Mathematical Sciences (MASC)

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

College of Science

Mathematics

www.math.vt.edu/ Peter Haskell, Head Associate Head for Undergraduate Students, C. J. Parry Director for Undergraduate Programs: R. Rogers Graduate Director: J. Borggaard 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; W. J. Floyd; E. L. Green; W. Greenberg; G. A. Hagedorn; P. E. Haskell; T. L. Herdman; J. U. Kim; M. Klaus; W. E. Kohler; R. Laubenbacher; T. Lin; P. A. Linnell; G. Lloyd; C. J. Parry; C. L. Prather; F. Quinn; R. C. Rogers; J. F. Rossi; D. L. Russell; M. Shimozono; R. L. Snider; S. Sun; J. Turner; R. L. Wheeler Associate Professors: E. de Sturler; D. Gao; S. Gugercin; T. Iliescu; J. E. Shockley; P. Wapperom; M. Williams Assistant Professors: A. Elgart; N. Loehr; H. Mortveit; A. Norton; P. Yue; L. Zietsman Senior Instructors: D. Agud; S. Anderson; T. A. Bourdon; S. Hagen; C. Hodges; A. Kohler; L. Powers; E. T. Shugart; C. Stephens Advanced Instructors: M. P. McQuain Instructors: M. Cothren; L. L. Hanks; H. Hart; L. Peters; B. Reynolds; J. Schmale; D.B. Smith Career Advisor: Lizette Zietsman (231-2767)

Overview Bachelor of Science in Mathematics Minor in Mathematics Advanced Placement Satisfactory Progress Undergraduate Courses

Overview

Mathematics is essential to a clear and complete understanding of virtually all phenomena. 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 appropriately the usual mathematical treatment for the problem to accommodate for the deviation. In this case mathematical training provides a practical preparation for a career in today's changing world. Moreover, it is especially valuable since 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 and Computational Mathematics Option (ACM); 3) the Applied Discrete Mathematics Option (ADM); and 4) the Mathematics Education Option (MAED).

The Traditional Option, as its name implies, yields a broad and flexible background in mathematics. The other three options are more specialized. The ACM is designed for students who are confident that they want to have an applied mathematics career in an area closely associated with physics or some form of engineering. The ADM is designed for students who plan to have an applied mathematics career in an area closely associated with computer science, statistics, or actuarial 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 $30,000 in Hatcher, Morris, Layman, Rollins, Steeneck, Caldwell, Wells, Oehring, Eckert and Roselle scholarships is awarded annually to mathematics majors at Virginia Tech: $13,000 for incoming freshmen and $17,000 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.

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) and Pi Mu Epsilon (the national mathematics honorary society). 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 an Eastern U.S. Regional Mathematics competition. Each spring all freshmen and sophomores at Tech are invited to compete for prizes in a local 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. Individual undergraduate research projects are available to talented students, and a research prize is awarded. 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 nearly 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:

The following is a sketch of the requirements for the four undergraduate options. For more details, obtain a handbook from the Department of Mathematics. All four options require 120 hours, satisfaction of the Curriculum for Liberal Education, and the following mathematics courses: 1114, 1205-1206, 1224, 2214, 2224, 3034, and 3224. Special requirements for each option are as follows:

1. Traditional: 3124, 3144, 3214, 12 hours 4000-level electives; computer programming, 16 hours of approved application-area courses.
   
2. Education: 3124, 4334, 4044, 4625, 4644, 4654, 4664, 6 hours 3/4000-level electives; STAT 4705, CS 1044; EDCI 3024, 3144, 3724, 4124, 4404, 4744, and 4754.
   
3. ACM: 3144, 3214, 4414, 4425-4426, 4445-4446, 6 hours 4000-level electives; CS 1044; 12 hours approved applied-area courses.
   
4. ADM: 3124, 3134, 3144, 3214, 4134, 4164, 6 hours 4000-level electives; CS 1044, 1705, 1706, 2204, 2605, 2606, 4104; STAT 4714.

Those courses listed in the catalog under the subtitles "Basic Sequences for Students Not in Engineering and Science Curricula" and "Electives for All Students Except 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 mathematics student must either (a) obtain a C or better in the final attempt of each of 1114, 1205, 1206, 1224, and (2224 or 2214); or (b) have at least a 2.2 GPA in these five courses with at most one grade of C- and no D's in the last attempt in each. Similarly, a student wishing to transfer into mathematics from another discipline must meet the same standards in these courses (or as far in these courses as she/he has progressed).

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 mathematics courses as follows: Calculus (1205-1206, 1224, 2224); Linear Algebra & ODE's: (2214, 1114); and one course from either 3124, 3134, 3144, 3214, 3224, 4134, 4164, 4214, 4225, 4234, 4254, 4324, 4425, 4445, 4446 or 4514). The remaining hours are 3000- and 4000-level elective courses (except 2534 is allowed and 4044 is prohibited), and any 5000-level Mathematics course may be substituted in the above list. 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 may obtain advanced-placement or advanced-standing credit from high school work for 1015, 1205, or 1206. 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:

1. Within the previous two semesters, the student must pass at least one mathematics course which is used in the in-major GPA calculation.
2. Upon having attempted 72 semester credits (including transfer, advanced placement, advanced standing, credit by examination, course withdrawal), students must have completed:
   

   MATH 1205-1206, 1224, 2224: Calculus	11
   MATH 1114, 2214: Linear Algebra and ODE's	5
   MATH 3034: Proofs and Algebraic Systems	3
   Total Credits	(19)

3. Upon having attempted 96 semester credits, students must have an in-major grade-point average of 2.0 or above.

Undergraduate Courses (MATH)

• Basic Sequences for Students in Engineering, Building Construction, Chemistry, Computer Science, Geological Sciences, Mathematics, Physics, and Statistics
• Basic Sequences for Students in Agriculture, Architecture, Biology, Business, and Liberal Arts and Human Sciences
• Electives (may not be taken by Mathematics Majors)
• Electives (including Mathematics Majors)

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. Partially duplicates 2524. 1525 may not be taken after taking 1114. 2 units of high school algebra, 1 unit of geometry, 1/2 unit each of trigonometry and precalculus required. (2H,2C)

1114H: ELEMENTARY LINEAR ALGEBRA
(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 1016 and 1526. 1206 partially duplicates 2015. Pre (for 1205): Either a grade of C or better in 1504, or 2 units of H.S. algebra, 1 unit of geometry, 1/2 unit each of trigonometry and precalculus, and placement by Math Dept. (3H,3C) 1205: I,II,III; 1206:

1205H-1206H: CALCULUS
(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. Co: 1206, 1114. (2H,2C) I,II,III,IV.

1224H: VECTOR GEOMETRY
(2H,2C)

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 2514 and 4544. Pre: (1114 or 1114H), (1206 or 1206H or 2015). (3H,3C) I,II,III,IV.

2214H: INTRO DIFF EQUATIONS
Pre: (1114 or 1114H), (1206 or 1206H or 2015). (3H,3C)

2224: MULTIVARIABLE CALCULUS
Partial differentiation, multiple integration, and infinite series. Partially duplicates 2016. Pre: (1206 or 1206H or 2015), (1224 or 1224H). (3H,3C) I,II,III,IV.

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

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

1015-1016: ELEMENTARY CALCULUS WITH TRIGONOMETRY I
1015: College algebra, functions, exponentials and logarithms, matrices, sequences and series. 1016: Calculus including limits, derivatives, applications of derivatives, trigonometric functions. 1015 partially duplicates 1525. 1016 partially duplicates 1526 and 1205. 2 units of high school algebra and 1 of plane geometry required. (3H,3C) I,II,III,IV.

1525-1526: ELEMENTARY CALCULUS WITH MATRICES
1525: Modeling, linear, quadratic, exponential and logarithmic functions, limits, derivatives. 1526: Integration with applications, matrix algebra and solving systems of equations, multivariable calculus. Credit for 1525 precludes credit for 1016 or 1205. Credit for 1526 precludes credit for 1114. Credit for 2015 or 1206 precludes credit for 1526. Pre: 2 units of high school algebra and 1 unit of plane geometry required. (3H,3C)

1535,1536: GEOMETRY AND MATHEMATICS OF DESIGN
1535: Review of Euclidean geometry and trigonometry. Descriptive and projective geometry applied to drawing. Similarity, proportion, and the golden mean. Applications of graph theory. 1536: Calculus with applications to max/min, areas, volumes, and centroids. Polygons, patterns, and tilings of the plane. Polyhedra and vectors applied to 3-dimensional design. Pre: 2 units of high school algebra and 1 unit of high school geometry for each of 1535 and 1536. (3H,3C)

2015-2016: ELEMENTARY CALCULUS WITH TRIGONOMETRY II
Continuation of Math 1015-1016. 2015: Trigonometric calculus, indefinite integrals, definite integration, areas and volumes, multivariable differential calculus. 2016: Multiple integrals, indeterminate forms, infinite sequences and series, differential equations. 2015 partially duplicates 1206. 2016 partially duplicates 2224. Pre: 1016 for 2015; 2015 for 2016. (3H,3C) 2015: I,II,IV; 2016:

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. I (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) II.

2514: ELEMENTARY DIFFERENTIAL EQUATIONS
Linear and nonlinear ordinary differential equations. Use of differential equations to model observed physical phenomena and experiments. Analytic and numerical solution methods. Graphical interpretation and slope fields. This course partially duplicates 2214 and 4544. Pre: 2015. (3H,3C)

2524: MATRICES, MODELING, AND LINEAR PROGRAMMING
An introductory course in mathematical modeling, matrix algebra, and linear programming. Includes the construction of several models to emphasize the diverse use of these topics in the study of other disciplines. Partially duplicates 1114. 1525 cannot be taken after 2524. Pre: 1205 or 1016 or 1526. (3H,3C)

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. (3H,3C)

2624: ALGEBRA AND COMPUTING FOR TEACHERS
The beginning portions of the course are designed to provide, from an advanced standpoint, a comprehensive view of school algebra curricula. The remainder of the course is devoted to the investigation of algebraic relationships and algorithms through the use of the computer. Pre: 1624. (3H,3C)

4514: APPLIED ALGEBRA
Binary relations, groups, semigroups, monoids, rings, fields, Boolean algebras, and Polya's theory of Enumerations. Partially duplicates 3124. Pre: 1114, 1206. (3H,3C)

4525-4526: PRINCIPLES OF ADVANCED CALCULUS
4525: Elementary calculus review, real number system, continuity of single variable functions, integration, sequences and infinite series. 4526: Differential calculus of several variable functions, multiple integration, curves and surfaces, line and surface integrals. For non-mathematics majors: 4525 partially duplicates 3224, 4526 partially duplicates 3214. Pre: 2224 for 4525; 4525 for 4526. (3H,3C)

4544: ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS
Ordinary differential equations: first order, second and higher order linear equations, power series and Laplace transform methods. Partial differential equations: Fourier series, separation of variables for heat, wave and potential equations. Sturm-Liouville theory. Bessel functions. Legendre polynomials. Partially duplicates 2214 and 4564. Pre: 1114, 1206. (5H,5C)

4554: NUMERICAL METHODS FOR ENGINEERS
Root-finding, interpolation, linear algebraic systems, numerical integration, and numerical solution of ordinary and partial differential equations. Duplicates 3414 and 4404, and can not be taken by mathematics majors. Computer programming required. Pre: 2214. (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. Duplicates 4544. Pre: 2214. (3H,3C) I,II,III,IV.

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. (3H,3C) I,II,III.

4584 (STAT 4584): ADVANCED CALCULUS FOR STATISTICS
Introduction to those topics in advanced calculus and linear algebra needed by statistics majors. Infinite sequences and series. Orthogonal matrices, projections, quadratic forms. Extrema of functions of several variables. Multiple integrals, including convolution and nonlinear coordinate changes. Pre: 1114, 1205, 1206, 2224. (3H,3C)

Electives (including Mathematics Majors)

1504: PRECALCULUS
College algebra, functions, exponential and logarithmic functions, trigonometry. This course is designed specifically to meet the needs of students who intend to be in a curriculum that requires 1205, but who are not yet ready to begin 1205. 1504 partially duplicates 1015 and 1525. 2 units of high school algebra and 1 unit of plane geometry required. (3H,3C) I,III.

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, MATH 1114. (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. (1H,1C)

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

2974: INDEPENDENT STUDY
Variable credit course.

2984: 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 and 3434. 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). (3H,3C) I,II,III.

3034H: HONORS INTRODUCTION TO PROOFS
(3H,3C)

3124: MODERN ALGEBRA
Introductory course in groups, rings and fields. This course partially duplicates 4514. Pre: 3034 or 3434. (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, (2534 or 3034). (3H,3C) I,II,IV.

3144: LINEAR ALGEBRA I
Introductory course in linear algebra. Abstract vector spaces, linear transformations, algorithms for solving systems of linear equations, matrix analysis. Pre: 3034 or 3434. (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. Duplicates 4526. Pre: 2224. (3H,3C) II,IV.

3224: ADVANCED CALCULUS
Theory of limits, continuity, differentiation, integration, series. This course duplicates 4525. Course involves mathematical proofs; one may benefit by first taking 3034. Pre: 2224. (3H,3C) I,II,III.

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. A grade of C or better required in CS prerequisite 1044 or 1705. Pre: 2214, 2224, CS 1044 or CS 1705. (3H,3C) II.

3434: INTRODUCTION TO DISCRETE APPLIED MATHEMATICS
Finite and infinite sets, mathematical induction, functions, equivalence relations, order relations, recurrence relations, discrete dynamical systems. Partially duplicates 2534 and 3034. In order to enroll in 3434, a 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). (3H,3C)

4024: AXIOMATIC SET THEORY
Peano postulates, cardinal arithmetic, axiom of choice, well orderings, ordinal arithmetic, continuum hypothesis. Pre: 2224, (3034 or 3134). (3H,3C)

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

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) I,II,III.

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

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. I 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)

4214: LINEAR ANALYSIS
Linear algebra of infinite dimensional spaces, inner product spaces, normed spaces, completeness, orthonormal bases, bounded and unbounded linear operators, adjoints, spectra, compact operators. Pre: (3214 or 4526), (3224 or 4525), (4425 or 4564). (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) I,II.

4234: ELEMENTARY COMPLEX ANALYSIS
Analytic functions, complex integration, series representation of analytic functions, residues, conformal mapping, applications Pre: 4525 or 3224. (3H,3C) II.

4245-4246: INTERMEDIATE DIFFERENTIAL EQUATIONS
Solution techniques, linear systems, the matrix exponential, existence theorems, stability, non-linear systems, eigenvalue problems. Pre: 3224 or 4525. (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 or 4525. (3H,3C)

4324: ELEMENTARY TOPOLOGY
Basic concepts of topological spaces, continuous functions, connected spaces, compact spaces, and metric spaces. I 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, 1206. (3H,3C) I,II.

4344: TOPICS IN GEOMETRY
Selected topics in geometry for advanced undergraduates. Pre: 1114, 1206.
(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 4554 and 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. (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. 4425: I, II. 4426: II. Pre: (2214, 2224), (3224 or 4525) for 4425; (2214, 2224), (3224 or 4525), 4425 for 44 26. (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) 4445: I,III; 4446:

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)

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) I.

4654: CAPSTONE THESIS AND SEMINAR
Students will review and discuss current research, state and national policies, and curriculum materials and trends in secondary mathematics education. Students will apply research findings and policy initiatives to the development of a major project in which they write and present unified plans for the teaching of a particular mathematical topic. Admission to the Graduate School and instructor approval required. Pre: 3034, EDCI 3724. (1H,1C)

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.

4984: SPECIAL STUDY
Variable credit course.

4994: UNDERGRADUATE RESEARCH
Variable credit course.

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