factor analysis; analytic rotations; canonical correlations.

Prerequisites: MATH 4030/6030 and 8050 or permission of instructor.**MATH 8080: Reliability and Life Testing, 3 cr. (3 and 0)**

Probability models and statistical methods relevant to parametric and nonparametric analysis of reliability and life testing data.

Prerequisites: MATH 4000/6000 and 4010/6010 or equivalent.**MATH 8090: Time Series Analysis, Forecasting and Control, 3 cr. (3 and 0)**

Modeling and forecasting random processes; autocorrelation functions and spectral densities; model identification, estimation and diagnostic checking; transfer function models; feedforward and feedback control schemes.

Prerequisites: MATH 6000 and 6050, or MATH 8000 and 6050 or equivalent.**MATH 8810: Mathematical Statistics, 3 cr. (3 and 0)**

Fundamental concepts of sufficiency, hypothesis testing and estimation; robust estimation; resampling (jackknife, bootstrap, etc.) methods; asymptotic theory; two-stage and sequential sampling problems; ranking and selection procedures.

Prerequisite: MATH 4030/6030 or equivalent.

**MATH 8840: Statistics for Experimenters, 3 cr. (3 and 0)**

Statistical methods for students who are conducting experiments; introduction to descriptive statistics, estimation, and hypothesis testing as they relate to design of experiments; higher-order layouts, factorial and fracitonal factorial designs, and response surface models. Offered Fall semester only.

Prerequisite: MATH 2060 or equivalent.**MATH 8850: Advanced Data Analysis, 3 cr. (3 and 0)**

Continuation of MATH 8050, covering alternatives to ordinary least squares, influence and diagnostic considerations, robustness, special statistical computation methods.

Prerequisites: MATH 6030, 8000 and 8050.

**MATH 9010: Probability Theory I, 3 cr. (3 and 0)**

Axiomatic theory of probability; distribution functions; expectation; Cartesian product of infinitely many probability spaces, and the Kolmogorov consistency theorem; models of convergence; weak and strong laws of large numbers.

Prerequisite: MATH 4000 and 8220, or MATH 8000 and 8220 or consent of instructor.

**MATH 9020: Probability Theory II, 3 cr. (3 and 0)**

Continuation of MATH 901; characteristic functions, infinitely divisible distributions, central limit theorems, laws of large numbers, conditioning, and limit properties of sums of dependent random variables, conditioning, matingales.

Prerequisite: MATH 9010.

**MATH 9810: Selected Topics in Mathematical Statistics and Probability, 1-3 cr. (1-3 and 0)**

Advanced topics in mathematical statistics and probability of current interest. May be repeated for credit, but only if different topics are covered.

Course Substitution Policy. . . back to Statistics page

**For courses outside the Depatment of Mathematical Sciences:**

You must get prior approval from the Graduate Coordinator for courses to be taken outside the department if they are to be counted towards the course requirements for the degree (MS or PhD). Your committee chair should send the Graduate Coordinator a brief statement (email is fine) that is endorsed by the whole committee.

**For courses within the Department of Mathematical Sciences:**

If you plan to count a course in one area for a required course in another area, your advisor should send the Graduate Coordinator an email justifying the reason. For example, 803 is not a statistics course and therefore does not satisfy the statistics breadth requirement.

If you have any further questions, please refer to the Long Range Plan on the web or contact the Graduate Corrdinator at mthgrad@clemson.edu .