Mathematics
Linyuan Lu, Chair
The department offers a program leading to the degree of Bachelor of Science in mathematics and a special five-year program leading to a Bachelor of Science degree and a Master of Science degree in mathematics. In addition, the department serves many of the disciplines within the University through course offerings which provide basic mathematical skills necessary to the pursuit of studies in these disciplines.
General Mathematics Courses
MATH 111 is a course in basic mathematics intended for students who plan to take MATH 122 or MATH 170 and who need more thorough development in algebraic methods.
MATH 111I is an intensive version of MATH 111. This course is intended for students who plan to take MATH 122 or MATH 170 and desire additional support—in the form of smaller classes and more contact hours—to develop the necessary algebraic skills.
MATH 112 is the basic trigonometry course for students who plan to take MATH 141 and have adequate preparation in algebra but need more thorough development in trigonometry. This course may not be used for mathematics credit in the College of Engineering and Computing.
MATH 115 is the basic precalculus course for students who plan to take MATH 141 and need more thorough development in algebra and trigonometry before entering MATH 141. This course may not be used for mathematics credit in the College of Engineering and Computing.
MATH 122 is intended for students in business, the social sciences, pharmacy, and other disciplines which require an introduction to computational mathematics and calculus and is open to all interested students who satisfy the general requirements listed below.
MATH 141, MATH 142, MATH 241 constitute the normal calculus sequence for students in the College of Arts and Sciences and the College of Engineering and Computing. These courses are open to all students who satisfy the general requirements listed below.
MATH 170 is a basic course in finite mathematics. It may be used to satisfy the University’s core requirements and is open to all interested students who satisfy the general requirements listed below.
Freshman Placement in Mathematics
MATH 111: Qualification through placement.
MATH 111I: Qualification through placement.
MATH 112: Qualification through placement or credit for MATH 111, either by successful completion of the course with a grade of C or better, transfer credit from another university, or successful completion of the test in MATH 111, available from the testing service.
MATH 115: Qualification through placement.
MATH 122: Qualification through placement or credit for MATH 111, either by successful completion of the course with a grade of C or better, transfer credit from another university, or successful completion of the test in MATH 111, available from the testing service.
MATH 141: Qualification through placement or credit for MATH 112 or MATH 115, either by successful completion of the course with a grade of C or better, transfer credit from another university, or successful completion of the test in MATH 115, available from the testing service.
Students who do not qualify for MATH 141 under paragraph 1 are strongly encouraged to try to obtain credit for MATH 115 either by taking the course or the examination during the summer preceding their first fall semester.
MATH 170: Qualification through placement or credit for MATH 111 or MATH 115, either by successful completion of the course with a grade of C or better, transfer credit from another university, or successful completion of the test in MATH 111 or MATH 115 which is available from the testing service.
Incoming students who wish to obtain bypass credit for certain mathematics courses may do so as follows:
MATH 111: CLEP Subject Examination titled “College Algebra” available from the testing service.
MATH 112: CLEP Subject Examination titled “Trigonometry” available from the testing service.
MATH 115: CLEP Subject Examination titled “College Algebra-Trigonometry” available from the testing service.
MATH 141: CLEP Subject Examination titled “Calculus with Analytic Geometry” available from the testing service.
Advanced Placement Test in Mathematics: The Advanced Placement Mathematics tests may be used to gain credit and advanced placement in calculus. Information is available from the testing service.
Courses
Basic college algebra; linear and quadratic equations, inequalities, functions and graphs of functions, exponential and logarithm functions, systems of equations.
An intensive treatment of the topics covered in MATH 111.
Topics in algebra and trigonometry specifically needed for MATH 141, MATH 142, MATH 241. Subsets of the real line, absolute value; polynomial, rational, inverse, logarithmic, exponential functions; circular functions; analytic trigonometry.
Essential algebra and trigonometry topics for Calculus, including working with equations that involve polynomials, rational functions, exponential and logarithmic functions, and trigonometric and inverse trigonometric functions. Intended for students with prior experience in Precalculus, but not ready for MATH 141.
Derivatives and integrals of elementary algebraic, exponential, and logarithmic functions. Maxima, minima, rate of change, motion, work, area under a curve, and volume.
Carolina Core: ARP
Functions, limits, derivatives, introduction to integrals, the Fundamental Theorem of Calculus, applications of derivatives and integrals. Four classroom hours and one laboratory hour per week.
Carolina Core: ARP
Methods of integration, sequences and series, approximations. Four classroom hours and one laboratory hour per week.
Carolina Core: ARP
Small study group practice in applications of calculus. For elective credit only.
Elementary matrix theory; systems of linear equations; permutations and combinations; probability and Markov chains; linear programming and game theory.
Carolina Core: ARP
Biological modeling with differential and difference equations; techniques of model modifications; analytic, numerical, and graphical solution methods; equilibria, stability, and long-term system behavior; geometric series; vectors, matrices, eigenvalues, and eigenvectors. Applications principally to population dynamics and compartment models.
Carolina Core: ARP
Logic, number theory, sequences, series, recursion, mathematical induction, set theory, enumeration, functions, relations, graphs and trees. Connections to computers and to programming are emphasized when possible.
Carolina Core: ARP
An overview of different areas of mathematical research and career opportunities for mathematics majors. Pass/fail only.
Graduation with Leadership Distinction: GLD: Research
The meaning of number, fundamental operations of arithmetic, the structure of the real number system and its subsystems, elementary number theory. Open only to students in elementary or early childhood teacher certification.
Informal geometry and basic concepts of algebra. Open only to students in elementary or early childhood teacher certification.
Vector algebra, geometry of three-dimensional space; lines, planes, and curves in space; polar, cylindrical, and spherical coordinate systems; partial differentiation, max-min theory; multiple and iterated integration, line integrals, and Green’s theorem in the plane.
Ordinary differential equations of first order, higher order linear equations, Laplace transform methods, series methods; numerical solution of differential equations. Applications to physical sciences and engineering.
Rigor of mathematical thinking and proof writing via logic, sets, and functions. Intended to bridge the gap between lower-level (computational-based) and upper-level (proof-based) mathematics courses.
General solutions of systems of linear equations, vector spaces and subspaces, linear transformations, determinants, orthogonality, characteristic polynomials, eigenvalues and eigenvectors, singular value decomposition, and generalized inverse. MATH 344L is an optional laboratory course where additional applications will be discussed.
Computer based applications of linear algebra for science and engineering students. Topics include numerical analysis of matrices, direct and indirect methods for solving linear systems, and least squares method (regression). Typical applications include practical issues related to discrete Markov processes, image compression, and linear programming. Credit not allowed for both MATH 344L and 544L.
Contract approved by instructor, advisor, and department chair is required for undergraduate students.
Graduation with Leadership Distinction: GLD: Research
Academic counterpart to a professional work experience in which mathematics plays a central role. Introduction to the uses of problem formulation and problem solving in a working environment. Introduction to career possibilities for a student trained in mathematics. Restricted to MATH major with 3.0 or better GPA and completion of at least 60 credits.
Research on a specific mathematical subject area. The specific content of the research project must be outlined in a proposal that must be approved by the instructor and the Undergraduate Director. Intended for students pursuing the B.S. in Mathematics with Distinction. Pass-Fail grading only.
Graduation with Leadership Distinction: GLD: Research
Probability and independence; discrete and continuous random variables; joint, marginal, and conditional densities, moment generating functions; laws of large numbers; binomial, Poisson, gamma, univariate, and bivariate normal distributions.
Cross-listed course: STAT 511
Probability spaces. Random variables. Mean and variance. Geometric Brownian Motion and stock price dynamics. Interest rates and present value analysis. Pricing via arbitrage arguments. Options pricing and the Black-Scholes formula.
Cross-listed course: STAT 522
Convex sets. Separating Hyperplane Theorem. Fundamental Theorem of Asset Pricing. Risk and expected return. Minimum variance portfolios. Capital Asset Pricing Model. Martingales and options pricing. Optimization models and dynamic programming.
Cross-listed course: STAT 523
Laplace transforms, two-point boundary value problems and Green’s functions, boundary value problems in partial differential equations, eigenfunction expansions and separation of variables, transform methods for solving PDE’s, Green’s functions for PDE’s, and the method of characteristics.
Applications of differential and difference equations and linear algebra modeling the dynamics of populations, with emphasis on stability and oscillation. Critical analysis of current publications with computer simulation of models.
Descent methods, conjugate direction methods, and Quasi-Newton algorithms for unconstrained optimization; globally convergent hybrid algorithm; primal, penalty, and barrier methods for constrained optimization. Computer implementation of algorithms.
Matrix algebra, Gauss elimination, iterative methods; overdetermined systems and least squares; eigenvalues, eigenvectors; numerical software. Computer implementation. Credit may not be received for both MATH 526 and MATH 544. Three lectures and one laboratory hour per week.
Interpolation and approximation of functions; solution of algebraic equations; numerical differentiation and integration; numerical solutions of ordinary differential equations and boundary value problems; computer implementation of algorithms.
Cross-listed course: CSCE 561
The study of geometry as a logical system based upon postulates and undefined terms. The fundamental concepts and relations of Euclidean geometry developed rigorously on the basis of a set of postulates. Some topics from non-Euclidean geometry.
Projective geometry, theorem of Desargues, conics, transformation theory, affine geometry, Euclidean geometry, non-Euclidean geometries, and topology.
Finite structures useful in applied areas. Binary relations, Boolean algebras, applications to optimization, and realization of finite state machines.
Vectors, vector spaces, and subspaces; geometry of finite dimensional Euclidean space; linear transformations; eigenvalues and eigenvectors; diagonalization. Throughout there will be an emphasis on theoretical concepts, logic, and methods. MATH 544L is an optional laboratory course where additional applications will be discussed.
Computer-based applications of linear algebra for mathematics students. Topics include numerical analysis of matrices, direct and indirect methods for solving linear systems, and least squares method (regression). Typical applications include theoretical and practical issues related to discrete Markov processes, image compression, and linear programming. Credit not allowed for both MATH 344L and 544L.
Rings, ideals, polynomial rings, unique factorization domains; structure of finite groups; topics from: fields, field extensions, Euclidean constructions, modules over principal ideal domains (canonical forms).
Vector fields, line and path integrals, orientation and parametrization of lines and surfaces, change of variables and Jacobians, oriented surface integrals, theorems of Green, Gauss, and Stokes; introduction to tensor analysis.
Complex integration, calculus of residues, conformal mapping, Taylor and Laurent Series expansions, applications.
Least upper bound axiom, the real numbers, compactness, sequences, continuity, uniform continuity, differentiation, Riemann integral and fundamental theorem of calculus.
Riemann-Stieltjes integral, infinite series, sequences and series of functions, uniform convergence, Weierstrass approximation theorem, selected topics from Fourier series or Lebesgue integration.
Syntax and semantics of formal languages; sentential logic, proofs in first order logic; Godel’s completeness theorem; compactness theorem and applications; cardinals and ordinals; the Lowenheim-Skolem-Tarski theorem; Beth’s definability theorem; effectively computable functions; Godel’s incompleteness theorem; undecidable theories.
Mathematical models; mathematical reasoning; enumeration; induction and recursion; tree structures; networks and graphs; analysis of algorithms.
Divisibility, primes, congruences, quadratic residues, numerical functions. Diophantine equations.
Design of secret codes for secure communication, including encryption and integrity verification: ciphers, cryptographic hashing, and public key cryptosystems such as RSA. Mathematical principles underlying encryption. Code-breaking techniques. Cryptographic protocols.
Cross-listed course: CSCE 557
A review of literature in specific subject areas involving student presentations. Content varies and will be announced in the Master Schedule of Classes by title. For undergraduate credit only.
Recent developments in pure and applied mathematics selected to meet current faculty and student interest.
This course is designed for middle-level pre-service mathematics teachers. This course covers geometric reasoning, Euclidean geometry, congruence, area, volume, similarity, symmetry, vectors, and transformations. Dynamic software will be utilized to explore geometry concepts. This course cannot be used for credit toward a major in mathematics.
This course introduces basic concepts in number theory and modern algebra that provide the foundation for middle level arithmetic and algebra. Topics include: algebraic reasoning, patterns, inductive reasoning, deductive reasoning, arithmetic and algebra of integers, algebraic systems, algebraic modeling, and axiomatic mathematics. This course cannot be used for credit towards a major in mathematics.
A thorough study of the topics to be presented in AP calculus, including limits of functions, differentiation, integration, infinite series, and applications. Not intended for degree programs in mathematics.