Mathematical Sciences (MASC)

The departments of computer science, mathematics, and statistics have joined together to offer the following introductory, interdisciplinary courses in mathematical sciences:

1024: MATHEMATICS, A LIBERAL ARTS APPROACH
This is the first course in a sequence that is intended to give those students who will not make extensive use of the Mathematical Sciences in their specialties some insight into Mathematics, Computer Science, and Statistics in an integrated setting. Topics include set theory, number theory, and modular arithmetic. (3H,3C)

1044: COMPUTER SCIENCE, A LIBERAL ARTS APPROACH
Intended to provide those students who will not make extensive use of the mathematical sciences in their specialties some insight into the concepts of computer science. Topics include introduction to computer architecture, operating systems, programming languages, and algorithms; history of computing; computer applications in the modern world. Prior credit for any of the following precludes credit for 1044: CS 1104, 1704, or any computer science course at the 2000 level or higher. (3H,3C)


College of Science

Mathematics

www.math.vt.edu/

Peter Haskell, Chair
Associate Chair for Undergraduate Students, R. C. Rogers
Director for Undergraduate Programs: L. Zietsman
Graduate Director: S. Sun
John K. Costain Faculty Chair and Professor:T. Warburton
Hatcher Professor of Mathematics: J. A. Burns
Virginia Tech Class of 1950 Mathematics Professors: M. Renardy and Y. Renardy
Alumni Distinguished Professor: E. Brown
Professors: S. Adjerid; J. A. Ball; C. A. Beattie; J. Borggaard; M. V. Day; E. de Sturler; M. Embree ;  W. J. Floyd; S. Gugercin; P. E. Haskell; T. L. Herdman; T. Iliescu; J. U. Kim; M. Klaus; W. E. Kohler; T. Lin; P. A. Linnell; C. M. Reidys; R. C. Rogers; J. F. Rossi; D. L. Russell; M. Shimozono; S. Sun; J. Turner
Associate Professors: S. Ciupe; A. Elgart; N. Loehr; C. Mihalcea; H. Mortveit;  A. Norton; P. Wapperom; P. Yue; L.  Zietsman
Assistant Professors: J. Chung; M. Chung; N. Glatt-Holtz; E. Johnson; H. Liu; D. Orr;  M. Wawro
Patricia Ann Caldwell Post-Doctoral Fellow and Visiting Assistant Professor: J. Fillman
Senior Instructors: D. Agud; S. Anderson; T. A. Bourdon; S. Hagen; C. Stephens
Advanced Instructors: H. Hart; J. Hurdus; M. P. McQuain; E. Savel'ev; J. Schmale; D. B. Smith
Instructors: R. Arnold; T.  Asfaw; M. Chung; J. Clemons; G. Dillon; N. Gildersleeve; L. L. Hanks; M. Heitzman; E. Jasso Hernandez; S. McIntyre; S. Miller; B. Ordonez-Delgado; E. Rappold; N. Robbins; E. Saenz Maldonado; E. Ufferman; J. Wilson; J. Wong
Lecturers: E. Adkins; A. Sibol
Career Advisor: S. Ciupe
Scholarship Chair: J. Kim
Math Emporium

Overview

    Mathematics is essential to a clear and complete understanding of virtually all phenomena. Its precision, depth, and generality support the development of critical thinking and problem-solving skills. The study of mathematics provides the ability to describe applied problems quantitatively and to analyze these problems in a precise and logical manner. This is a principal reason behind the strong demand for mathematicians in government and industry. Essentially all complex problems, whether physical, social, or economic, are solved by designing a mathematical model, analyzing the model, and determining computational algorithms for an efficient and accurate approximation of a solution. Each of these phases is mathematical in nature. For example, if a problem deviates from a standard form, a mathematician should be able to adjust the usual mathematical treatment of the problem to accommodate the deviation. In this case mathematical training provides a practical preparation for a career in today's changing world. Moreover, it is especially valuable because it is an education that equips one to continue to adapt to new situations.

    Mathematicians typically are employed as applied mathematicians in their specialty areas. Our recent mathematics graduates have been approximately equally divided among government and industry, graduate school, and teaching. There are four different paths or options that a student may follow towards a B.S. in Mathematics: 1) the Traditional Option; 2) the Applied Computational Mathematics Option (ACM); 3) the Applied Discrete Mathematics Option (ADM); and 4) the Mathematics Education Option (MSTR).

    The Traditional Option, as its name implies, yields a broad and flexible background in mathematics. The other three options are more specialized. The ACM option is designed for students primarily interested in computational mathematics and its applications to engineering and the natural and social sciences. The ADM option is designed for students primarily interested in areas of applied mathematics closely associated with computer science. The Mathematics Education Option is designed for students who want to be certified to teach secondary mathematics.

    Often students will begin their studies in the Traditional Option and later change to one of the other three options when they become more sure of the path they wish to pursue. One, however, can acquire many aspects of the three specialized options within the Traditional Option, because it also requires some degree of specialization in an applications area and provides career development features. The three specialized options are each less general, but bring particular career paths into sharper focus. Each of the four options provides an excellent foundation for graduate study, either in mathematics or in an applications area. Handbooks for each of the options, as well as mathematics career information, are available upon request.

    Approximately $45,000 in Hatcher, Morris, Layman, Rollins, Steeneck, Caldwell, Wells, Oehring, Eckert, Persinger, Kim, Kimball, and Roselle scholarships is awarded annually to mathematics majors at Virginia Tech: $5,000 for incoming freshmen and $40,000 for continuing undergraduates. Information on the scholarships is available from the scholarship chairman in mathematics.

    The Cooperative Education Program is also available to qualified candidates, and students wishing to mix practical experience with their formal course studies are encouraged to investigate this option. For more information, contact Career Services at Virginia Tech.

    The Mathematics Department firmly believes that mathematics is not only useful and beautiful, but also fun. The department sponsors student chapters of MAA (Mathematical Association of America), SIAM (Society for Industrial and Applied Mathematics),  Pi Mu Epsilon (the national mathematics honorary society), and  AWM (Association for Women in Mathematics). As well as social activities, these groups sponsor speakers to talk on how mathematics is used in their work. Each fall, Virginia Tech also sponsors  the Virginia Tech Regional Mathematics Contest. In addition, students (not all of whom are mathematics majors) annually receive organized preparation and compete in the nationwide William Lowell Putnam Competition and the international Mathematical Contest in Modeling. Individual undergraduate research projects are available to talented students, and a Layman Prize is awarded for the best research project. An overall outstanding senior, as well as an outstanding senior for each option, is recognized each year.

    The Honors Program in Mathematics provides outstanding undergraduate majors the opportunity for an enriched academic environment. Through honors courses, an honors project, individual association with the faculty and honors advisors, and other perquisites, the honors student in mathematics enjoys a valuable advantage in the undergraduate experience. Moreover, in coordination with the head of Mathematics and the dean of Science, the honors student may design her/his own individual set of graduation requirements.

    In addition to the four undergraduate-degree options, the department also offers the M.S. and Ph.D. Moreover, for qualified students, a combined program is available that leads to both a B.S. and an M.S. in Mathematics. This program saves a year from the usual time required for a B.S. and an M.S. done separately. Students in the Education Option obtain a B.S. in Math and an M.A. in Education by completing four years of undergraduate study and a fifth year in education for a full secondary certification.

    The minor is designed to provide recognition for those students who take a program of study in mathematics above the normal requirements of their disciplines.

Bachelor of Science in Mathematics

    Requirements:

    Note that  the Calculus  curriculum is in transition and there are two possible paths through Calculus. We distinguish the two paths as follows : Path 1 for  students who have received credit for MATH 1205 prior to fall 2014 and Path 2 for students who have not received credit for MATH 1205 prior to fall 2014.

    Degree Requirements:

    The graduation requirements in effect at the time of graduation apply. When choosing the degree requirements information, always choose the year of your expected date of graduation. Requirements for graduation are referred to via university publications as “Checksheets”. The number of credit hours required for degree completion varies among curricula. Students must satisfactorily complete all requirements and university obligations for degree completion.

    The university reserves the right to modify requirements in a degree program. However, the university will not alter degree requirements less than two years from the expected graduation year unless there is a transition plan for students already in the degree program.

    Please visit the University Registrar website at http://www.registrar.vt.edu/graduation/checksheets/index.html for degree requirements.

    Those courses listed in the catalog under the subtitles "Basic Sequences for Students in Agriculture, Architecture, Biology, Business, and Liberal Arts and Human Sciences" and "Electives (may not be taken by Mathematics Majors)" may not be used for graduation in mathematics. Special exceptions to this exclusion must have the approval of the head of the department of mathematics.

    In order to enroll in 3034, a student must (obtain a C or better in the final attempt of each of 1205, 1206, 1114, and (2224 or 2214)) or (obtain a C or better in the final attempt of each of 1225, 1226, 1114 and (2204 or 2214)) or (obtain a C or better in the final attempt of MATH 2114). Math students with one C- in the above courses should confer with their advisor.

    Each student is required to participate in the department's Outcomes Assessment procedures as determined by each year's Undergraduate Program Committee and approved by the department head.

    Prospective Student Website

    A great deal of further information on the Mathematics Program and on mathematical careers can be found on our website at www.math.vt.edu/

Minor in Mathematics

    Requirements:

    A total of 25 semester hours of  the following mathematics courses for students who follow  Path 1 : Calculus (1205-1206, 1224, 2224); Linear Algebra & ODE's: (1114, 2214); and 9 hours of approved mathematics courses numbered 3000 or higher. Students who  follow Path 2, should take a total of 26 semester hours of the following mathematics courses  Calculus (1225-1226, 2204) ; Linear Algebra &ODEs (2114, 2214) ; and 9 hours of approved mathematics courses numbered 3000 or higher or  selections from CMDA 3605, 3606, and 4604. Duplications are prohibited. The student must have a 2.00 average in courses used for the minor, none of which may be taken pass/fail.

Advanced Placement

    A student following Path 1 may obtain advanced placement  credit for 1205, or 1206, and students following Path 2 may obtain advanced placement credit for 1225 or 1226. The Mathematics Department strongly encourages calculus students to take the C.E.E.B. advanced placement test in calculus.

Satisfactory Progress

    University policy requires that students who are making satisfactory progress toward a degree meet minimum criteria toward the Curriculum for Liberal Education (see Academics chapter in this catalog), toward the College of Science Core (see first part of this chapter), and toward the degree in mathematics.

    Satisfactory progress toward the B.S. in mathematics requires that:

University policy requires that students who are making satisfactory progress toward a degree meet minimum criteria toward the General Education (Curriculum for Liberal Education) (see "Academics") and toward the degree in Economics.

Satisfactory progress requirements toward the B.S. in Mathematics can be found on the major checksheet by visiting the University Registrar website at http://www.registrar.vt.edu/graduation/checksheets/index.html.

Undergraduate Courses (MATH)


Basic Sequences for Students in Engineering, Building Construction, Chemistry, Computer Science, Geological Sciences, Mathematics, Physics, and Statistics

1114: ELEMENTARY LINEAR ALGEBRA
Euclidean vectors, complex numbers, and topics in linear algebra including linear systems, matrices, determinants, eigenvalues, and bases in Euclidean space. This course, along with 1205-1206 and 1224, constitutes the freshman science and engineering mathematics courses. 2 units of high school algebra, 1 unit of geometry, ½ unit each of trigonometry and pre-calculus required. Partially duplicates MATH 2114. (2H,2C)

1205-1206: CALCULUS
Unified calculus course including techniques and applications of differentiation and integration of functions of a single variable. Limits, continuity, differentiation, integration, and transcendental functions. This sequence, together with 1114 and 1224, constitutes the first-year science and engineering mathematics courses. 1205 partially duplicates MATH 1016, 1025, 1225, and 1525. 1206 partially duplicates MATH 1026, 1226, and 2015. Pre 1205: 2 units of high school algebra, 1 unit of geometry, 1/2 unit each of trigonometry and precalculus and placement by Math Dept.; or a grade of B or better in one of 1015, 1016, or 1536; or a passing grade on the Calculus Readiness Exam; Pre: 1205 for 1206. (3H,3C)

1224: VECTOR GEOMETRY
Topics in analytic geometry and conic sections, and the calculus of vector-valued functions. This course, along with 1114 and 1205-1206, constitutes the freshman science and engineering mathematics courses. Pre: 1205 or 1225. Co: 1206, 1114. (2H,2C)

1225-1226: CALCULUS OF A SINGLE VARIABLE
Unified calculus course covering techniques of differential and integral calculus for functions of one variable. This sequence constitutes the standard first-year mathematics courses for science and engineering. 1225: limits, continuity, differentiation, transcendental functions, applications of differentiation, introduction to integration. Assumes 2 units of high school algebra, 1 unit of geometry, 1/2 unit each of trigonometry and precalculus, and placement by Math Dept. 1226: techniques and applications of integration, trapezoidal and Simpson's rules, improper integrals, sequences and series, power series, parametric curves and polar coordinates, software-based techniques. 1225 partially duplicates 1205, 1016, 1025, and 1525. 1226 partially duplicates 1026, 1206, and 2015. Pre: 1225 for 1226. (4H,4C)

2114: INTRODUCTION TO LINEAR ALGEBRA
Vector and matrix algebra, systems of linear equations, linear equations, linear independence, bases, orthonormal bases, rank, linear transformations, diagonalization, implementation with contemporary software. Math 1226 or a grade of at least B in VT 1225. Pre: 1225 or 1226. (3H,3C)

2114H: INTRODUCTION TO LINEAR ALGEBRA
Vector and matrix algebra systems of linear equations, linear equations, linear independence, bases, orthonormal bases, rank, linear transformations, diagonalization, implementation with contemporary software. Math 1226 or a grade of at least B in VT 1225. Pre: 1225 or 1226. (3H,3C)

2204: INTRODUCTION TO MULTIVARIABLE CALCULUS
Calculus for functions for several variables. Planes and surfaces, continuity, differentiation, chain rule, extreme values, Lagrange multipliers, double and triple integrals and applications, software-based techniques. Partially duplicates MATH 2016, 2024 and 2224. Pre: 1226. (3H,3C)

2204H: INTRODUCTION TO MULTIVARIABLE CALCULUS
Calculus for functions of several variables. Planes and surfaces, continuity, differentiation, chain rule, extreme values, Lagrange multipliers, double and triple integrals and applications, software-based techniques. Partially duplicates 2016, 2024, and 2224. Pre: 1226. (3H,3C)

2214: INTRODUCTION TO DIFFERENTIAL EQUATIONS
Unified course in ordinary differential equations. First-order equations, second- and higher-order linear equations, systems of first-order linear equations, and numerical methods. Partially duplicates 4544. Pre: (1114 or 1114H or 2114 or 2114H), (1206 or 1206H or 1226 or 2015 or 1026). (3H,3C)

2214H: INTRODUCTION TO DIFFERENTIAL EQUATIONS
Pre: (1114 or 1114H or 2114 or 2114H), (1206 or 1206H or 1226 or 2015 or 1026). (3H,3C)

2224: MULTIVARIABLE CALCULUS
Partial differentiation, multiple integration, and infinite series. Partially duplicates MATH 2204, 2024, and 2016. Pre: (1206 or 1206H or 2015 or 1026), (1224 or 1224H). (3H,3C)

2224H: MULTIVARIABLE CALCULUS
Pre: (1206 or 1206H or 2015 or 1026), (1224 or 1224H). (3H,3C)

2405H-2406H: MATHEMATICS IN A COMPUTATIONAL CONTEXT
Unified course covering topics from linear algebra, differential equations, and calculus for functions of several variables. Comprises the standard second year mathematics courses for science and engineering. 2405H: Vector and matrix algebra, systems of linear equations, linear independence, bases, orthonormal bases, rank, linear transformations and diagonalization. Ordinary linear homogeneous differential equations, implementation with contemporary software. 2406H: Ordinary nonhomogeneous differential equations, calculus for functions of several variables, planes and surfaces, continuity, differentiation, chain rule, extreme values, Lagrange multipliers, double and triple integrals and applications, with software-based techniques. MATH 2405H partially duplicates 2114, 2214. MATH 2406H partially duplicates 2204, 2214. Pre: 1226 for 2405H; 2405H for 2406H. (5H,5C)

Basic Sequences for Students in Agriculture, Architecture, Biology, Business, and Liberal Arts and Human Sciences

1014: PRECALCULUS WITH TRANSCENDENTAL FUNCTIONS
Precalculus college algebra, basic functions (algebraic, exponential, logarithmic, and trigonometric), conic sections (parabolas, circles, ellipses, hyperbolas), graphing techniques, basic probability. Use of spreadsheet software. Two units of high school algebra and one of plane geometry are required. Partially duplicates MATH 1015. (3H,3C)

1025-1026: ELEMENTARY CALCULUS
1025: Differential calculus, graphing, applications for the life sciences. Use of spreadsheet software. Assumes 2 units of high school algebra, 1 unit of geometry, 1/2 unit of trigonometry, and 1/2 unit of precalculus. 1026: Integral calculus, numerical techniques, elementary differential equations, applications for the life sciences. Use of spreadsheet software. 1025 partially duplicates MATH 1016, 1205, and 1225. 1026 partially duplicates MATH 2015,1206, and 1226. (3H,3C)

1525-1526: ELEMENTARY CALCULUS WITH MATRICES
1525: Linear, quadratic, exponential and logarithmic functions. Differential calculus with graphical interpretation. Terminology and applications for business, including spreadsheet software. 1526: Integration, substitution and approximation methods. Matrix algebra and solving systems of equations. Partial derivatives and optimization for functions of several variables. Applications for business, including spreadsheet software. 1525 partially duplicates MATH 1016, 1025, 1205, and 1225. 1526 partially duplicates MATH 1114,2114,2015, 1026,1206, and 1226. Assumes 2 unit of high school algebra and 1 unit of plane geometry. (3H,3C)

1535,1536: GEOMETRY AND MATHEMATICS OF DESIGN
1535: Euclidean geometry, isometries, congruencies, similarities. Circles and trigonometry, sequences and the golden ratio. Graph theory, tilings of the plane, polygons and polyhedra. Applications for 2- and 3-dimensional design, including geometric software. 1536: Vectors in the plane and space, lines and planes, and cross product. Descriptive and projective geometry. Differential and integral calculus. Applications for 2- and 3-dimensional design. Assumes 2 units of high school algebra and 1 unit of high school geometry. (3H,3C)

2024: INTERMEDIATE CALCULUS
Continuation of Math 1025-1026. Calculus for functions of several variables, differential equations, sequences and series. Applications for the life sciences. Use of spreadsheet software. Partially duplicates MATH 2016, 2224, 2204, and 2214. Pre: 1026 or 2015. (3H,3C)

Electives (may not be taken by Mathematics Majors)

1614: NUMBER AND COMPUTING FOR TEACHERS
A study of the nature and structure of number, number theory, number systems, properties, operations and problem solving which are part of the foundation of the K-8 mathematics curriculum. Computer component includes an emphasis on using spreadsheets to construct mathematical models. (4H,4C)

1624: GEOMETRY AND COMPUTING FOR TEACHERS
A study of key geometry concepts from multiple perspectives including transformational, coordinate, Euclidean and analytical geometry. Geometric and spatial reasoning are part of the foundation of the mathematical curriculum for grades K-8. Computer component integrates the Geometer's Sketchpad, Logo programming language, and other geometry based software. (4H,4C)

2534: INTRODUCTION TO DISCRETE MATHEMATICS
Emphasis on topics relevant to computer science. Topics include logic, propositional calculus, set theory, relations, functions, mathematical induction, elementary number theory and Boolean algebra. Does not carry credit for mathematics majors, but may be used as though it were a 3000-level elective course for the mathematics minor. Partially duplicates 3034. Two units of high school algebra, one unit of geometry, one-half unit each of trigonometry and precalculus mathematics required. Pre: CS 1114 or ECE 1574. (3H,3C)

4574: VECTOR AND COMPLEX ANALYSIS FOR ENGINEERS
Vector Analysis: Green's theorem, potential theory, divergence, and Stokes' theorem. Complex Analysis: Analyticity, complex integration, Taylor series, residues, conformal mapping, applications. Pre: 2224 or 2204 or 2204H. (3H,3C)

Electives (including Mathematics Majors)

2004 (ME 2004): ENGINEERING ANALYSIS USING NUMERICAL METHODS
Numerical methods applied to engineering analysis. Linear systems. Root finding. Numerical integration. Ordinary differential equations. Programming using a software package such as Matlab. Pre: ENGE 1016, (MATH 1206 or MATH 1226), (MATH 1114 or MATH 2114 or MATH 2114H). (2H,2C)

2644: MATHEMATICS TUTORING
An introduction to mathematics tutoring. Course activities include the development of listening and questioning skills, assessment of a student's mathematical difficulties, and an exploration of teaching and learning processes. In a weekly journal, students will reflect on their tutoring experiences to develop and refine teaching goals and skills. A concurrent mathematics tutoring experience is required. Pre: 1206 or 1226. (1H,1C)

2964: FIELD STUDY
Pass/Fail only. Variable credit course.

2974: INDEPENDENT STUDY
Variable credit course.

2974H: INDEPENDENT STUDY
Honors section.
Variable credit course.

2984: SPECIAL STUDY
Variable credit course.

2984H: SPECIAL STUDY
Variable credit course.

2994: UNDERGRADUATE RESEARCH
Variable credit course.

3034: INTRODUCTION TO PROOFS
Practice in writing mathematical proofs. Exercises from set theory, number theory, and functions. Specific topics include set operations, equivalence relations, mathematical induction, the division algorithm and images and pre-images of sets. Partially duplicates 2534. In order to enroll in Math 3034, student must obtain (1) a C or better in each of 1114, 1205, 1206, 1224 and (either 2214 or 2224), or (2) at most one C- and a GPA of at least 2.2 in the courses mentioned in (1). Pre: (1205, 1206, 1114), (2224 or 2214) or (1225, 1226, 1114) or (2204, 2214) or 2114. (3H,3C)

3054: PROGRAMMING FOR MATHEMATICAL PROBLEM SOLVING
An Introduction to computer programming designed for mathematics majors. Variable types, data structures, control flow and program structure. Procedural, functional and objective-oriented programming paradigms for solution of a variety of mathematical problems. Co: 2214. (3H,3C)

3124: MODERN ALGEBRA
Introductory course in groups, rings and fields. Pre: 3034. (3H,3C)

3134: APPLIED COMBINATORICS AND GRAPH THEORY
Emphasis on concepts related to computational theory and formal languages. Includes topics in graph theory such as paths, circuits, and trees. Topics from combinatorics such as permutations, generating functions, and recurrence relations. Pre: (1206 or 1226), (2534 or 3034). (3H,3C)

3144: LINEAR ALGEBRA I
Introductory course in linear algebra. Abstract vector spaces, linear transformations, algorithms for solving systems of linear equations, matrix analysis. This course involves mathematical proofs; it is strongly recommended that students take 3034 first. Pre: 2214 or 2214H. (3H,3C)

3214: CALCULUS OF SEVERAL VARIABLES
Fundamental calculus of functions of two or more variables. Implicit function theorem, Taylor expansion, line integrals, Green's theorem, surface integrals. Pre: 2224 or 2224H or 2204 or 2204H. (3H,3C)

3224: ADVANCED CALCULUS
Theory of limits, continuity, differentiation, integration, series. 3224 duplicates 4525. Pre: (2224 or 2224H or 2204 or 2204H), 3034. (3H,3C)

3414 (CS 3414): NUMERICAL METHODS
Computational methods for numerical solution of non-linear equations, differential equations, approximations, iterations, methods of least squares, and other topics. Partially duplicates Math 4554 and 4404. A grade of C or better required in CS prerequisite 1044 or 1705. Pre: (CS 1044 or CS 1705 or CS 1114 or CS 1124), (MATH 2214 or MATH 2214H), (MATH 2224 or MATH 2224H or MATH 2204 or MATH 2204H). (3H,3C)

3624: EARLY TEACHING EXPERIENCE IN MATHEMATICS
An early field experience designed for mathematics students in the mathematics education option. Principles for school mathematics. Secondary school classroom experience and experience-based research. Pre: Junior standing and permission of the instructor. (4H,4C)

4044: HISTORY OF MATHEMATICS
Historical development of mathematics from antiquity to modern times. Senior standing in mathematics required. (3H,3C)

4124: INTRODUCTION TO ABSTRACT ALGEBRA
An introduction to the theory of groups and rings. Topics include normal subgroups, permutation groups, Sylow's Theorem, Abelian groups, Integral Domains, Ideals, and Polynomial Rings. Pre: 3124. (3H,3C)

4134: NUMBER THEORY
Divisibility, congruencies, multiplicative functions, primitive roots, quadratic reciprocity. Pre: 2534 or 3034 or 3134. (3H,3C)

4144: LINEAR ALGEBRA II
Second course in linear algebra. Similarity invariants, Jordan canonical form, inner product spaces, self-adjoint operators, selected applications. Pre: 3144. (3H,3C)

4164: ADVANCED DISCRETE MATHEMATICS
Advanced topics in discrete mathematics with applications. Includes counting techniques, generating functions, recurrence relations, combinatorial designs, semigroups, words and rewriting rules, matroids, and selected additional topics (e.g., Ramsey theory, Polya theory, Young tableaux). Knowledge of a programming language (e.g., C, fortran, Pascal) required. Pre: 3034, 3134. (3H,3C)

4175-4176: CRYPTOGRAPHY
4175: Elementary concepts in cryptography; classical cryptosystems; modern symmetric cryptography; public key cryptography; digital signatures, authentication schemes; modular arithmetic, primitive roots, primality testing. At least one mathematics course at or above the 3000 level and facility with either a programming language or a computer algebra system is required. 4176: Discrete logs; pseudoprime tests; Pollard rho factoring; groups; quadratic residues; elliptic curve cryptosystems and factoring; coding theory; quantum cryptography. (3H,3C)

4225-4226: ELEMENTARY REAL ANALYSIS
Real number system, point set theory, limits, continuity, differentiation, integration, infinite series, sequences and series of functions. Pre: 3224 for 4225; 4225 for 4226. (3H,3C)

4234: ELEMENTARY COMPLEX ANALYSIS
Analytic functions, complex integration, series representation of analytic functions, residues, conformal mapping, applications Pre: 3224. (3H,3C)

4245-4246: INTERMEDIATE DIFFERENTIAL EQUATIONS
Solution techniques, linear systems, the matrix exponential, existence theorems, stability, non-linear systems, eigenvalue problems. Pre: 3224. (3H,3C)

4254: CHAOS AND DYNAMICAL SYSTEMS
Survey of basic concepts in chaotic dynamical systems. Includes material on bifurcation theory, conjugacy, stability, and symbolic dynamics. Pre: 3224. (3H,3C)

4324: ELEMENTARY TOPOLOGY
Basic concepts of topological spaces, continuous functions, connected spaces, compact spaces, and metric spaces. Pre: 3124, 3224. (3H,3C)

4334: COLLEGE GEOMETRY
Transformational approach to Euclidean geometry including an in-depth study of isometries and their application to symmetry, geometric constructions, congruence, coordinate geometry, and non-Euclidean geometries. Pre: (1114 or 2114 or 2114H), (1206 or 1226). (3H,3C)

4404 (AOE 4404): APPLIED NUMERICAL METHODS
Interpolation and approximation, numerical integration, solution of equations, matrices and eigenvalues, systems of equations, approximate solution of ordinary and partial differential equations. Applications to physical problems. Partially duplicates 3414. Mathematics majors or minors cannot take both 4404 and 3414. Pre: 4564, ESM 2074. (3H,3C)

4414 (CS 4414): ISSUES IN SCIENTIFIC COMPUTING
Theory and techniques of modern computational mathematics, computing environments, computational linear algebra, optimization, approximation, parameter identification, finite difference and finite element methods and symbolic computation. Project-oriented course; modeling and analysis of physical systems using state-of-the-art software and packaged subroutines. Pre: 2214, 3214, (CS 2114 or MATH 3054). (2H,3L,3C)

4425-4426: FOURIER SERIES AND PARTIAL DIFFERENTIAL EQUATIONS
Separation of variables for heat, wave, and potential equations. Fourier expressions. Application to boundary value problems. Bessel functions. Integral transforms and problems on unbounded domains. Pre: 2214, (2224 or 2204 or 2204H), 3224 for 4425; 2214, (2224 or 2204 or 2204H), 3224, 4425 for 4426. (3H,3C)

4445,4446: INTRODUCTION TO NUMERICAL ANALYSIS
4445: Vector spaces and review of linear algebra, direct and iterative solutions of linear systems of equations, numerical solutions to the algebraic eigenvalue problem, solutions of general non-linear equations and systems of equations. 4446: Interpolation and approximation, numerical integration and differentiation, numerical solutions of ordinary differential equations. Computer programming skills required. Pre: 2214, 2224. (3H,3C)

4454: APPLIED MATHEMATICAL MODELING
Analysis of classical and modern applications of mathematics in the physical, biological and social sciences. Emphasis on problem formulating, modeling, solving, simulating, and analyzing results. Programming language required. (3H,3C)

4564: OPERATIONAL METHODS FOR ENGINEERS
Laplace transformations, Fourier series, partial differential equations and separation of variables, boundary value problems, and Sturm-Liouville theory. Pre: 2214. (3H,3C)

4625,4626: MATHEMATICS FOR SECONDARY TEACHERS
Course activities will emphasize the curricular themes of problem solving, reasoning and proof, communication, connections, and representation. 4625: Topics in discrete mathematics and algebra from a secondary teaching perspective. 4626: Topics in trigonometry, geometry, measurement, statistics, and probability from a secondary teaching perspective. Pre: 3034. (3H,3C)

4644: SECONDARY SCHOOL MATHEMATICS WITH TECHNOLOGY
Use and impact of technology in secondary mathematics curriculum. Various technologies including graphing calculators, calculator based laboratory and probes (CBLs), computer algebra systems, spreadsheets, dynamic geometry software and the Internet will be used to explore secondary mathematical concepts from an advanced viewpoint. Pre: 3034. (3H,3C)

4664: SENIOR MATH EDUCATION SEMINAR
A review of basic principles and problem-solving techniques in the eleven topics covered by the Praxis II (Mathematics Content Knowledge) examination. Passing the Praxis II examination prior to student teaching is a state requirement for all students seeking secondary licensure. Passing Praxis I required. Pre: 3124. (2H,2C)

4754: INTERNSHIP
May be repeated for a maximum of 12 credits.
Pass/Fail only. Variable credit course.

4964: FIELD STUDY
Pass/Fail only. Variable credit course.

4974: INDEPENDENT STUDY
Variable credit course.

4974H: INDEPENDENT STUDY
Honors section.
Variable credit course.

4984: SPECIAL STUDY
Variable credit course.

4994: UNDERGRADUATE RESEARCH
Variable credit course.

4994H: UNDERGRADUATE RESEARCH
Honors section.
Variable credit course.

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