### Mathematics

**Chair:** Eric de Sturler

**Associate Chair:** R. C. Rogers

**Director for Undergraduate Programs:** L. Zietsman

**Graduate Director:** T. Warburton

**Class of 1950 Professor in Mathematics:** S. Gugercin

**John K. Costain Faculty Chair and Professor:** J. Borggaard

**Hatcher Professor of Mathematics:** J. A. Burns

**Professors:** S. Adjerid, C. A. Beattie, J. Borggaard, S. Ciupe, E. de Sturler, A. Elgart, M. Embree, P. E. Haskell, T. L. Herdman, T. Iliescu, M. Klaus, T. Lin, P. A. Linnell, N. Loehr, G. Matthews, C. Mihalcea, A. Norton, R. C. Rogers, M. Shimozono, S. Sun, J. Turner, T. Warburton, and P. Yue

**Associate Professors:** N. Abaid, J. Chung, Ma. Chung, D. Orr, P. Wapperom, M. Wawro, and L. Zietsman

**Assistant Professors:** L. Childs, R. Hewett, E. Johnson, H. Liu, E. Martin, T. Morrison, E. Palsson, O. Saucedo, W. Sun, and Y. Yang

**Collegiate Assistant Professors:** R. Arnold, E. Ufferman, and J. Wilson

**Visiting Assistant Professors:** A. Biswas, B. Cook, P. Manoharan, S. Pantic, K. Saglam, R. Steiner, and T. Topcu

**Patricia Ann Caldwell Post-Doctoral Fellow and Visiting Assistant Professor:** R. Singh

**Senior Instructors:** D. Agud, S. Anderson, T. A. Bourdon, S. Hagen, H. Hart, J. Hurdus, and J. Schmale

**Advanced Instructors:** J. Clemons, E. Jasso Hernandez, N. Robbins, and S. Yasuda

**Instructors:** H. Abobaker, T. Asfaw, S. Aslan, T. Balkew, S. Barreto, J. Brooks, G. Cerezo, D. Callie, R. Carracedo Rodriguez, My. Chung, F. Elsrrawi, J. England, H. Farhat, N. Gildersleeve, S. Hammer, T. Juste, K. Karcher, K. Kasebian, C. Letona, C. Lungstrum, N. Malik, C. Nicolas, M. Ouliaei-Nia, K. Perera, S. Pidaparthi, E. F. Rabby, K. Robinson, Rappold, Y. Shen, S. Silber, J. St.Clair, E. Sullivan, J. Thompson, J. Truman, J. Wells, C. Withrow, and K. Zachrich

**Postdoctoral Associates:** Pranial, L. Strube, and M. Vishnubhotia

**Lecturers:** V. Kairamkonda, W. Reilly, A. Sibol, and E. Widdowson

**Career Advisor:** L. Zietsman

**Scholarship Chair:** L. Childs

Web: www.math.vt.edu

### 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 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 on the Math Department advising website.

Approximately $50,000 in Caldwell, Eckert, Gaskins, Hatcher, Kim, Kimball, Layman, Rollins, Roselle, Steeneck, and Wells scholarships is awarded annually to mathematics majors at Virginia Tech, the majority for continuing undergraduates. Information on the scholarships is available from the scholarship committee chairperson 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.

Several academic departments in the College of Science, including the Mathematics Department, offer special Honors sections of their courses. In addition to special sections of some courses, the Honors Office sponsors a colloquia series each semester. The colloquia explore special topics not in the University Curriculum. Details about the Honors College can be found at http://www.honorscollege.vt.edu. Students may also contact the Mathematics Department Honors Advisor, Dr. Nick Loehr.

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

#### Degree Requirements

The graduation requirements in effect during the academic year of admission to Virginia Tech apply. When choosing the degree requirements information, always choose the year you started at Virginia Tech. 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.

Please visit the University Registrar's website at https://www.registrar.vt.edu/graduation-multi-brief/checksheets.html for degree requirements.

In order to enroll in 3034, a student must obtain a C or better in MATH 2114 or obtain a C or better MATH 2114H or obtain a C or better in MATH 2405H.

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 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 may obtain advanced placement credit for 1225 and/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 General Education (Pathways) (see "Academics") and toward the degree.

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

### Undergraduate Course Descriptions (MASC)

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 CS 1114 or any other Computer Science course at the 2000 level or higher precludes credit for 1044. (3H,3C)

### Undergraduate Course Descriptions (MATH)

1004: DISCOVERING MATHEMATICS I Introduction to the scope and applicability of mathematics and its many sub-disciplines. Introduction to the process of thinking, learning, and writing as a mathematician through topics such as logic systems, recreational mathematics, LaTeX programming, history, ethics, open problems, and research in mathematics. Also includes advising topics such as planning a Virginia Tech course of study. P/F only. Math majors. Pass/Fail only. (1H,1C)

1014: PRECALCULUS WITH TRANSCENDENTAL FUNCTIONS Precalculus college algebra, basic functions (algebraic, exponential, logarithmic, and trigonometric), conic sections, graphing techniques, basic probability. Usage of mathematical models, analytical calculations, and graphical or numerical representations of data to analyze problems from multiple disciplines that address intercultural and global challenges in areas such as chemistry, environmental science, the life sciences, finance, and statistics. Use of spreadsheet software. Two units of high school algebra and one of plane geometry are required. (3H,3C)

1025-1026: ELEMENTARY CALCULUS Quantitative and computational thinking to address relevant global issues. Unified calculus course covering techniques and applications of differential and integral calculus for functions of one variable. Constitutes the standard first-year mathematics courses for the life sciences. 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 precalculus. 1026: Integral calculus, numerical techniques, elementary differential equations, applications for the life sciences, use of spreadsheet and scientific software. A student can earn credit for at most one of 1025 and 1225. A student can earn credit for at most one of 1026 and 1226. (3H,3C)

1044: DISCOVERING MATHEMATICS II Introduction to the scope and applicability of mathematics and its many sub-disciplines. Introduction to the process of thinking, learning, and writing as a mathematician through topics in pure and applied mathematics and a brief experience with mathematical research. Also includes advising topics such as planning a Virginia Tech course of study. Math majors. (2H,2C)

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, 1/2 unit each of trigonometry and pre-calculus required. A student cannot earn credit for 1114 if taken after earning credit for 2114. (2H,2C)

1225-1226: CALCULUS OF A SINGLE VARIABLE Quantitative and computational thinking to address relevant intercultural and global issues. Unified calculus course covering techniques of differential and integral calculus for functions of one variable. 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. A student can earn credit for at most one of 1025 and 1225. A student can earn credit for at most one of 1525 and 1225. A student can earn credit for at most one of 1026 and 1226. Pre: Grade of at least C- in 1225 for 1226. (4H,4C)

1454: INTRODUCTION TO MATHEMATICAL PROBLEM-SOLVING An introduction to mathematical problem-solving strategies involving complex problems and subproblems. Examination of supporting data. Implementation of solution strategies through computer programming. Topics include logic, iterative process and recursion, Monte Carlo integration, random walks, visualization, computational geometry and graph theory. Graphical representation of mathematical information. Co: 1225. (3H,3C)

1524: BUSINESS CALCULUS Differential calculus techniques for functions of one and two variables. Emphasis on graphs, rates of change, and optimization of linear, quadratic, exponential, and logistic functions. Terminology and applications for business, including spreadsheet software. Mathematical models of real-world business problems, including discrete and continuous models, that address intercultural and global challenges in such areas as finance, marketing, and accounting. Assumes 2 units of high school algebra and 1 unit of geometry. (4H,4C)

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. Assumes 2 unit of high school algebra and 1 unit of plane geometry. A student can earn credit for at most one of 1525 and 1225. A student cannot earn credit for 1525 if taken after earning credit for 1025. A student cannot earn credit for 1526 if taken after earning credit for 1026, 1114, 1226, or 2114. (3H,3C)

1535-1536: GEOMETRY AND MATHEMATICS OF DESIGN A standard first-year mathematics sequence for architecture majors. Mathematical models of real-world problems, including discrete and continuous models, that address relevant global challenges in such areas as urban planning, building construction, and home design. 1535: Euclidean geometry, trigonometry, sequences and the golden ratio, graph theory, tilings, polygons and polyhedra, applications for 2- and 3-dimensional design and construction, use of geometric software. 1536: vectors in the plane and space, descriptive and projective geometry, differential and integral calculus, applications for 2- and 3-dimensional design and construction, including areas, volumes, centroids, and optimization. Assumes 2 unites of high school algebra and 1 unit of high school geometry. (3H,3C)

1614: NUMBERS AND OPERATIONS FOR TEACHERS Study of the nature and structure of numbers for prospective elementary and middle school teachers; number theory, number systems, operations and algebraic thinking, problem solving, and mathematical modeling. 1614 may not be taken by math majors for credit. (3H,3C)

1624: GEOMETRY FOR TEACHERS Study of key geometry concepts for prospective elementary and middle school teachers; multiple perspectives including transformational, coordinate, Euclidean and analytical geometry; geometric modeling; geometric and spatial reasoning. 1624 may not be taken by math majors for credit. Pre: 1614. (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. A student can earn credit for at most one of 2024 and 2204. A student cannot earn credit for 2024 if taken after earning credit for 2214. Pre: 1026. (3H,3C)

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 MATH 1225. A student can earn credit for at most one of 2114 and 2405H. 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 MATH 1225. A student can earn credit for at most one of 2114H and 2405H. 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. A student can earn credit for at most one of 2204 and 2406H. A student can earn credit for at most one of 2024 and 2204. A student can earn credit for at most one of 2204 and CMDA 2005. 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. A student can earn credit for at most one of 2204H and 2406H. A student can earn credit for at most one of 2024 and 2204H. A student can earn credit for at most one of 2204H and CMDA 2005. Pre: 1226. (3H,3C)

2214: INTRODUCTION TO DIFFERENTIAL EQUATIONS Unified course in ordinary differential equations. First-order equations, second-and-higher-order constant coefficient linear equations, systems of first-order linear equations, and numerical methods. Mathematical models describing motion and cooling, predator-prey population models, SIR-models, mechanical vibrations, electric circuits, rates of chemical reactions, radioactive decay. Quantitative and computational thinking to address relevant intercultural and global issues. A student can earn credit for at most one of 2214 and 2406H. A student can earn credit for at most one of 2214 and CMDA 2006. Pre: (1114 or 2114 or 2114H or 2405H or ISC 2105), MATH 1226. (3H,3C)

2214H: INTRODUCTION TO DIFFERENTIAL EQUATIONS Unified course in ordinary differential equations. First-order equations, second-and-higher-order constant coefficient linear equations, systems of first-order linear equations, and numerical methods. Mathematical models describing motion and cooling, predator-prey population models, SIR-models, mechanical vibrations, electric circuits, rates of chemical reactions, radioactive decay. Quantitative and computational thinking to address relevant intercultural and global issues. A student can earn credit for at most one of 2214H and 2406H. A student can earn credit for at most one of 2214H and CMDA 2006 Pre: (1114 or 2114 or 2114H or 2405H or ISC 2105), MATH 1226. (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. A student can earn credit for at most one of 2114, 2114H, and 2405H. A student can earn credit for at most one of 2204, 2204H, and 2406H. A student can earn credit for at most one of 2214, 2214H, and 2406H. Pre: 1226 for 2405H; 2405H for 2406H. (5H,5C)

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. Two units of high school algebra, one unit of geometry, one-half unit each of trigonometry and precalculus mathematics required. 2534 may not be taken by math majors for credit without special permission. A student can earn credit for at most one of 2534 and 3034. Pre: CS 1114 or ECE 1574 or ECE 1004. (3H,3C)

2644: MATHEMATICS TUTORING Introduction to professional, culturally respectful mathematics tutoring. Development of listening and questioning skills, assessment of students mathematical difficulties. Exploration of teaching and learning processes, effectively utilizing technology, and adjusting instruction to diversity in students mathematical reasoning. Concurrent mathematics tutoring experience required. May be repeated twice with different leadership expectations for a maximum of 3 credits. Pre: 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.

2994H: UNDERGRADUATE RESEARCH Variable credit course.

3034: INTRODUCTION TO PROOFS Practice in writing mathematical proofs. Exercises from set theory, number theory, and functions. Propositional logic, set operations, equivalence relations, methods of proof, mathematical induction, the division algorithm and images and pre-images of sets. A student can earn credit for at most one of 2534 and 3034. Pre: 2114 or 2114H or 2405H. (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: MATH 2214 or MATH 2214H or MATH 2406H or CMDA 2006. (3H,3C)

3124: MODERN ALGEBRA Introduction to abstract algebraic structures (groups, rings, and fields) and structure-preserving maps (homomorphisms) for these structures. Proof-intensive course illustrating the rigorous development of a mathematical theory from initial axioms. 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: 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. Pre: (3034 or 2534), (2114 or 2114H or 2405H). (3H,3C)

3214: CALCULUS OF SEVERAL VARIABLES Fundamental calculus of functions of two or more variables. Implicit function theorem, Taylor expansion, line integrals, Greens theorem, surface integrals. Pre: 2204 or 2204H or 2406H or CMDA 2005. (3H,3C)

3224: ADVANCED CALCULUS Theory of limits, continuity, differentiation, integration, series. 3224 duplicates 4525. Pre: (2204 or 2204H or 2406H or CMDA 2005), MATH 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. A grade of C or better required in CS prerequisite 1044 or 1705. A student can earn credit for at most one of 3414 and 4404. Pre: (CS 1044 or CS 1705 or CS 1114 or CS 1124), MATH 2406H or (CMDA 2005, CMDA 2006) or (MATH 2214 or MATH 2214H), (MATH 2204 or MATH 2204H). (3H,3C)

3574: APPLIED COMPLEX VARIABLES Arithmetic of complex numbers. Geometry of the complex plane. Geometry of exponentiation and roots. Complex exponential, trigonometric and hyperbolic functions. Continuity and differentiability. Analytic and harmonic functions. Pre: 2204 or 2204H. (1H,1C)

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 or mathematics education required. (3H,3C)

4124: INTRODUCTION TO ABSTRACT ALGEBRA An introduction to the theory of groups and rings. Topics include normal subgroups, permutation groups, Sylows 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)

4175,4176: CRYPTOGRAPHY 4175: Introduction to classical and modern symmetric-key cryptography; alphabetic ciphers, block ciphers and stream ciphers; background in modular arithmetic and probability; perfect secrecy; linear and differential cryptanalysis; Advanced Encryption Standard; hashing. 4176: Introduction to modern public-key cryptography and cryptanalysis; RSA algorithm, ElGamal algorithm, Diffie-Hellman algorithm; digital signatures; background in group theory and number theory; algorithms for primality testing, factoring, and discrete logarithms; elliptic curves. Pre: (3034 or 3124 or 3134 or 3144 or 3224 or 4134 or CMDA 3605) for 4175; 4175 or CM DA 3606 or (MATH 3034, MATH 3124) or (MATH 3034, MATH 3134) or (MATH 3034, MATH 3144) or (MATH 3034, MATH 3224) or (MATH 3034, MATH 4134) or (MATH 3124, MATH 3134) or (MA TH 3124, MATH 3144) or (MATH 3124, MATH 3224) or (MATH 3124, MATH 4134) or (MATH 3134 , MATH 3144) or (MATH 3134, MATH 3224) or (MATH 3134, MATH 4134) or (MATH 3144, MATH 3224) or (MATH 3144, MATH 4134) or (MATH 3224, MATH 4134) for 4176. (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 or 2405H, 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. A student can earn credit for at most one of 3414 and 4404. 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 or 2214H or 2406H or CMDA 2006), MATH 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: 2406H or CMDA 2006 or MATH 2214 or MATH 2214H, MATH 3224 for 4425; 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: 2406H or (CMDA 2005, CMDA 2006) or (MATH 2214 or MATH 2214H), (MATH 2204 or MATH 2204H). (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. Pre: 3214. (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 or 2214H) or 2406H or CMDA 2006. (3H,3C)

4574: VECTOR AND COMPLEX ANALYSIS FOR ENGINEERS Vector Analysis: Greens theorem, potential theory, divergence, and Stokes theorem. Complex Analysis: Analyticity, complex integration, Taylor series, residues, conformal mapping, applications. 4574 may not be taken by math majors for credit. Pre: 2204 or 2204H. (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.