Mathematics
Manfred Stoll, Chair of the Department
Professors
Colin Bennett, Ph.D., University of Newcastle upon Tyne, 1971
Susanne C. Brenner, Ph.D., University of Michigan, 1988
Assistant Chair
Ronald A. DeVore, Ph.D., Ohio State University, 1967
Robert L. Sumwalt Professor of Mathematics
Stephen J. Dilworth, Ph.D., Cambridge University, 1985
Michael A. Filaseta, Ph.D., University of Illinois, 1984
Jerrold R. Griggs, Ph.D., Massachusetts Institute of Technology, 1977
Ralph E. Howard, Ph.D., California Institute of Technology, 1982
Andrew Kustin, Ph.D., University of Illinois, 1979
George F. McNulty, Ph.D., University of California, Berkeley, 1972
Matthew Miller, Ph.D., University of Illinois, 1979
Peter J. Nyikos, Ph.D., CarnegieMellon University, 1971
Konstantin Oskolkov, Ph.D., Steklov Institute, 1978
Pencho Petrushev, Ph.D., University of Sofia, 1977
James W. Roberts, Ph.D., Rutgers University, 1970
Anton R. Schep, Ph.D., University of Leiden, 1977
Graduate Director
Robert C. Sharpley, Ph.D., University of Texas, 1972
Robert M. Stephenson Jr., Ph.D., Tulane University, 1967
Manfred Stoll, Ph.D., Pennsylvania State University, 1971
László A. Székely, Ph.D., Eötvös University, 1983
Vladimir Temlyakov, Ph.D., Steklov Institute, 1981
Associate Professors
Howard S. Becker, Ph.D., University of California, Los Angeles, 1979
Daniel B. Dix, Ph.D., University of Chicago, 1988
Maria Girardi, Ph.D., University of Illinois, 1990
Peter W. Harley III, Ph.D., University of Georgia, 1966
Richard H. Hudson, Ph.D., Duke University, 1971
George W. Johnson III, Ph.D., University of Tennessee, 1971
Marek B. Kossowski, Ph.D., University of North Carolina, 1982
Douglas B. Meade, Ph.D., CarnegieMellon University, 1989
Undergraduate Director
David P. Sumner, Ph.D., University of Massachusetts, 1971
Liyeng Sung, Ph.D., State University of New York at Stony Brook, 1983
Ognian T. Trifonov, Ph.D., University of Sofia, 1990
Hong Wang, Ph.D., University of Wyoming, 1992
Xian Wu, Ph.D., Harvard University, 1986
Assistant Professors
George Androulakis, Ph.D., University of Texas, 1996
Adela Vraciu, Ph.D., University of Michigan, 2000
Gang Yu, Ph.D., University of Georgia, 2000
Senior Instructor
Mary Ellen O’Leary, M.A., University of Michigan, 1967
Instructor
Robert F. Murphy, B.S., University of Illinois, 1989
Distinguished Professors Emeriti
Thomas L. Markham, Ph.D., Auburn University, 1967
H. Edward Scheiblich, Ph.D., University of Texas, 1966
James H. Wahab, Ph.D., University of North Carolina, 1951
Faculty Emeriti
Richard R. Croxton, M.Ed., University of South Carolina, 1947
Karl H. Matthies, Dr. Rerum Naturalium, University of Freiburg, 1956
Charles A. Nicol Jr., Ph.D., University of Texas, 1954
Paul L. Sperry, Ph.D., New Mexico State University, 1963
Overview
The department offers a program leading to the degree of Bachelor of Science in mathematics and a special fiveyear 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 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 Information Technology.
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 Information Technology.
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, 142, 241 constitute the normal calculus sequence for students in the College of Science and Mathematics and the College of Engineering and Information Technology. 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 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 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 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 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 entitled "College Algebra" available from the testing service.
MATH 112: CLEP Subject Examination entitled "Trigonometry" available from the testing service.
MATH 115: CLEP Subject Examination entitled "College AlgebraTrigonometry" available from the testing service.
MATH 141: CLEP Subject Examination entitled "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.
Degree Requirements
(128 hours)
1. General Education Requirements (4354)
The following courses fulfill some of the general education requirements, as well as some of the requirements of certain cognates and minors. These courses must be completed for the B.S. degree in mathematics: MATH 141, 142, and 241 (each with a grade of C or better); CSCE 145. In addition, one of the following sequences must also be completed: (i) STAT 511 {= MATH 511} and STAT 512; (ii) either STAT 509 or STAT 515, and either STAT 516 or CSCE 146. Mathematics majors may use MATH 141, 142, and CSCE 145 to fulfill Group II of the general education requirements, MATH 511 {= STAT 511} (with a grade of C or better) for major credit, and STAT 509, 512, 515, and 516 for cognate or minor credit. Only one of STAT 509 and STAT 515 may be used for cognate or minor credit.
For an outline of other general education requirements, see "College of Science and Mathematics."
2. Major Requirements
A grade of C or better is required in each major course and in each of MATH 141, 142, and 241. Students may enroll in each major course and in each of MATH 141, 142, and 241 a maximum of two times. Either MATH 526 or MATH 544 may be used for credit but not both. The student may repeat a maximum of three mathematics courses.
General Mathematics Option
MATH 520, 544 (or 526), 546, 554, 574, plus three approved MATH electives numbered above 500, to include at least one of MATH 534, 550, 552 (2426 hours).
Applied Mathematics Option
MATH 520, 524, 526, 546, 554, 570 or 527, 574, plus one elective chosen from MATH 521, 527, 550, 552, 570, 575 (2526 hours).
Actuarial Mathematics Option^{1}
MATH 511, 520, 526 (or 544), 546, 554, 574, 570 (or 524), plus 3 hours in mathematics at 500 level (2425 hours).
A minimum of 24 hours in business administration and statistics as follows:
statistics (612 hours): STAT 512, 513, and 06 hours from STAT 510, 520 {=MGSC 520}.
business administration (1218 hours): ACCT 222, ECON 224, FINA 363 {=ECON 363}, FINA 341 or 444, and 06 hours from FINA 342, 346, 443, 444, 445, MGSC 392, 393, 520 {=STAT 520}, 594, ECON 420, 594, BADM 499. For the minor in risk management and insurance (18 hours), of the additional 6 hours, an additional 3 hours must be chosen from FINA 342, 443, 444, or 445.
computing (78 hours): CSCE 145, plus one elective from CSCE 146, MGSC 390, STAT 517.
Mathematics Education
MATH 520, 531 or 532, 544 (or 526), 546, 554, 574, 580, plus 3 additional hours chosen from 511, 531, 532, 550, 552 (2425 hours). For the cognate, students must take the required EDUC courses in the College of Education.
Intensive Major
Any major above, plus an additional four approved MATH electives numbered above 500 (3637 hours).
^{1}Pending S.C. Commission on Higher Education approval
3. Cognate or Minor (1218 hours)
See "College of Science and Mathematics"
4. Electives, see "College of Science and Mathematics"
The applied mathematics major is highly recommended for students who intend to pursue a professional career in mathematics or a mathematicsrelated discipline in government or industry. Such students are strongly encouraged to select a cognate or minor in computer science.
The actuarial mathematics major is highly recommended for students who intend to pursue a career in the actuarial profession in the insurance and financial securities industries. In addition to the required courses in business administration and statistics, students are also encouraged to choose appropriate electives from the social sciences and humanities.
The mathematics education major is intended for students planning on pursuing the fiveyear M.T. program and careers in secondary mathematics teaching.
Students intending ultimately to pursue graduate studies in mathematics or mathematicsrelated disciplines are encouraged to supplement either the general mathematics major or the applied mathematics major with courses having substantial theoretical content.
FiveYear Program
The fiveyear program of study is designed to permit outstanding mathematics students to obtain both a Bachelor of Science and a Master of Science degree in mathematics in five years. To be eligible for the program, students must have completed one of the preparatory sequences MATH 546547 or MATH 554555 by the end of the junior year and have earned 102 credit hours toward the bachelor's degree by the beginning of the senior year. Students will normally be considered for admission to the program at the end of their junior year on the recommendation of their undergraduate advisor. It is expected that such students will have attained a 3.50 overall GPA and a 3.50 GPA on all mathematics courses taken.
Upon admission to the program, students will be eligible for financial assistance from the department. During the fall semester of the senior year, students will receive consideration for employment as undergraduate assistants, and during the spring semester of the senior year, they will receive consideration for quartertime graduate assistantships. In the fifth year and the first and second summer of graduate study, students will be eligible for halftime graduate assistantships.
Cognate or Minor for Nonmajors
Students with majors in other departments may effectively supplement their major program of study by selecting a cognate or minor in mathematics.
Cognate in Mathematics. Most courses in mathematics numbered 241 and above may be used for cognate credit.
Minor in Mathematics. The minor consists of MATH 241 together with at least 15 hours of mathematics courses selected from MATH 242 or 500level MATH courses. At least 6 of the 15 hours must be chosen from MATH 520, 526, 544, 546, 554, 574. At most, one of MATH 526, 544 may be used for minor credit.
Minor in Actuarial Mathematics and Statistics. The minor consists of the prerequisite courses MATH 141, 142, 241 plus 18 hours of mathematics and statistics courses chosen as follows: MATH 511 {=STAT 511}, STAT 512, 513, one of STAT 510 and 520, one of MATH 526 and 544, and one of MATH 570 and 574.
Course Descriptions (MATH)
 111  Basic College Mathematics. (3) (Prereq: qualification through placement) Basic college algebra; linear and quadratic equations, inequalities, functions and graphs of functions, exponential and logarithm functions, systems of equations. Credit may not be received for both MATH 111 and 115.
 112  Trigonometry. (2) (Prereq: qualification through placement or a grade of C or better in MATH 111) Topics in trigonometry specifically needed for MATH 141, 142, 241. Circular functions, analytic trigonometry, applications of trigonometry. Credit may not be received for both MATH 112 and 115.
 115  Precalculus Mathematics. (4) (Prereq: qualification through placement) Topics in algebra and trigonometry specifically needed for MATH 141, 142, 241. Subsets of the real line, absolute value; polynomial, rational, inverse, logarithmic, exponential functions; circular functions; analytic trigonometry. Credit may not be received for both MATH 111 and 115 or both MATH 112 and 115.
 122  Calculus for Business Administration and Social Sciences. (3) (Prereq: qualification through placement or a grade of C or better in MATH 111 or 115) Derivatives and integrals of elementary algebraic, exponential, and logarithmic functions. Maxima, minima, rate of change, motion, work, area under a curve, and volume.
 141  Calculus I. (4) (Prereq: qualification through placement or a grade of C or better in MATH 112 or 115) Limits, continuity; derivatives, chain rule, rates of change, curve sketching, maxmin problems; definite integral, antiderivatives, and the Fundamental Theorem.
 142  Calculus II. (4) (Prereq: qualification through placement or a grade of C or better in MATH 141) Techniques of integration, exponential, and inverse trigonometric functions; numerical methods, and applications of the integral; sequences, power and Taylor series.
 151  Calculus Workshop I. (2) (Coreq: MATH 141) Small study group practice in applications of calculus. For elective credit only. Two 2hour sessions per week.
 152  Calculus Workshop II. (2) (Coreq: MATH 142) Small study group practice in applications of calculus. For elective credit only. Two 2hour sessions per week.
 170  Finite Mathematics. (3) (Prereq: qualification through placement or a grade of C or better in MATH 111 or 115) Elementary matrix theory; systems of linear equations; permutations and combinations; probability and Markov chains; linear programming and game theory.
 172  Mathematical Modeling for the Life Sciences. (3) Modeling with difference equations; vectors, trigonometry, polar coordinates, matrices, eigenvalues and eigenvectors; addition and multiplication in combinatorics, permutations, combinations, introduction to probability theory (discrete, continuous); techniques of integration, symmetry. Credit may not be received for both MATH 172 and either MATH 170 or 174.
 174  Discrete Mathematics for Computer Science. (3) (Prereq: qualification through placement or a grade of C or better in MATH 112 or 115) Induction, complexity, elementary counting, combinations and permutations, recursion and recurrence relations, graphs and trees; discussion of the design and analysis of algorithmswith emphasis on sorting and searching.
 221  Basic Concepts of Elementary Mathematics I. (3) (Prereq: qualification through placement or a grade of C or better in MATH 111 or 115) 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.
 222  Basic Concepts of Elementary Mathematics II. (3) (Prereq: MATH 221) Informal geometry and basic concepts of algebra. Open only to students in elementary or early childhood teacher certification.
 241  Vector Calculus. (3) (Prereq: qualification through placement or a grade of C or better in MATH 142) Vector algebra, geometry of threedimensional space; lines, planes, and curves in space; polar, cylindrical, and spherical coordinate systems; partial differentiation, maxmin theory; multiple and iterated integration, line integrals, and Green's theorem in the plane.
 242  Elementary Differential Equations. (3) (Prereq: qualification through placement or a grade of C or better in MATH 142) 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.
 300  Transition to Advanced Mathematics. (3) (Prereq: MATH 142) Rigor of mathematical thinking and proof writing via logic, sets, and functions. Intended to bridge the gap between lowerlevel (computationalbased) and upperlevel (proofbased) mathematics courses.
 374  Discrete Structures. (3) (Prereq: MATH 142 and CSCE 146) Propositional and predicate logic; proof techniques; recursion and recurrence relations; sets, combinatorics, and probability; functions, relations, and matrices; algebraic structures.
 399  Independent Study. (39) Contract approved by instructor, advisor, and department chair is required for undergraduate students.
 401  Conceptual History of Mathematics. (3) (Prereq: MATH 122, or 141, or consent of the department) Topics from the history of mathematics emphasizing the 17th century to the present. Various mathematical concepts are discussed and their development traced. For elective or Group II credit only.
 511  Probability. {= STAT 511} (3) (Prereq: MATH 241 with a grade of C or higher) 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.
 520  Ordinary Differential Equations. (3) (Prereq: MATH 544 or 526; or consent of department) Differential equations of the first order, linear systems of ordinary differential equations, elementary qualitative properties of nonlinear systems.
 521  Boundary Value Problems and Partial Differential Equations. (3) (Prereq: MATH 520 or 241 and 242) Laplace transforms, twopoint 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.
 522  Wavelets. (3) (Prereq: MATH 544 or 526 or consent of department) Basic principles and methods of Fourier transforms, wavelets, and multiresolution analysis; applications to differential equations, data compression, and signal and image processing; development of numerical algorithms. Computer implementation.
 524  Nonlinear Optimization. (3) (Prereq: MATH 526 or 544 or consent of department) Descent methods, conjugate direction methods, and QuasiNewton algorithms for unconstrained optimization; globally convergent hybrid algorithm; primal, penalty, and barrier methods for constrained optimization. Computer implementation of algorithms.
 525  Mathematical Game Theory. (3) (Prereq: MATH 526 or 544) Twoperson zerosum games, minimax theorem, utility theory, nperson games, market games, stability.
 526  Numerical Linear Algebra. (4) (Prereq: MATH 241) Matrix algebra, Gauss elimination, iterative methods; overdetermined systems and least squares; eigenvalues, eigenvectors; numerical software. Computer implementation. Three lectures and one laboratory hour per week. Credit may not be received for both MATH 526 and MATH 544.
 527  Numerical Analysis. {=CSCE 561} (3) (Prereq: MATH 242 or 520) 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.
 531  Foundations of Geometry. (3) (Prereq: MATH 241) 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 nonEuclidean geometry.
 532  Modern Geometry. (3) (Prereq: MATH 241) Projective geometry, theorem of Desargues, conics, transformation theory, affine geometry, Euclidean geometry, nonEuclidean geometries, and topology.
 533  Elementary Geometric Topology. (3) (Prereq: MATH 241) Topology of the line, plane, and space, Jordan curve theorem, Brouwer fixed point theorem, Euler characteristic of polyhedra, orientable and nonorientable surfaces, classification of surfaces, network topology.
 534  Elements of General Topology. (3) (Prereq: MATH 241) Elementary properties of sets, functions, spaces, maps, separation axioms, compactness, completeness, convergence, connectedness, path connectedness, embedding and extension theorems, metric spaces, and compactification.
 540  Modern Applied Algebra. (3) (Prereq: MATH 241) Finite structures useful in applied areas. Binary relations, Boolean algebras, applications to optimization, and realization of finite state machines.
 541  Algebraic Coding Theory. (3) (Prereq: MATH 526 or MATH 544 or consent of department) Errorcorrecting codes, polynomial rings, cyclic codes, finite fields, BCH codes.
 544  Linear Algebra. (3) (Prereq: MATH 241) Matrix algebra, solution of linear systems; notions of vector space, independence, basis, dimension; linear transformations, change of basis; eigenvalues, eigenvectors, Hermitian matrices, diagonalization; CayleyHamilton theorem. Credit may not be received for both MATH 526 and MATH 544.
 546  Algebraic Structures I. (3) (Prereq: MATH 241) Permutation groups; abstract groups; introduction to algebraic structures through study of subgroups, quotient groups, homomorphisms, isomorphisms, direct product; decompositions; introduction to rings and fields.
 547  Algebraic Structures II. (3) (Prereq: MATH 546) 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).
 550  Vector Analysis. (3) (Prereq: MATH 241) 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.
 551  Introduction to Differential Geometry. (3) (Prereq: MATH 241) Parametrized curves, regular curves and surfaces, change of parameters, tangent planes, the differential of a map, the Gauss map, first and second fundamental forms, vector fields, geodesics, and the exponential map.
 552  Applied Complex Variables. (3) (Prereq: MATH 241) Complex integration, calculus of residues, conformal mapping, Taylor and Laurent Series expansions, applications.
 554  Analysis I. (3) (Prereq: MATH 241) Least upper bound axiom, the real numbers, compactness, sequences, continuity, uniform continuity, differentiation, Riemann integral and fundamental theorem of calculus.
 555  Analysis II. (3) (Prereq: MATH 554 or consent of department) RiemannStieltjes integral, infinite series, sequences and series of functions, uniform convergence, Weierstrass approximation theorem, selected topics from Fourier series or Lebesgue integration.
 561  Introduction to Mathematical Logic. (3) (Prereq: MATH 241) 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 LowenheimSkolemTarski theorem; Beth's definability theorem; effectively computable functions; Godel's incompleteness theorem; undecidable theories.
 562  Theory of Computation. {=CSCE 551} (3) (Prereq: CSCE 350 or MATH 526 or 544 or 574) Basic theoretical principles of computer science as modeled by formal languages and automata; computability and computational complexity. Major credit may not be received for both CSCE 355 and CSCE 551.
 570  Discrete Optimization. (3) (Prereq: MATH 526 or 544) Discrete mathematical models. Applications to such problems as resource allocation and transportation. Topics include linear programming, integer programming, network analysis, and dynamic programming.
 574  Discrete Mathematics I. (3) (Prereq: MATH 142) Mathematical models; mathematical reasoning; enumeration; induction and recursion; tree structures; networks and graphs; analysis of algorithms.
 575  Discrete Mathematics II. (3) (Prereq: MATH 574) A continuation of MATH 574. Inversion formulas; Polya counting; combinatorial designs; minimax theorems; probabilistic methods; Ramsey theory; other topics.
 576  Combinatorial Game Theory. (3) (Prereq: MATH 526, 544, or 574) Winning in certain combinatorial games such as Nim, Hackenbush, and Domineering. Equalities and inequalities among games, SpragueGrundy theory of impartial games, games which are numbers.
 580  Elementary Number Theory. (3) (Prereq: MATH 241) Divisibility, primes, congruences, quadratic residues, numerical functions. Diophantine equations.
 587  Introduction to Cryptography. {=CSCE 557} (3) (Prereq: CSCE 145, MATH 241, and either CSCE 355 or MATH 574) 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. Codebreaking techniques. Cryptographic protocols.
 599  Topics in Mathematics. (13) Recent developments in pure and applied mathematics selected to meet current faculty and student interest.
 650  AP Calculus for Teachers. (3) (Prereq: current secondary high school teacher certification in mathematics and at least 6 hours of calculus) 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.)
College of Science and Mathematics
