MATH   001. Elementary Algebra. 0 Hours.

Semester course; 3 lecture or 3 laboratory/tutorial hours. No credit. Prerequisite: permission of the department chair. The purpose of this course is to provide laboratory and tutorial instruction for those seeking remediation or review of high school algebra. Topics include basic properties of real numbers, operations with algebraic expressions, solution of equations and inequalities, exponents and radicals, introduction to functions and graphing.

MATH   121. Perspective Geometry. 1 Hour.

Short course (5 weeks); 3 lecture hours. 1 credit. Students will examine ways in which Renaissance artists who developed linear perspective in geometry in order to paint scenes realistically infuenced the development of mathematics and geometry. Topics covered will include the foundations of projective geometry. Pascal's mystic hexagram, Brianchon"s Theorem and duality. A need for higher mathematics will also be introduced and explained. MATH   121-122-123 fulfills the math requirement for art students. The sequence can be taken in any order.

MATH   122. Tessellations. 1 Hour.

Short course (5 weeks); 3 lecture hours. 1 credit. Students will examine ways in which mathematics is rooted in both natural philosophy and art by examining tiling theory. Course topics include Penrose tilings, symmetries and various other tessellations. MATH   121-122-123 fulfills the math requirement for art students. The sequence can be taken in any order.

MATH   123. Visualization. 1 Hour.

Short course (5 weeks); 3 lecture hours. 1 credit. Students will examine ways in which mathematics has been visualized artistically and will develop their own way to express a mathematical idea. Topics covered will include fractals, knots, minimal surfaces, non-Euclidean geometry and the fourth dimension. MATH   121-122-123 fulfills the math requirement for art students. The sequence can be taken in any order.

MATH   131. Introduction to Contemporary Mathematics. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: satisfactory score on the VCU Mathematics Placement Test within the one-year period immediately preceding the beginning of the course. An exception to this policy is made in the case where the stated alternative prerequisite course has been completed at VCU. Topics include optimization problems, data handling, growth and symmetry, and mathematics with applications in areas of social choice. Major emphasis is on the process of taking a real-world situation, converting the situation to an abstract modeling problem, solving the problem and applying what is learned to the original situation. Does not serve as a prerequisite for MATH   151 or other advanced mathematical sciences courses.

MATH   141. Algebra with Applications. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: one year of high school algebra and satisfactory score on the VCU Mathematics Placement Test within the one-year period immediately preceding the beginning of the course. An exception to this policy is made in the case where the stated alternative prerequisite course has been completed at VCU. Topics include sets, functions, exponents, logarithms, matrix algebra, systems of linear equations, inequalities, binomial theorems, sequences, series, complex numbers and linear programming.

MATH   151. Precalculus Mathematics. 4 Hours.

Semester course; 3 lecture and 1 mathematics laboratory/recitation hours. 4 credits. Prerequisite: MATH   141 or satisfactory score on the VCU Mathematics Placement Test within the one-year period immediately preceding the beginning of the course. An exception to this policy is made in the case where the stated alternative prerequisite course has been completed at VCU. Concepts and applications of algebra and trigonometry. Topics include graphics, transformations and inverses of functions; linear, exponential, logarithmic, power, polynomial, rational and trigonometric functions.

MATH   191. Topics in Mathematics. 1-3 Hours.

Semester course; 1-3 credits. May be repeated for credit. A study of selected topics in mathematics. For a course to meet the general education requirements it must be stated in the Schedule of Classes. See the Schedule of Classes for specific topics to be offered each semester and prerequisites.

MATH   200. Calculus with Analytic Geometry. 4 Hours.

Continuous courses; 4 lecture hours. 4-4 credits. Prerequisite for MATH   200: MATH   151 or satisfactory score on the VCU Mathematics Placement Test within the one-year period immediately preceding the beginning of the course. Prerequisite for MATH   201: completion of MATH   200. Limits, continuity, derivatives, differentials, antiderivatives and definite integrals. Applications of differentiation and integration. Selected topics in analytic geometry. Infinite series.

MATH   201. Calculus with Analytic Geometry. 4 Hours.

Continuous courses; 4 lecture hours. 4-4 credits. Prerequisite for MATH   200: MATH   151 or satisfactory score on the VCU Mathematics Placement Test within the one-year period immediately preceding the beginning of the course. Prerequisite for MATH   201: completion of MATH   200. Limits, continuity, derivatives, differentials, antiderivatives and definite integrals. Applications of differentiation and integration. Selected topics in analytic geometry. Infinite series.

MATH   211. Mathematical Structures. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: calculus-level placement on the VCU Mathematics Placement Test within the one-year period immediately preceding enrollment in the course or MATH   151, MATH   200, MATH   201 or MGMT 212. An alternative prerequisite course may be approved at the discretion of the academic adviser. An introduction to mathematical logic and set theory, including applications in Boolean algebras and graph theory.

MATH   230. Mathematics in Civilization. 3 Hours.

Semester course; 3 lecture hours. 3 credits. For Honors College students only. The growth, development and far-reaching applications of trigonometry, navigation, cartography, logarithms and algebra through ancient, medieval, post-Renaissance and modern times are explored. Will include methods to solve mathematical problems using various historical procedures and will involve collaboration through group projects.

MATH   255. Introduction to Computational Mathematics. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   201. (A core course for mathematics/applied mathematics majors.) An introduction to computer algebra systems (CAS) and their use in mathematical, scientific and engineering investigations/computations. Introductory mathematical computer programming using a CAS, including implementation of problem-specific algorithms.

MATH   291. Topics in Mathematics. 1-3 Hours.

Semester course; 1-3 credits. May be repeated for credit. A study of selected topics in mathematics. See the Schedule of Classes for specific topics to be offered each semester and prerequisites.

MATH   300. Introduction to Mathematical Reasoning. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   201. (A core course for mathematics/applied mathematics majors.) An introduction to basic concepts of mathematical reasoning and the writing of proofs in an elementary setting. Direct, indirect and induction proofs. Illustrations of the concepts include basic proofs from mathematical logic, elementary set theory, elementary number theory, number systems, foundations of calculus, relations, equivalence relations, functions and counting with emphasis on combinatorial proofs.

MATH   301. Differential Equations. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   201. Solutions of ordinary differential equations of first order. Solutions of higher order linear differential equations with constant coefficients and variable coefficients by the methods of undetermined coefficients and variation of parameters, solutions by Laplace transforms and applications.

MATH   302. Numerical Calculus. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   255 (or knowledge of a programming language/mathematical software package) and MATH   201, or permission of the instructor. An introduction to numerical algorithms for solving systems of linear equations, finding zeroes, numerical differentiation and definite integration, optimization.

MATH   303. Investigations in Geometry. 3 Hours.

Semester course; 2 lecture and 3 laboratory hours. 3 credits. Prerequisite: MATH   361. Restricted to students majoring in the liberal studies for early and elementary education in the Bachelor of Interdisciplinary Studies program. A study of topics in Euclidean geometry to include congruence, similarity, measurement, coordinate geometry, symmetry and transformation in both two and three dimensions. These topics will be investigated using manipulatives and computer software.

MATH   305. Elementary Number Theory. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   300. Divisibility, congruences, Euler phi-function, Fermat's Theorem, primitive roots, Diophantine equations.

MATH   307. Multivariate Calculus. 4 Hours.

Semester course; 4 lecture hours. 4 credits. Prerequisite: MATH   201. The calculus of vector-valued functions and of functions of more than one variable. Partial derivatives, multiple integrals, line integrals, surface integrals and curvilinear coordinates. Lagrange multipliers; theorems of Green, Gauss and Stokes. Applications.

MATH   310. Linear Algebra. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   201. Systems of linear equations, vector spaces, linear dependence, bases, dimensions, linear mappings, matrices, determinants, quadratic forms, orthogonal reduction to diagonal form, eigenvalues and geometric applications.

MATH   350. Introductory Combinatorics. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   201 with a minimum grade of C. An introduction to basic combinatorial concepts such as combinations, permutations, binomial coefficients, Fibonacci numbers and Pascal’s triangle; basic theorems such as the pigeonhole principle and Newton’s binomial theorem; algorithms such as bubble sort and quicksort; and discussion of basic applications such as chessboard problems, combinatorial games, magic squares and Latin squares.

MATH   351. Applied Abstract Algebra. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   300. A survey of several areas in applied abstract algebra which have applications in computer science such as groups, codes, matrix algebra, finite fields and advanced graph theory.

MATH   353. Experimental Mathematics. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   201 with a minimum grade of C. An introduction to a mathematical computing package, computer manipulation of lists and sets, and symbolic computing. Numerical computation will be used to investigate mathematical objects, such as integers, prime numbers, graphs, matrices and to identify properties and patterns among these objects. Random methods will be used to explore properties and patterns in long sequences and large collections.

MATH   356. Graphs and Algorithms. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   201 with a minimum grade of C. An introduction to basic graph theoretic concepts such as trees, colorings and matchings; basic theorems such as the handshaking lemma and the Gallai identities; algorithms such as Dijkstra’s and Kruskal’s; and discussion of famous open problems such as finding shortest tours for a traveling salesman.

MATH   361. Numbers and Operations. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: TEDU   101 and either MATH   131 or satisfactory score on the VCU Mathematics Placement Test within the one-year period immediately preceding the beginning of the course. Ways of representing numbers, relationships between numbers, number systems, the meanings of operations and how they relate to one another, and computation within the number systems as a foundation for algebra. Structured observations and tutoring of elementary-level students. Restricted to students majoring in the liberal studies concentration for early and elementary education in the Bachelor of Interdisciplinary Studies program.

MATH   362. Algebra and Functions. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   361. Topics include algebraic concepts, linear, quadratic, exponential, logarithmic, trigonometric functions including graphical modeling of physical phenomena. Attention will be given to the use of graphing technology, the transition from arithmetic to algebra, working with quantitative change, and the description and prediction of change. Structured observations and tutoring of elementary-level students. Restricted to B.I.S. students in the liberal studies for early and elementary education concentration.

MATH   380. Introduction to Mathematical Biology. 4 Hours.

Semester course; 3 lecture and 2 laboratory hours. 4 credits. Prerequisites: MATH   200 and BIOL   151, or permission of instructor. An introduction to mathematical biology. Various mathematical modeling tools will be covered and implemented in a range of biological areas. Additionally, the collaborative research process will be presented and discussed. Crosslisted as: BNFO   380/BIOL   380.

MATH   391. Topics in Mathematics. 1-3 Hours.

Semester course; 1-3 credits. May be repeated for credit. A study of selected topics in mathematics. See the Schedule of Classes for specific topics to be offered each semester and prerequisites.

MATH   401. Introduction to Abstract Algebra. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300 and MATH   310, each with a minimum grade of C. An introduction to groups, rings and fields from an axiomatic point of view. Coset decomposition and basic morphisms.

MATH   404. Algebraic Structures and Functions. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300 and MATH   310, each with a minimum grade of C; one additional mathematical sciences course; and permission of instructor. Semigroups, groups, rings, integral domains and fields. Exponential, logarithmic and trigonometric functions. Graphing in parametric and polar coordinates. Arithmetic and geometric sequences and series.

MATH   407. Advanced Calculus. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   300. Theoretical aspects of calculus. Topics include properties of real numbers, countable and uncountable sets, sequences and series, limits, continuity, derivatives, and Riemann integration.

MATH   409. General Topology. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   407 with a minimum grade of C. Foundations and fundamental concepts of point-set topology. Topological spaces, continuity, convergence, connected sets, compactness, product spaces, quotient spaces, function spaces, separation properties.

MATH   415. Numerical Methods. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   255, MATH   301 and MATH   310, each with a minimum grade of C. Numerical methods for interpolation, solving systems of linear equations and initial value problems (ordinary differential equations) and the exploration of computational error.

MATH   427. Excursions in Analysis: Real. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   307, MATH   310 and MATH   407. May be repeated once for credit with a different emphasis and permission of the instructor. Intensive study of ideas and applications from real analysis.

MATH   428. Excursions in Analysis: Complex. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   307, MATH   310 and MATH   407. May be repeated once for credit with a different emphasis and permission of the instructor. Intensive study of ideas and applications from complex analysis.

MATH   429. Excursions in Analysis: Applied. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   301, MATH   307, MATH   310 and MATH   407. May be repeated once for credit with a different emphasis and permission of the instructor. Intensive study of ideas and applications from applied analysis.

MATH   430. The History of Mathematics. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300, MATH   307, MATH   310, and either MATH   301 or OPER   327, all with a minimum grade of C. Surveys major trends in the development of mathematics from ancient times through the 19th century and considers the cultural and social contexts of mathematical activity.

MATH   431. Expositions in Modern Mathematics. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300, MATH   307, MATH   310, and either MATH   301 or OPER   327, all with a minimum grade of C. Descriptively studies several major ideas relevant to present-day mathematics, such as the advent of pure abstraction, difficulties in the logical foundations of mathematics, the impact of mathematics and statistics in the 20th century and the computer revolution.

MATH   432. Ordinary Differential Equations. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300, MATH   301, MATH   307 and MATH   310, each with a minimum grade of C. Existence and uniqueness of solutions, linearization and stability analysis, Lyapunov stability theory, periodic solutions, and bifurcations. Applications and simulations are emphasized.

MATH   433. Partial Differential Equations. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300, MATH   301, MATH   307 and MATH   310, each with a minimum grade of C. Parabolic (heat), hyperbolic (wave) and elliptic (steady-state) partial differential equations are studied. Solution techniques such as separation of variables, reflection methods, integral transform methods and numerical methods are demonstrated. Practical problems and applications are emphasized.

MATH   434. Discrete Dynamical Systems. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300, MATH   301, MATH   307 and MATH   310, each with a minimum grade of C. Theory and applications of difference equations including existence and uniqueness of solutions, linearization and stability, periodic solutions, and bifurcations.

MATH   454. Using Technology in the Teaching of Mathematics. 3 Hours.

Semester course; 2 lecture and 2 laboratory hours. 3 credits. Prerequisites: MATH   200 and STAT   212, each with a minimum grade of C; six additional credits in the mathematical sciences; and permission of the instructor. Using graphing calculators, calculator-based labs and computer software packages in teaching topics in algebra, geometry, trigonometry, statistics, finance and calculus.

MATH   480. Methods of Applied Mathematics for the Life Sciences: Discrete. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   301, MATH   307, MATH   310 and MATH   380, each with a minimum grade of C. Focuses on the use of discrete dynamical system models to describe phenomena in biology and medicine. Students will explore the theoretical mathematics necessary to analyze these models. Computational solutions to these models will be developed and implemented to validate the models and to further explore the biological phenomena.

MATH   481. Methods of Applied Mathematics for the Life Sciences: ODE. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   301, MATH   307, MATH   310 and MATH   380, each with a minimum grade of C. Focuses on the use of ordinary differential equation models to describe phenomena in biology and medicine. Students will explore the theoretical mathematics necessary to analyze these models. Computational solutions to these models will be developed and implemented to validate the models and to further explore the biological phenomena.

MATH   482. Methods of Applied Mathematics for the Life Sciences: PDE. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   301, MATH   307, MATH   310 and MATH   380, each with a minimum grade of C. Focuses on the use of partial differential equation models to describe phenomena in biology and medicine. Students will explore the theoretical mathematics necessary to analyze these models. Computational solutions to these models will be developed and implemented to validate the model and to further explore the biological phenomena.

MATH   490. Mathematical Expositions. 3 Hours.

Semester course; 2 lecture hours. 2 credits. Prerequisites: UNIV   200 or HONR   200. Restricted to seniors in mathematical sciences with at least 85 credit hours taken toward the degree. Required for all majors in the Department of Mathematics and Applied Mathematics. A senior capstone course in the major designed to help students attain proficiency in expository mathematical writing and oral presentation, which require the efficient and effective use of mathematics and the English language. Students will learn a variety of topics in mathematics, write reviews of selected award-winning mathematics papers and write a senior paper.

MATH   492. Independent Study. 1-4 Hours.

Semester course; variable hours. 1-4 credits. Maximum 4 credits per semester; maximum total of 6 credits. Generally open only to students of junior or senior standing who have acquired at least 12 credits in the departmental discipline. Determination of the amount of credit and permission of instructor and department chair must be procured prior to registration for the course. The student must submit a proposal for investigating some area or problem not contained in the regular curriculum. The results of the student's study will be presented in a report.

MATH   493. Mathematical Sciences Internship. 3 Hours.

Semester course; the equivalent of at least 15 work hours per week for a 15-week semester. 3 credits. Mathematical sciences majors only with junior or senior standing. Admission by permission from the department chair. Through placement in a position in business, industry, government or the university, the student will serve as an intern in order to obtain a broader knowledge of the mathematical sciences and their applications.

MATH   501. Introduction to Abstract Algebra. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300 and MATH   310, or their equivalents. An introduction to groups, rings and fields from an axiomatic point of view. Coset decomposition and basic morphisms.

MATH   504. Algebraic Structures and Functions. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   200-201, MATH   300 and one additional mathematical science course and permission of instructor. Semigroups, groups, rings, integral domains and fields. Exponential, logarithmic and trigonometric functions. Graphing in parametric and polar coordinates. Arithmetic and geometric sequences and series. Not applicable toward M.S. in Mathematical Sciences.

MATH   505. Modern Geometry. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300, and MATH   307 or MATH   310. Topics in Euclidean, projective and non-Euclidean geometries from a modern viewpoint.

MATH   507. Bridge to Modern Analysis. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Enrollment restricted to student with graduate standing. Metric spaces, normed vector spaces, inner-product spaces and orthogonality, sequences and series of functions, convergence, compactness, completeness, continuity, contraction mapping theorem, and inverse and implicit function theorems.

MATH   508. Analysis II. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   307, 310 and MATH   507. Theoretical aspects of calculus, sequences, limits and continuity in higher dimensions, infinite series, series of functions, integration, differential geometry.

MATH   509. General Topology I-II. 3 Hours.

Continuous courses; 3 lecture hours. 3-3 credits. Prerequisites: MATH   300 and MATH   307. Foundations and fundamental concepts of point-set topology. Topological spaces, convergence, connected sets, compactness, product spaces, quotient spaces, function spaces, separation properties, metrization theorems, mappings and compactifications.

MATH   510. General Topology I-II. 3 Hours.

Continuous courses; 3 lecture hours. 3-3 credits. Prerequisites: MATH   300 and MATH   307. Foundations and fundamental concepts of point-set topology. Topological spaces, convergence, connected sets, compactness, product spaces, quotient spaces, function spaces, separation properties, metrization theorems, mappings and compactifications.

MATH   511. Applied Linear Algebra. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   310. The algebra of matrices, the theory of finite dimensional vector spaces and the basic results concerning eigenvectors and eigenvalues, with particular attention to applications.

MATH   512. Complex Analysis for Applications. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   307, and MATH   300 or knowledge equivalent to MATH   300. The algebra and geometry of complex numbers, analytic functions, integration, series, contour integration, analytic continuation, conformal mapping, with particular attention to applications.

MATH   515. Numerical Analysis. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Enrollment restricted to student with graduate standing. Knowledge of a programming language or mathematical software package recommended. Theoretical derivation and implementation of numerical methods. Topics to include direct methods, data fitting, differentiation, integration and solutions to ordinary differential equations.

MATH   516. Numerical Analysis II. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   255 and 301. Numerical solution of initial value problems in ordinary differential equations, two-point boundary value problems. Introduction to numerical techniques for solving partial differential equations. Selected algorithms may be programmed for solution on computers.

MATH   517. Methods of Applied Mathematics. 3 Hours.

Continuous courses; 3 lecture hours. 3-3 credits. Prerequisites: MATH   301, MATH   307 and MATH   300 or knowledge equivalent to MATH   300. Vector analysis, matrices, complex analysis, special functions, Legendre and Hermite polynomials. Fourier series, Laplace transforms, integral equations, partial differential equations, boundary-value and initial-value problems.

MATH   518. Methods of Applied Mathematics. 3 Hours.

Continuous courses; 3 lecture hours. 3-3 credits. Prerequisites: MATH   301, MATH   307 and MATH   300 or knowledge equivalent to MATH   300. Vector analysis, matrices, complex analysis, special functions, Legendre and Hermite polynomials. Fourier series, Laplace transforms, integral equations, partial differential equations, boundary-value and initial-value problems.

MATH   520. Game Theory and Linear Programming. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   310. The mathematical basis of game theory and linear programming. Matrix games, linear inequalities and convexity, the mini-max theorems in linear programming, computational methods and applications. Crosslisted as: OPER   520.

MATH   521. Introduction to Algebraic Number Theory. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   501. Introduction to algebraic numbers and algebraic number fields with emphasis on quadratic and cyclotomic fields. Units, primes, unique factorization.

MATH   525. Introduction to Combinatorial Mathematics. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   211 or 300, or permission of instructor. Topics include basic counting, binomial theorems, combinations and permutations, recurrence relations, generating functions, and basic graph theory with emphasis to applications.

MATH   530. The History of Mathematics. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: 17 credits at the 200 level or above in mathematical sciences or permission of instructor. Surveys major trends in the development of mathematics from ancient times through the 19th century and considers the cultural and social contexts of mathematical activity. Either MATH   530 or MATH   531 (but not both) may be applied to the M.S. in Mathematical Sciences or Computer Science. Both MATH   530 and MATH   531 may be applied to the M.Ed. in Curriculum and Instruction with a concentration in secondary education/mathematics.

MATH   531. Expositions in Modern Mathematics. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: 6 credits at the 400 level or above in mathematical sciences. Studies descriptively several major ideas relevant to present-day mathematics, such as the advent of pure abstraction, difficulties in the logical foundations of mathematics, the impact of mathematics and statistics in the 20th century, and the computer revolution. Either MATH   530 or MATH   531 (but not both) may be applied to the M.S. in Mathematical Sciences or Computer Science. Both MATH   530 and MATH   531 may be applied to the M.Ed. in Curriculum and Instruction with a concentration in secondary education/mathematics.

MATH   532. Ordinary Differential Equations I. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300, 301, 307 and 310. An introduction to the theory of ordinary differential equations; existence, uniqueness and extension of solutions; stability and linearization; Lyapunov stability theory; invariance theorem; applications.

MATH   533. Partial Differential Equations I. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300, 301, 307 and 310, or permission of instructor. Parabolic (heat), hyperbolic (wave) and elliptic (steady-state) partial differential equations are studied. Solution techniques such as separation of variables, reflection methods, integral transform methods and numerical methods are demonstrated. Practical problems and applications are emphasized.

MATH   534. Applied Discrete Dynamical Systems. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   300, MATH   301, MATH   307 and MATH   310. Theory and applications of difference equations, graphs, networks, agent-based models and Markov processes. Methods of analysis and simulations will be discussed.

MATH   535. Introduction to Dynamical Systems. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Enrollment restricted to students with graduate standing. Theoretical and computational introduction to continuous and discrete dynamical systems with applications. Topics include existence and uniqueness of solutions, stability and bifurcations.

MATH   554. Using Technology in the Teaching of Mathematics. 3 Hours.

Semester course; 2 lecture and 2 laboratory hours. 3 credits. Prerequisites: MATH   200 and STAT   212 and six additional credits of mathematical science courses and permission of the instructor. Using graphing calculators, CBLs (calculator based labs) and computer software packages in teaching topics in algebra, geometry, trigonometry, statistics, finance and calculus. Not applicable toward M.S. in Mathematical Sciences.

MATH   555. Dynamics and Multivariable Control I. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   301 and 310 or the equivalent. Systems of differential equations with controls, linear control systems, controllability, observability, introduction to feedback control and stabilization. Crosslisted as: ENGR 555.

MATH   556. Fundamentals of Graph Theory I. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   310 and MATH   300 or MATH   211, or permission of instructor. Introduction to graph classes, graph invariants, graph algorithms, graph theoretic proof techniques and applications.

MATH   580. Methods of Applied Mathematics for the Life Sciences: Discrete. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   301, 307, 310 and 380. This course will focus on the use of discrete dynamical system models to describe phenomena in biology and medicine. Students will explore the theoretical mathematics necessary to analyze these models. Computational solutions to these models will be developed and implemented to validate the models and to further explore the biological phenomena.

MATH   581. Methods of Applied Mathematics for the Life Sciences: ODE. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   301, 307, 310 and 380. This course will focus on the use of ordinary differential equation models to describe phenomena in biology and medicine. Students will explore the theoretical mathematics necessary to analyze these models. Computational solutions to these models will be developed and implemented to validate the models and to further explore the biological phenomena.

MATH   582. Methods of Applied Mathematics for the Life Sciences: PDE. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   301, 307, 310 and 380. This course will focus on the use of partial differential equation models to describe phenomena in biology and medicine. Students will explore the theoretical mathematics necessary to analyze these models. Computational solutions to these models will be developed and implemented to validate the model and to further explore the biological phenomena.

MATH   585. Biomathematics Seminar:____. 1 Hour.

Semester course; 2 lecture hours. 1 credit. Prerequisite: MATH   301 or permission of instructor. May be repeated with different thematic content. Opportunity for students to develop their understanding of the connection between mathematics and the areas of biology and medicine. Activities include reading of classical and contemporary research literature, attending seminar talks and class discussions.

MATH   591. Topics in Mathematics. 1-3 Hours.

Semester course; 1-3 credits. May be repeated for credit with different topics. Prerequisite: permission of the instructor. Open to qualified undergraduates. A study of selected topics in mathematical sciences. See the Schedule of Classes for specific topics to be offered each semester and prerequisites.

MATH   593. Internship in Mathematical Sciences. 3,6 Hours.

Semester course; variable hours. 1-6 credits. May be repeated for credit. Student participation in a planned educational experience under the supervision of a mathematical sciences faculty member. The internship may include supervised teaching, statistical consulting or participation in theoretical or applied research projects. A grade of P may be assigned students in this course. May be applied toward the degree in mathematical sciences only with the permission of the graduate affairs committee.

MATH   601. Abstract Algebra I. 3 Hours.

Continuous course; 3 lecture hours. 3 credits. Prerequisite: MATH   501. A study of algebraic structures (including groups, rings and fields), Galois theory, homomorphisms, subalgebras, direct products, direct decompositions, subdirect decompositions, free algebras, varieties of algebras.

MATH   602. Abstract Algebra II. 3 Hours.

Continuous course; 3 lecture hours. 3 credits. Prerequisite: MATH   602. A study of algebraic structures (including groups, rings and fields), Galois theory, homomorphisms, subalgebras, direct products, direct decompositions, subdirect decompositions, free algebras, varieties of algebras.

MATH   603. Advanced Probability Theory. 3 Hours.

Continuous courses; 3 lecture hours. 3-3 credits. Prerequisites: MATH   507, and STAT 503 or BIOS/STAT   513. Completion of MATH   603 to enroll in 604. A measure-theoretic approach to the theory of probability. Borel sets, probability measures and random variables. Special topics include characteristic functions, modes of convergence and elements of stochastic processes.

MATH   604. Advanced Probability Theory. 3 Hours.

Continuous courses; 3 lecture hours. 3-3 credits. Prerequisites: MATH   507, and STAT 503 or BIOS/STAT   513. Completion of MATH   603 to enroll in 604. A measure-theoretic approach to the theory of probability. Borel sets, probability measures and random variables. Special topics include characteristic functions, modes of convergence and elements of stochastic processes.

MATH   607. Measure and Integration Theory. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: Math 507. Measurable sets and functions, sets of measure zero, Borel sets, Lebesgue measure and integral, fundamental convergence theorems, Lp spaces, and foundations of probability theory.

MATH   608. Real Analysis II. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequiste: MATH   607. Lebesgue integral, integration of positive as well as complex functions, the monotone and dominated convergence theorems, L<sup>p</sup>-spaces, duality, bounded linear functionals on the L<sup>p</sup>, the Radon-Nikodym theorem and the Riesz representation theorem.

MATH   610. Advanced Linear Algebra. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Vector spaces, bases and dimension, change of basis. Linear transformations, linear functionals. Simultaneous triangularization and diagonalization. Rational and Jordan canonical forms.

MATH   615. Numerical Analysis. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   515 or MATH   516. Theoretical development of solutions to large linear and nonlinear systems by iterative methods with consideration given to optimal implementation.

MATH   620. Theory of Partial Differential Equations. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   301 and MATH   508. Classification of partial differential equations; elliptic, hyperbolic and parabolic equation; potential theory, techniques of solving various partial differential equations; application to electromagnetism and solid mechanics.

MATH   632. Ordinary Differential Equations II. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   507 and MATH   532. Linear systems theory; existence, uniqueness and continuous dependence for nonlinear systems; invariant manifolds; stable manifold theorem; Hartman-Grobman theorem; Lyapunov stability theory; Hamiltonian and gradient systems.

MATH   633. Asymptotic and Perturbation Methods. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   532. Asymptotic solution of algebraic and transcendental equations, Taylor's remainder estimate, regular perturbation expansions, two-point boundary value problems, boundary layers and matched asymptotic expansions, Poincare-Lindstedt technique, method of multiple scales, asymptotic approximation of integrals (Laplace, WKB and stationary phase methods).

MATH   634. Partial Differential Equations. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   532 and 608. Classification of partial differential equations, initial and boundary value problems, well-posedness; first-order equations and methods of characteristics; wave equation in several dimensions; heat equation, transform methods, maximum principle, energy methods; Laplace's equation, Dirichlet problem for a disc; survey of nonlinear equations.

MATH   640. Mathematical Biology I. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   532. Mathematical modeling in the biological and medical sciences. Topics will include continuous and discrete dynamical systems describing interacting and structured populations, resource management, biological control, reaction kinetics, biological oscillators and switches, and the dynamics of infectious diseases.

MATH   655. Dynamics and Multivariable Control II. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   555 and MATH   507 recommended, or permission of instructor. Control problems for nonlinear systems of ordinary differential equations, methods of feedback control to achieve control objectives. Crosslisted as: ENGR 655.

MATH   661. Number and Operations. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Ways of representing numbers, relationships between numbers, number systems, the meanings of operations and how they relate to one another, and computation within the number system as a foundation for algebra; episodes in history and development of the number system; and examination of the developmental sequence and learning trajectory as children learn number concepts. A core course for preparation as a K-8 mathematics specialist. Not applicable to M.S. in Mathematical Sciences.

MATH   662. Geometry and Measurement. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Explorations of the foundations of informal measurement and geometry in one, two and three dimensions. The van Hiele model for geometric learning is used as a framework for how children build their understanding of length, area, volume, angles and geometric relationships. Visualization, spatial reasoning and geometric modeling are stressed. As appropriate, transformational geometry, congruence, similarity and geometric constructions will be discussed. A core course of preparation as a K-8 mathematics specialist. Not applicable to M.S. in Mathematical Sciences.

MATH   663. Functions and Algebra. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Examination of representation and analysis of mathematical situations and structures using generalization and algebraic symbols and reasoning. Attention will be given to the transition from arithmetic to algebra, working with quantitative change, and the description of and prediction of change. A core course for preparation as a K-8 mathematics specialist. Not applicable to M.S. in Mathematical Sciences.

MATH   664. Statistics and Probability. 3 Hours.

Semester course; 3 lecture hours. 3 credits. An introduction to probability, descriptive statistics and data analysis; exploration of randomness, data representation and modeling. Descriptive statistics will include measures of central tendency, dispersion, distributions and regression. Analysis of experiments requiring hypothesizing, experimental design and data gathering. A core course for preparation as a K-8 mathematics specialist. Not applicable to M.S. in Mathematical Sciences.

MATH   665. Rational Numbers and Proportional Reasoning. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Basic number strands in fractions and rational numbers, decimals and percents; ratios and proportions in the school curriculum. Interpretations, computations and estimation with a corrdinated program of activities that develop both rational number concepts and skills and proportional reasoning. A core course for preparation as a K-8 mathematics specialist. Not applicable to M.S. in Mathematical Sciences.

MATH   667. Functions and Algebra II. 3 Hours.

Semester course; 3 lecture hours, 3 credits. Prerequisite: Math 663 or equivalent. Examination of the K-8 strands related to algebra. A study of linear, exponential and quadratic functions. Use of number lines, coordinate axes, tables, graphing calculators and manipulatives to understand core algebraic ideas and real-world contexts. Course provides preparation for K-8 mathematics specialists. Not applicable to M.S. in Mathematical Sciences.

MATH   690. Research Seminar. 1 Hour.

Semester course; 1 credit. May be repeated for credit. Prerequisite: graduate standing. Discussion of topics in the mathematical sciences as stimulated by independent reading in selected areas and at least one oral presentation by each student.

MATH   691. Special Topics in Mathematics. 1-3 Hours.

Semester course; 1-3 lecture hours. 1-3 credits. May be repeated for credit. Prerequisite: permission of instructor. A detailed study of selected topics in mathematics. Possible topics include commutative rings and algebras, topological groups, special functions, Fourier analysis, abstract harmonic analysis, operator theory, functional analysis, differential geometry, Banach algebras and control theory.

MATH   697. Directed Research. 1-3 Hours.

Semester course; variable hours. 1-3 credits per semester. May be repeated for credit. Prerequisite: graduate standing. Supervised individual research and study in an area not covered in the present curriculum or in one which significantly extends present coverage. Research culminates with an oral presentation and submission of a written version of this presentation to the supervising faculty member.

MATH   698. Thesis. 1-3 Hours.

Hours to be arranged. 1-3 credits. A total of 3 or 6 credits may be applied to the M.S. in Mathematical Sciences/Applied Mathematics or to the M.S. in Mathematical Sciences/Mathematics. May be repeated for credit. Prerequisite: graduate standing. Independent research culminating in the writing of the required thesis as described in this bulletin. Grade of S/U/F may be assigned in this course.

MATH   707. Functional Analysis I. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   607. Banach and Hilbert spaces, bounded linear maps, Hahn-Banach theorem, open mapping theorem, dual spaces, weak topologies, Banach-Alaoglu theorem, reflexive spaces, compact operators, spectral theory in Hilbert spaces.

MATH   711. Complex Analysis I. 3 Hours.

Continuous course; 3 lecture hours. 3 credits. Prerequisite: MATH   508, 512 or permission of instructor. Complex derivative, analyticity, Cauchy's theorem and integral formula, Taylor and Laurent series, poles, residues, analytic continuation, Riemann surfaces, periodic functions, conformal mapping, meromorphic functions and applications.

MATH   712. Complex Analysis II. 3 Hours.

Continuous course; 3 lecture hours. 3 credits. Prerequisite: MATH   711. Complex derivative, analyticity, Cauchy's theorem and integral formula, Taylor and Laurent series, poles, residues, analytic continuation, Riemann surfaces, periodic functions, conformal mapping, meromorphic functions and applications.

MATH   715. Numerical Solutions for Differential Equations. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   533 and either MATH   515 or 516. Students will use the finite difference method and the finite element method to solve ordinary and partial differential equations. Course will explore the theoretical underpinnings of the techniques and implement the methods to solve a variety of equations.

MATH   719. Operational Methods. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   508 or permission of instructor. Transform methods applied to existence theory, explicit solutions to problems of mathematical physics, Schrodinger operators, distributions of Schwartz and Gelfand-Silov, locally complex spaces, duality, kernel theorems of Schwartz, symmetries and the mathematical framework of quantum field theory.

MATH   721. Boundary Value Problems. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   517-518 or permission of instructor. Survey of boundary value problems, approximate analytic solutions such as Galerkin methods of approximating solutions of elliptic boundary value problems in one and several dimensions and the Ritz method; application to heat transfer, fluid mechanics and potential theory. Initial boundary-value problems on nonlinear solid and fluid thermomechanics.

MATH   732. Ordinary Differential Equations III. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   632. Center manifold theory; normal form theory; oscillations in nonlinear systems; local bifurcation theory of equilibria and periodic orbits.

MATH   740. Mathematical Biology II. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH 637 and 640. Mathematical models of spatial processes in biology including pattern formation, applications of traveling waves to population dynamics, epidemiology and chemical reactions, and models for neural patterns will be examined.

MATH   750. Combinatorics I-II. 3 Hours.

Continuous courses; 3 lecture hours. 3-3 credits. Prerequisites: MATH   525 and permission of the instructor. A two-semester advanced introduction to combinatorial theory. In the first course, basic counting techniques and some classical results will be discussed. Topics for 750 include pigeonhole principle, exclusion-inclusion principle, unimodality of binomial coefficients, the multinomial theorem, Newton�s binomial theorem, recurrence relations, generating functions, special counting sequences, Ramsey theory, and combinatorial designs and codes. The second part focuses on tools from probability and linear algebra, optimization problems in combinatorics and applications to other fields. Topics for 751 include probabilistic methods, linear algebra methods, extremal problems, partially ordered sets and symmetric functions. Other topics may vary depending on the interest of the students and the instructor.

MATH   751. Combinatorics I-II. 3 Hours.

Continuous courses; 3 lecture hours. 3-3 credits. Prerequisites: MATH   525 and permission of the instructor. A two-semester advanced introduction to combinatorial theory. In the first course, basic counting techniques and some classical results will be discussed. Topics for 750 include pigeonhole principle, exclusion-inclusion principle, unimodality of binomial coefficients, the multinomial theorem, Newton�s binomial theorem, recurrence relations, generating functions, special counting sequences, Ramsey theory, and combinatorial designs and codes. The second part focuses on tools from probability and linear algebra, optimization problems in combinatorics and applications to other fields. Topics for 751 include probabilistic methods, linear algebra methods, extremal problems, partially ordered sets and symmetric functions. Other topics may vary depending on the interest of the students and the instructor.

MATH   756. Graph Theory I. 3 Hours.

Continuous course; 3 lecture hours. 3 credits. Prerequisite: MATH   525 or permission of the instructor. The first course lays a rigorous theoretical foundation for further advanced study in graph theory. Topics include trees, bipartiteness, connectivity, metric properties, matching, planarity, coloring and Hamiltonian cycles. The second course builds on the first but explores more specialized areas. Topics include extremal graph theory, infinite graphs and minors. Other topics may vary depending on the interest of the instructor or students.

MATH   757. Graph Theory II. 3 Hours.

Continuous course; 3 lecture hours. 3 credits. Prerequisite: MATH   756. The first course lays a rigorous theoretical foundation for further advanced study in graph theory. Topics include trees, bipartiteness, connectivity, metric properties, matching, planarity, coloring and Hamiltonian cycles. The second course builds on the first but explores more specialized areas. Topics include extremal graph theory, infinite graphs and minors. Other topics may vary depending on the interest of the instructor or students.

MATH   759. Graph Enumeration. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisites: MATH   750 and 756 or approval of instructor. Enumeration of labeled graphs, unlabeled graphs and digraphs, and other categories of graph and digraph structures (such as graph imbedding). Polya's theorem of enumeration, the power group method, the superposition method, Redfield's enumeration theorems and recent developments in graph enumeration.

MATH   769. Special Topics in Mathematical Life Sciences. 3 Hours.

Semester course; 3 lecture hours. 3 credits. May be repeated with different topics for credit. A detailed study of selected topics in mathematical life sciences. Possible topics include mathematical ecology, mathematical physiology, biofluids, neural networks, cardio-electrophysiology and other topics in the mathematical life sciences.

MATH   770. Fourier Analysis. 3 Hours.

Semester course; 3 lecture hours. 3 credits. Prerequisite: MATH   608. The Fourier transform on the circle and line, convergence of Fejer means; Parseval's relation and the square summable theory, convergence and divergence at a point; conjugate Fourier series, the conjugate function and the Hilbert transform, the Hardy-Littlewood maximal operator, Hardy spaces and wavelets.

MATH   787. Special Topics in Discrete Mathematics. 3 Hours.

Semester course; 3 lecture hours. 3 credits. May be repeated with different topics for credit. A detailed study of selected topics in discrete mathematics. Possible topics include algebraic graph theory, algorithmic graph theory, coding theory, cryptography, combinatorial designs, combinatorial topology, graph drawing, graph homomorphism, graph products, topological graph theory, WZ algorithms and other topics in discrete mathematics.