# Department of Mathematics

## Course Descriptions for (MATH) 5xx and 7xxl

511 - Probability (3) Probability and independence; discrete and continuous random variables; joint, marginal, and conditional densities, moment generating functions; laws of large numbers; binomial, Poisson, gamma, univariate, and bivariate normal distributions.
Prerequisites: C or higher or concurrent enrollement in MATH 241 or consent of the Undergraduate Director

514 - Financial Mathematics I (3) Probability spaces. Random variables. Mean and variance. Geometric Brownian Motion and stock price dynamics. Interest rates and present value analysis. Pricing via arbitrage arguments. Options pricing and the Black-Scholes formula.
Prerequisites: C or higher or concurrent enrollement in MATH 241 or consent of the Undergraduate Director

515 - Financial Mathematics II (3) Convex sets. Separating Hyperplane Theorem. Fundamental Theorem of Asset Pricing. Risk and expected return. Minimum variance portfolios. Capital Asset Pricing Model. Martingales and options pricing. Optimization models and dynamic programming.
Prerequisites: C or better in MATH 514 or STAT 522 or consent of the Undergraduate Director

520 - Ordinary Differential Equations (3) Differential equations of the first order, linear systems of ordinary differential equations, elementary qualitative properties of nonlinear systems.
Prerequisites: C or better in MATH 344 or 544; or consent of the Undergraduate Director

521 - Boundary Value Problems and Partial Differential Equations (3) Laplace transforms, two-point boundary value problems and Green’s functions, boundary value problems in partial differential equations, eigenfunction expansions and separation of variables, transform methods for solving PDE’s, Green’s functions for PDE’s, and the method of characteristics.
Prerequisites: C or better in MATH 520 or MATH 241 and 242 or consent of the Undergraduate Director

522 - Wavelets (3) Basic principles and methods of Fourier transforms, wavelets, and multiresolution analysis; applications to differential equations, data compression, and signal and image processing; development of numerical algorithms. Computer implementation.
Prerequisites: C or better in MATH 344 or 544 or consent of the Undergraduate Director

523 - Mathematical Modeling of Population Biology (3) Applications of differential and difference equations and linear algebra modeling the dynamics of populations, with emphasis on stability and oscillation. Critical analysis of current publications with computer simulation of models.
Prerequisites: C or better in MATH 142, BIOL 301, or MSCI 311 recommended

524 - Nonlinear Optimization (3)  Descent methods, conjugate direction methods, and Quasi-Newton algorithms for unconstrained optimization; globally convergent hybrid algorithm; primal, penalty, and barrier methods for constrained optimization. Computer implementation of algorithms.
Prerequisites: C or better in MATH 344 or 544 or consent of the Undergraduate Director

525 - Mathematical Game Theory (3) Two-person zero-sum games, minimax theorem, utility theory, n-person games, market games, stability.
Prerequisites: C or better in MATH 544 or in both MATH 300 and 344, or consent of the Undergraduate Director

526 - Numerical Linear Algebra (4) Matrix algebra, Gauss elimination, iterative methods; overdetermined systems and least squares; eigenvalues, eigenvectors; numerical software. Computer implementation. Credit may not be received for both MATH 526 and MATH 544.
Prerequisites: Concurrent enrollment in or C or better in MATH 142 or consent of the Undergraduate Director

527 - Numerical Analysis (3) Interpolation and approximation of functions; solution of algebraic equations; numerical differentiation and integration; numerical solutions of ordinary differential equations and boundary value problems; computer implementation of algorithms.
Prerequisites: C or better MATH 520 or in both MATH 242 and 344, or consent of the Undergraduate Director

531 - Foundations of Geometry (3) The study of geometry as a logical system based upon postulates and undefined terms. The fundamental concepts and relations of Euclidean geometry developed rigorously on the basis of a set of postulates. Some topics from non-Euclidean geometry.
Prerequisites: C or better in MATH 300 or consent of the Undergraduate Director

532 - Modern Geometry (3) Projective geometry, theorem of Desargues, conics, transformation theory, affine geometry, Euclidean geometry, non-Euclidean geometries, and topology.
Prerequisites: C or better in MATH 300 or consent of the Undergraduate Director

533 - Elementary Geometric Topology (3) Topology of the line, plane, and space, Jordan curve theorem, Brouwer fixed point theorem, Euler characteristic of polyhedra, orientable and non-orientable surfaces, classification of surfaces, network topology.
Prerequisites: C or better in MATH 300 or consent of the Undergraduate Director

534 - Elements of General Topology (3) Elementary properties of sets, functions, spaces, maps, separation axioms, compactness, completeness, convergence, connectedness, path connectedness, embedding and extension theorems, metric spaces, and compactification.
Prerequisites: C or better in MATH 300 or consent of the Undergraduate Director

540 - Modern Applied Algebra (3) Finite structures useful in applied areas. Binary relations, Boolean algebras, applications to optimization, and realization of finite state machines.
Prerequisites: MATH 241

541 - Algebraic Coding Theory (3) Error-correcting codes, polynomial rings, cyclic codes, finite fields, BCH codes
Prerequisites: C or better in MATH 544 or in both MATH 300 and 344 or consent of the Undergraduate Director

544 - Linear Algebra (3) Vectors, vector spaces, and subspaces; geometry of finite dimensional Euclidean space; linear transformations; eigenvalues on theoretical concepts, logic, and meethods.
Prerequisites: C or better in MATH 300, or consent of the Undergraduate Director

544L - Linear Algebra Lab (1) Computer-based applications of linear algebra for mathematics students. Topics include numerical analysis of matrices, direct and indirect methods for solving linear systems, and least squares method (regression). Typical applications include theoretical and practical issues related to discrete Markov’s processes, image compression, and linear programming.
Prerequisites: Prereq or coreq: C or better or concurrent enrollment in MATH 544.

546 - Algebraic Structures I (3)
Permutation groups; abstract groups; introduction to algebraic structures through study of subgroups, quotient groups, homomorphisms, isomorphisms, direct product; decompositions; introduction to rings and fields.
Prerequisites: C or better in MATH 544 or consent of the Undergraduate Director

547 - Algebraic Structures II (3) Rings, ideals, polynomial rings, unique factorization domains; structure of finite groups; topics from: fields, field extensions, Euclidean constructions, modules over principal ideal domains (canonical forms).
Prerequisites: C or higher in MATH 546 or consent of the Undergraduate Director

548 - Geometry, Algebra, and Algorithms (3) Polynomials and affine space, Groebner bases, elimination theory, varieties, and computer algebra systems.
Prerequisites: Math 300 and Math 544 or consent of the Undergraduate Director.

550 - Vector Analysis (3) Vector fields, line and path integrals, orientation and parametrization of lines and surfaces, change of variables and Jacobians, oriented surface integrals, theorems of Green, Gauss, and Stokes; introduction to tensor analysis.
Prerequisites: C or higher in MATH 241 or consent of the Undergraduate Director

551 - Introduction to Differential Geometry (3) Parametrized curves, regular curves and surfaces, change of parameters, tangent planes, the differential of a map, the Gauss map, first and second fundamental forms, vector fields, geodesics, and the exponential map.
Prerequisites: C or better in MATH 300 or consent of the Undergraduate Director

552 - Applied Complex Variables (3)  Complex integration, calculus of residues, conformal mapping, Taylor and Laurent Series expansions, applications.
Prerequisites: C or better in MATH 241 or consent of the Undergraduate Director

554 - Analysis I (3) Least upper bound axiom, the real numbers, compactness, sequences, continuity, uniform continuity, differentiation, Riemann integral and fundamental theorem of calculus.
Prerequisites: C or better in MATH 300 and either at last one of 511, 520, 534, 550, or 552, or consent of the Undergraduate Director

555 - Analysis II (3)  Riemann-Stieltjes integral, infinite series, sequences and series of functions, uniform convergence, Weierstrass approximation theorem, selected topics from Fourier series or Lebesgue integration.
Prerequisites: C or better in MATH 554 or consent of the Undergraduate Director

561 - Introduction to Mathematical Logic (3) Syntax and semantics of formal languages; sentential logic, proofs in first order logic; Godel’s completeness theorem; compactness theorem and applications; cardinals and ordinals; the Lowenheim-Skolem-Tarski theorem; Beth’s definability theorem; effectively computable functions; Godel’s incompleteness theorem; undecidable theories.
Prerequisites: C or better in MATH 300 or consent of the Undergraduate Director

562 - Theory of Computation (3) Basic theoretical principles of computing as modeled by formal languages and automata; computability and computational complexity.
Prerequisites: C or better in CSCE 350 or MATH 344 or 544 or 574 or consent of the Undergraduate Director

570 - Discrete Optimization (3) Discrete mathematical models. Applications to such problems as resource allocation and transportation. Topics include linear programming, integer programming, network analysis, and dynamic programming.
Prerequisites: C or better in MATH 344 or 544, or consent of the Undergraduate Director

574 - Discrete Mathematics I (3) Mathematical models; mathematical reasoning; enumeration; induction and recursion; tree structures; networks and graphs; analysis of algorithms.
Prerequisites: C or better in MATH 300 or consent of the Undergraduate Director

575 - Discrete Mathematics II (3) A continuation of MATH 574. Inversion formulas; Polya counting; combinatorial designs; minimax theorems; probabilistic methods; Ramsey theory; other topics.
Prerequisites: C or better in MATH 574 or consent of the Undergraduate Director

576 - Combinatorial Game Theory (3) Winning in certain combinatorial games such as Nim, Hackenbush, and Domineering. Equalities and inequalities among games, Sprague-Grundy theory of impartial games, games which are numbers.
Prerequisites: C or better in MATH 344, 544, or 574, or consent of the Undergraduate Director

580 - Elementary Number Theory (3) Divisibility, primes, congruences, quadratic residues, numerical functions. Diophantine equations.
Prerequisites: C or better in MATH 300 or consent of the Undergraduate Director

587 - Introduction to Cryptography (3) Design of secret codes for secure communication, including encryption and integrity verification: ciphers, cryptographic hashing, and public key cryptosystems such as RSA. Mathematical principles underlying encryption. Code-breaking techniques. Cryptographic protocols.
Prerequisites: C or better in CSCE 145 or in MATH 241, and in either CSCE 355 or MATH 574, or consent of the Undergraduate Director

590 - Undergraduate Seminar (1-3) A review of literature in specific subject areas involving student presentations. Content varies and will be announced in the Master Schedule of Classes by suffix and title. Pass-fail grading. For undergraduate credit only.
Prerequisites: consent of instructor

599 - Topics in Mathematics (1-3) Recent developments in pure and applied mathematics selected to meet current faculty and student interest.

602 - An Inductive Approach to Geometry (3)
This course is designed for middle-level pre-service mathematics teachers. This course covers geometric reasoning, Euclidean geometry, congruence, area, volume, similarity, symmetry, vectors, and transformations. Dynamic software will be utilized to explore geometry concepts.
Prerequisites: C or better in MATH 122 or 141 or equivalent, or consent of the Undergraduate Director

603 - Inquiry Approach to Algebra (3) This course introduces basic concepts in number theory and modern algebra that provide the foundation for middle level arithmetic and algebra. Topics include: algebraic reasoning, patterns, inductive reasoning, deductive reasoning, arithmetic and algebra of integers, algebraic systems, algebraic modeling, and axiomatic mathematics. This course cannot be used for credit towards a major in mathematics.
Prerequisites: C or better in MATH 122 or 141 or equivalent, or consent of the Undergraduate Director

650 - AP Calculus for Teachers (3) A thorough study of the topics to be presented in AP calculus, including limits of functions, differentiation, integration, infinite series, and applications. (Not intended for degree programs in mathematics.)
Prerequisites: current secondary high school teacher certification in mathematics and a C or better in at least 6 hours of calculus, or consent of the Undergraduate Director

701I — Foundations of Algebra I. (3) An introduction to algebraic structures; group theory including subgroups, quotient groups, homomorphisms, isomorphisms, decomposition; introduction to rings and fields.
Prerequisites: none

702I — Foundations of Algebra II. (3)  Theory of rings including ideals, polynomial rings, and unique factorization domains; structure of finite groups; fields; modules.
Prerequisites: MATH 701-I or equivalent

703I — Foundations of Analysis I. (3) The real numbers and least upper bound axiom; sequences and limits of sequences; infinite series; continuity; differentiation; the Riemann integral.
Prerequisites: MATH 241 or equivalent

704I — Foundations of Analysis II. (3)
Sequences and series of functions; power series, uniform convergence; interchange of limits; limits and continuity in several variables.
Prerequisites: MATH 703-I or equivalent

712I — Probability and Statistics. (3) This course will include a study of permutations and combinations; probability and its application to statistical inferences; elementary descriptive statistics of a sample of measurements; the binomial, Poisson, and normal distributions; correlation and regression.
Prerequisites

736I — Modern Geometry. (3)
Synthetic and analytic projective geometry, homothetic transformations, Euclidean geometry, non-Euclidean geometries, and topology.
Prerequisites: MATH 241 or equivalent

752I — Complex Variables. (3)
Properties of analytic functions, complex integration, calculus of residues, Taylor and Laurent series expansions, conformal mappings.
Prerequisites:  MATH 241 or equivalent

780I — Theory of Numbers. (3)
Elementary properties of integers, Diophantine equations, prime numbers, arithmetic functions, congruences, and the quadratic reciprocity law.
Prerequisites: MATH 241 or equivalent