### Mathematics

**Chair:** Peter Haskell

**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, C. A. Beattie, J. Borggaard, 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, M. Shimozono, S. Sun, and J. Turner

**Associate Professors:** S. Ciupe, A. Elgart, N. Loehr, C. Mihalcea, H. Mortveit, A. Norton, P. Wapperom, M. Wawro, P. Yue, and L. Zietsman

**Assistant Professors:** L. Childs, J. Chung, M. Chung, E. Johnson, H. Liu, D. Orr, and E. Palsson

**Patricia Ann Caldwell Post-Doctoral Fellow and Visiting Assistant Professor:** J. Fillman

**Senior Instructors:** D. Agud, S. Anderson, T. A. Bourdon, S. Hagen, and C. Stephens

**Advanced Instructors:** J. Clemons, H. Hart, J. Hurdus, M. P. McQuain, E. Savel'ev, J. Schmale, and J. Wilson

**Instructors:** R. Arnold, T. Asfaw, E. Chang, M. Chung, S. Farmer, W. Galinaitis, N. Gildersleeve, M. Heitzman, E. Jasso Hernandez, S. Mason, T. McIntee, S. McIntyre, E. Rappold, N. Robbins, E. Saenz Maldonado, E. Ufferman, and S. Yasuda

**Lecturers:** E. Adkins and A. Sibol

**Career Advisor:** S. Ciupe

**Scholarship Chair:** J. Kim

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 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** or selections from CMDA 3605, 3606, and 4604. 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 General Education (Curriculum for Liberal Education) (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 http://www.registrar.vt.edu/graduation/checksheets/index.html.

### Undergraduate Course Descriptions (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)

### Undergraduate Course Descriptions (MATH)

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)

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.
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 Simpsonâs 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 (minimum grade of C-) for 1226
(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. 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)

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 Geometerâs
Sketchpad, Logo programming language, and other geometry
based software.
(4H,4C)

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 or MA
TH 2405H).
(2H,2C)

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)

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.
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.
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 1226).
(3H,3C)

2214H: 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 1226).
(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)

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)

2644: MATHEMATICS TUTORING

An introduction to mathematics tutoring. Course activities
include the development of listening and questioning skills,
assessment of a studentâs 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.
Pre: Grade of C or better in 2114 or 2114H or 2405H.
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

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: 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,
Greenâs theorem, surface integrals.
Pre: 2224 or 2224H or 2204 or 2204H or 2406H or CMDA 2005.
(3H,3C)

3224: ADVANCED CALCULUS

Theory of limits, continuity, differentiation, integration,
series. 3224 duplicates 4525.
Pre: (2224 or 2224H or 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.
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 2406H or (CMDA 2005, CMDA 2006)
or (MATH 2214 or MATH 2214H), (MATH 2224 or MATH 2224H or MATH 2204 or MATH 2204H).
(3H,3C)

3574: APPLIED COMPLEX VARIABLES

Arithmetic of complex numbers. Geometry of the complex
plane. Geomety of exponentiation and roots. Complex
exponential, trigonometric and hyperbolic functions.
Continuity and differentiability. Analytic and harmonic
functions.
Pre: 2204 or 2204H or 2224 or 2224H.
(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 required.
(3H,3C)

4124: INTRODUCTION TO ABSTRACT ALGEBRA

An introduction to the theory of groups and rings. Topics
include normal subgroups, permutation groups, Sylowâs
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: 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 or 2405H), (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 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 2005, CMDA 2006) or (MATH 2214 or MATH 2214H), (MATH 2224 or MATH
2224H or MATH 2204 or MATH 2204H), 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 2224 or MATH
2224H) or (MATH 2204 or MATH 2204H) for 4445; 2406H or (CMDA 2005, CMDA 2006) or (MA
TH 2214 or MATH 2214H), (MATH 2224 or MATH 2224H) for 4446.
(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: Greenâs theorem, potential theory,
divergence, and Stokesâ theorem. Complex Analysis:
Analyticity, complex integration, Taylor series, residues,
conformal mapping, applications.
Pre: 2224 or 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.