MS Math Courses

COURSES ACCEPTABLE FOR GRADUATE CREDIT FOR
THE MASTER OF SCIENCE WITH A MAJOR IN MATHEMATICS

The courses in this list can be counted toward the requirements of the Master of Science in Mathematics, with a concentration in Applied Mathematics. Other courses listed in the Graduate Catalog under Mathematics are for other graduate degrees. All requirements of the Department of Mathematical Sciences and the College of Graduate Studies of Georgia Southern University must be satisfied, including the distribution of the credit hours (namely that at least 1/2 of the required 36 credit hours be at the 7xxx level). All courses are 3 credits (unless otherwise indicated)
(Note that the following reflect updated prerequisites for some courses.)

Prerequisites: Permission of Instructor.
* Depending on the topics, this course may or may not be counted toward the degree. The approval of the advisor is needed.

MATH 5234G NUMBER THEORY
Introduction to the principal ideas of elementary number theory: divisibility, congruencies, linear Diophantine equations, Fermat's theorem, Euler's theorem, Pythagorean triple and t he distribution of primes.
Prerequisite: MATH 2332 (Mathematical Structures)

MATH 5330G OPERATIONS RESEARCH
Introduction to the analytical formulation and solution of decision problems. Mathematical methods of optimization of classical operations research models.
Prerequisites: MATH 3337 (Probability).

MATH 5332G ANALYSIS II
Continuation of the study of the fundamental concepts of calculus, such as continuity, differentiation and integrability in Euclidean n-space. Metric spaces. Function spaces.
Prerequisites: MATH 5331 (Analysis I).

MATH 5334G MODERN ALGEBRA II
A continuation of Modern Algebra I. Applications and deeper properties of the fundamental algebraic structures; isomorphisms of groups, rings and fields; quotient structures; vector spaces; and Euclidean constructions.
Prerequisites: MATH 3333 (Modern Algebra I).

MATH 5336G APPLIED NUMERICAL METHODS
Introduction to scientific computation on digital computers. Solution of nonlinear equations and systems of linear and nonlinear equations, polynomial interpolation, numerical differentiation and integration, data fitting and other numerical methods.
Prerequisites: MATH 2331 (Elementary Linear Algebra) and knowledge of a programming language.

MATH 5338G APPLIED MATHEMATICS
Theory and applications of mathematical methods such as power series solutions, Laplace Transforms, vector calculus, Fourier series, integrals, and partial differential equations.
Prerequisites: MATH 3230 (Differential Equations).

MATH 5430G INTRODUCTION TO MATHEMATICAL BIOLOGY
An introduction to applications of mathematics to various biological, ecological, physiological, and medical problems, which will be analyzed both analytically and numerically. Graduate students will be given additional assignments that will not be completed by undergraduate students. Prerequisite: MATH 3230 (Differential Equations) or permission of instructor.

MATH 5431G COMBINATORICS AND GRAPH THEORY
The course covers basic theory and applications of combinatorics and graph theory. Combinatorics is a study of different enumeration techniques of finite but large sets. Topics that will be studied include principle of inclusion and exclusion, generating functions and methods to solve difference equations. Graph theory is a study of graphs, trees and networks. Topics that will be discussed include Euler formula, Hamilton paths, planar graphs and coloring problem; the use of trees in sorting and prefix codes; useful algorithms on networks such as shortest path algorithm, minimal spanning tree algorithm and min-flow max-cut algorithm. Prerequisites: A minimum grade of "C" in MATH 2332 (Mathematical Structures) and MATH 3337 (Probability)

MATH 5433G DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES
Differential geometry uses tools from calculus and linear algebra to study the geometric properties of smooth curves and surfaces in Euclidean spaces. Topis include: arc length, surface area, geodesics, curvature, first and second fundamental forms, Gauss-Bonnett formula. Graduate students will be assigned additional assignments and/or project. Prerequisites: MATH 2243(Calculus III) and MATH 2331(Elementary Linear Algebra)

MATH 5434G FUNCTIONS OF A COMPLEX VARIABLE
Topics in complex variables including functions, limits, derivatives, integrals, the Cauchy-Riemann conditions, series representation of functions, and the Cauchy Integral formula.
Prerequisites: MATH 2332 (Mathematical Structures).

MATH 5435G INTRODUCTION TO TOPLOGY
An introduction to metric spaces, topological spaces, connectedness and compactness of topological spaces, and continuous functions on topological spaces. Graduate students enrolled in this course will complete one or more assignments that the undergraduate students will not be required to complete.
Prerequisites: A minimum grade of C in MATH 2332.

MATH 5436G INTRODUCTION TO FRACTALS
Fractals as nonlinear systems involving feedback and iteration. Classical fractals, Limits and self-similarity. Fractal dimensions. Encoding of fractals. Decoding of fractals. Iterated function systems.
Prerequisites: MATH 2243 (Calculus III), and MATH 2332 (Mathematical Structures), and MATH 5335 (Intermediate Linear Algebra).

MATH 5539G Mathematical Models
A study of model construction and types of models.
Prerequisites: MATH 3230 (Differential Equations), an introductory computing course, and at least 15 hours of upper level mathematics.

MATH 7090 SELECTED TOPICS IN APPLIED MATHEMATICS
Specialized study in a selected area of Applied Mathematics. (1 to 3 credits)
Prerequisites: Permission of Instructor.

MATH 7132 METHODS OF OPTIMIZATION
Selected methods for unconstrained and constrained optimization problems with applications.
Prerequisite: MATH 5330/5330G (OPERATIONS RESEARCH) or permission of instructor.

MATH 7231 ADVANCED NUMERICAL ANALYSIS I
An in-depth study of computer arithmetic, the solution of non-linear equations, the solution of systems of linear equations, eigenvalue problems, and interpolation. Algorithms and methods are developed and then implemented on a computer.
Prerequisites: MATH 5336 (Applied Numerical Methods).
Instructor may waive this prerequisite.

MATH 7232 ADVANCED NUMERICAL ANALYSIS II
An in-depth study of orthogonal polynomials, numerical integration, and numerical solutions of ordinary and partial differential equations. Development and computer implementation of algorithms and methods.
Prerequisites: MATH 7231 (ADVANCED NUMERICAL ANALYSIS I).
Prerequisite should be MATH 5336 (Applied Numerical Methods) or permission of instructor.

MATH 7234 ADVANCED LINEAR ALGEBRA
The study of linear maps on finite dimensional vector spaces. Topics include: diagonalization (direct sums, invariant subspaces and Cayley-Hamilton theorem for linear operators), inner product spaces (self-adjoint, orthogonal operators, orthogonal projections and the spectral theorem, bilinear and quadratic forms), canonical forms (Jordan and rational forms, minimal polynomials), special matrices (non-negative matrices), and the exponential of a linear operator. Prerequisites: MATH 5335(Intermediate Linear Algebra)

MATH 7235 ANALYTIC NUMBER THEORY
A study of topics from the classical analytic theory of numbers. Topics will be chosen from arithmetic functions, the distribution of primes, congruences, the Riemann-zeta functions, the prime number theorem, Eisenstein series, quadratic resides, Dirichlet series, Euler products, the Dedekind eta function, the Jacobi theta functions, integer partitions, and modular forms. Prerequisites: MATH 5234(Number Theory) and MATH 5434(Functions of a Complex Variable)

MATH 7236 ADVANCED ORDINARY DIFFERENTIAL EQUATIONS
Linear and nonlinear ordinary differential equations and their applications to physics and engineering. Topics include solution and stability of systems of equations, approximate solutions, and phase plane analysis.
Prerequisites: MATH 3230 (Differential Equations).

MATH 7237 MATHEMATICAL CONTROL THEORY
State-space techniques from modern control system theory. Topics include realization theory for MIMO systems, state-space techniques for feedback control, closed loop observer design, and state-space techniques in optimal control. Prerequisites: MATH 3230(Differential Equations) and MATH 5336(Applied Numerical Methods)

MATH 7330 FUNCTIONAL ANALYSIS
The study of normed linear spaces and linear operators. Topics include: Hilbert spaces (projection theorem, Riesz representation, Parseval relation); Banach spaces (convexity, duality, bounded and compact operators, theorems of Hahn-Banach, Banach-Steinhaus, open mapping, closed graph, Fredholm alternative); Stone-Weierstrass and Banach fixed point theorems. Prerequisites: MATH 5332(Analysis II) and MATH 5335(Intermediate Linear Algebra)

MATH 7331 REAL ANALYSIS
Theory of Lebesgue measure and integration, monotone convergence, the dominated convergence theorem, Fubini's Theorem, Radon-Nikodym theorem, Riesz representation theorem, Lp and lp spaces, functions of finite variation, Stieltjes integral, absolute continuity.
Prerequisite: MATH 5332/5332G (Analysis II).

MATH 7332 ADVANCED PARTIAL DIFFERENTIAL EQUATIONS
Theory of partial differential equations. Topics include Fourier series, boundary value problems of partial differential equations, applications of special functions, method of characteristics, and classification of second order equations.
Prerequisites: MATH 5338 (Applied Mathematics).

MATH 7333 COMPLEX ANALYSIS
An in-depth study of functions of one complex variable. Topics include: properties of holomorphic, harmonic, meromorphic and entire functions (open mapping, maximum modulus, mearn value, Poisson's, Rouche's, Liouville's, Picard's and Mittag-Leffler's theorems), residue theory (residue theorem, argument principle and applications), conformal mappings (Mobius and Christoffel- Schwarz canonical transformations, Riemann mapping theorem), analytic continuation (monodromy theorem, Schwarz reflection principle, Riemann surfaces and multi-valued functions). Prerequisites: MATH 5331(Analysis I) and MATH 5434(Functions of a Complex Variable)

MATH 7334 APPROXIMATION THEORY
The study of the approximation of functions in normed linear spaces. The course emphasizes the theory of interpolation and approximation by polynomials, rational functions and spline functions. Main topics include: best approximation, order of approximation, interpolation, existence and uniqueness of best approximants, theorems by Weierstrass, Haar, Chebyshev, Bernstein, Markov, Korovkin, Schoenberg, and applications. Prerequisites: MATH 5331(Analysis I) and MATH 5335(Intermediate Linear Algebra)

MATH 7432 DIFFERENTIAL GEOMETRY OF MANIFOLDS
The study and applications of calculus on manifolds. Topics include: atlases, tangent spaces, differentiable maps; immersions and submanifolds, submersions and quotient manifolds; matrix groups and their Lie algebras; vector fields and flows; differential forms, exterior derivative, Lie derivative. Prerequisites: MATH 5331(Analysis I) and MATH 5335(Intermediate Linear Algebra)

MATH 7435 ELEMENTS OF ALGEBRAIC TOPOLOGY
The study of the topology of geometric objects from the algebraic viewpoint, in particular using homotopy and homology groups. Main topics: Topological manifolds, homotopy, fundamental group, free groups, covering spaces, homology. Prerequisites: MATH 3333(Modern Algebra) and MATH 5435(Introduction to Topology)

MATH 7610 GRADUATE SEMINAR
Under supervision of one or more faculty members, each student will choose topics related to his or her concentration, or topics of interest to the class, read and research on them, then make presentations in front of the class or a larger audience. Students will also attend presentations of internal and external speakers on mathematical sciences. Prerequisites: MATH 5332(Analysis II), MATH 5335(Intermediate Linear Algebra), MATH 7231(Advanced Numerical Analysis I), and STAT 5531(Statistical Methods I) (Any two of the above four courses)

MATH 7890 DIRECTED STUDY IN APPLIED MATHEMATICS
Directed study under faculty supervision. (1 to 3 credit hours)
Prerequisites: Permission of Instructor and Department Chair.

MATH 7899 RESEARCH PROJECT IN APPLIED MATHEMATICS
Research project addressed toward a real world problem. (1 to 6 credit hours)
Prerequisites: Consent of project advisor and permission of Department Chair.

The following is approved, but is not usually taught
MATH 7130 MATHEMATICAL OPTIMIZATION THEORY
Necessity and sufficiency conditions for constrained optimization problems are derived. The derived conditions are used to help answer questions concerning whether a given optimization problem has a solution, whether a solution is unique, and how a solution can be found.
Prerequisites: MATH 5331 (Analysis I).