1981-1982
Senate Document #24
APC Document #21
Redefinition of Major/Change in Major
Requirements/Renumbering of Courses/Additional Courses
[4mDelete[0m: All [4mCatalog[0m material describing the mathematics
majors and mathematics courses found on pages 114-117.
[4mAdd[0m: The text on the following pages of this memorandum
[4mRationale[0m: The present mathematics major was defined several years
ago. Over the ensuing years, courses have been added and deleted
and course descriptions have been changed in a piecemeal fashion.
As a result the logic of the course numbering has been lost.
In this proposal we have redefined the major in line with what
the current mathematics faculty perceives as a solid foundation in
mathematics while allowing the student a great deal more
flexibility in choosing courses than was possible in the past.
There are greater numbers of students with a wider variety of
interests enrolled at UNCA than when the present major was defined.
New programs and tracks have necessitated the addition of new
courses to the curriculum. In restructuring the course offerings,
some courses have been changed in either number or content, some in
both. Some of the new courses are a blend of contents of old
courses, some are brand new and have never been offered at UNCA
before. The proposed changes are the result of a year-long study
by the department. We think that they are long overdue.
Page 2
MATHEMATICS
The mathematics major is designed to give the student a
substantial foundation in mathematics, while allowing enough
flexibility that each student, in consultation with his advisor,
may design a program of study tailored to his individual interests
and competencies. This allows for programs of study to be geared
toward different areas of secialization (e.g. pure mathematics,
applied mathematics or mathematics education).
In order to help each student prepare an optimal program of
study to meet his individual needs, the department requires that
upon the declaration of a major in mathematics, each student, in
consultation with his advisor, must present a proposed program of
study for approval by the department's Curriculum Committee. Any
subsequent changes in a program proposal must be approved by both
the advisor and the Curriculum Committee. The department
encourages each student to take an active role in the design of his
program.
Program A: Mathematics
I. Required courses in the major: 191, 192, 280, 291:
18 semester hours in courses numbered beyond 300,
including 361, 365, and 491. Completion of at least
one of the following two-course sequences: 361-362,
365-366, 431-434, 491-492, or Statistics 425-426.
II. Required courses outside the major: At least one of
the following two-course sequences: Chemistry 141-142,
Economics 200-201, Physics 221-222, or Statistics 325-
327. Some course work in Computer Science is strongly
recommended.
III. Foreign Language Requirement: competency on the
intermediate level in a classical or modern language.
French or German is strongly recommended.
IV. Other departmental requirements: A demonstration of
Competency examination in the senior year.
Program B: The Teaching of Mathematics
Students seeking secondary certification must complete the above
requirements and must choose 332 as one of the upper level courses.
In addition, they must complete Computer Science 141 and 243.
Page 3
[4mMATH 131[0m Mathematics for Elementary School Teachers (3)
An intuitive development of the real number system emphasizing
set theory, arithmetic properties of the real numbers, topics from
number theory, modular arithmetic, and analysis of basic algebraic
structures. Prerequisite: Developmental Studies 105 or
departmental approval.
[4mMATH 132[0m Geometry for Elementary School Teachers (3)
A continuation of MATH 131. Designed for elementary and
middle school teachers. Includes both an intuitive and a deductive
study of points, lines, planes, curves, surfaces, parallelism,
similarity, as well as linear, angular, area, and volume
measurement. Prerequisite: MATH 131.
[4mMATH 151[0m Finite Mathematics (3)
Content selected from symbolic logic, elementary set theory,
functions and relations, matrix arithmetic, linear functions, and
linear programming, probability models, and descriptive statistics.
Prerequisite: Developmental Studies 105 or departmental approval.
[4mMATH 161[0m Intermediate Algebra (3)
A comprehensive study of intermediate algebra and topics in
geometry and analytic geometry. Suggested as preparatory for MATH
163. Does not count toward the foreign language requirement.
Prerequisite: Developmental Studies 105 or departmental approval.
[4mMATH 163[0m College Algebra (3)
A study of linear, quadratic, exponential, logarithmic, and
rational functions and relations and their graphs. Prerequisite:
MATH 161 or three years of high school algebra and geometry and a
satisfactory performance on the departmental placement exam.
[4mMATH 164[0m Trigonometry (3)
Circular functions, identities, equations, complex numbers,
polar coordinates, and applications. Prerequisite: MATH 163 or
satisfactory performance on the departmental placement exam.
[4mMATH 191[0m Calculus I (4)
An introduction to limits, continuity, derivatives, and
integrals, with emphasis on the calculus of polynomial, rational,
and algebraic functions. A balanced presentation of the theory and
application of differential and integral calculus. Prerequisite:
MATH 163 or consent of the department and a satisfactory
performance on the departmental placement exam.
[4mMATH 192[0m Calculus II (4)
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The calculus of exponential, logarithmic, and trigonometric
functions, the Mean Value Theorem, indeterminate forms, improper
integrals, and infinite series. Prerequisite: MATH 191.
[4mMATH 266[0m Matrix Algebra with Applications (3)
Basic concepts of matrix algebra; the Gauss-Jordon method,
determinants, and inverses; linear systems of equations; linear
programming; and Markov chains. Prerequisite: MATH 163 or MATH
151.
[4mMATH 280[0m Introduction to the Foundations of Mathematics (3)
Set theory, functions and relations, the structure of the real
number system, deductive logic and the nature of proof, and
axiomatic systems. Corequisite: MATH 192.
[4mMATH 291[0m Calculus III (4)
Functions of several variables, with emphasis on partial
differentiation and multiple integrals. An introduction to vector
analysis, line and surface integrals, Green's Theorem, and Stokes'
Theorem. Prerequisite: MATH 192.
[4mMATH 332[0m Geometry (3)
Euclidean geometry and the fifth postulate. Hyperbolic and
elliptic geometries, the consistency of non-Euclidean geometries,
and projective geometry. Prerequisite: MATH 291.
[4mMATH 361[0m Modern Algebra I (3)
A study of the integers, integral domains, groups, rings, and
fields. Prerequisite: MATH 280 and MATH 291 or permission of the
department.
[4mMATH 362[0m Modern Algebra II (3)
A continuation of the study of topics in MATH 361.
Prerequisite: MATH 361.
[4mMATH 365[0m Linear Algebra I (3)
Vector spaces matrices and linear transformations.
Prerequisites: MATH 280 and MATH 291 or permission of the
department.
[4mMATH 366[0m Linear Algebra II (3)
A continuation of the study of topics in MATH 365.
Prerequisite: MATH 365.
[4mMATH 391[0m Advanced Calculus (3)
Page 5
Vector analysis and selected topics in calculus, especially
partial differential equations; gamma, beta, and Bessel functions;
and tensor analysis. The course has particular relevance for
students majoring in chemistry, engineering, physics, and applied
mathematics. Prerequisite: MATH 291.
[4mMATH 394[0m Differential Equations (3)
Existence and uniqueness of solutions of differential
equations; separable, homogeneous, and exact equations; the LaPlace
transform; elementary numerical and infinite series methods;
Fourier series; and various applications. Corequisite: MATH 291.
[4mMATH 392[0m Complex Variables (3)
Complex numbers and their geometrical representation, analytic
functions of a complex variable, integration, power series, and the
calculus of residues. Prerequisite: MATH 291.
[4mMATH 431[0m Topology I (3)
Metric spaces, topological spaces, separation axioms,
connectedness and compactness. Prerequisite: MATH 280 and MATH
291 or permission of department.
[4mMATH 432[0m Topology II (3)
A continuation of the study of topics in MATH 431.
Prerequisite: MATH 431.
[4mMATH 452[0m Introduction to Mathematical Models (3)
Content selected from difference equations, graph theory, game
theory, social choice and utility theory, and stochastic processes.
Prerequisite: MATH 266 and MATH 291 or consent of the department.
[4mMATH 462[0m Theory of Numbers (3)
Divisibility, prime numbers, congruences, linear and
non-linear Diophantine equations, quadratic residues,
representations as sums, and continued fractions. Prerequisite:
MATH 361.
[4mMATH 491[0m Analysis I (3)
The real number system, limits, sequences and functions,
continuity, derivatives, mean value theorems, and integration.
Prerequisite: MATH 280 and MATH 291 or permission of the
department.
[4mMATH 492[0m Analysis II (3)
A continuation of the study of topics in MATH 491.
Prerequisite: MATH 491.
Page 6
[4mSTAT 125[0m Introduction to Statistics and Data Analysis (4)
Basic probability theory, descriptive statistics, distribution
of random variables, simple statistics, hypothesis testing and
confidence intervals, analysis of variances, contengency table
analysis, correlation and regression, and methods of data analysis.
Prerequisite: Developmental Studies 105 or approval of the
instructor.
[4mSTAT 225[0m Intermediate Statistics and Data Analysis (4)
Organization and display of data; measures of central tendency
and dispersion; alternative formulations of probability;
distributions of random variables; the Central Limit Theorem;
statistical inference, confidence intervals, and hypothesis tests;
contingency table analysis; analysis of variances; and linear
correlation and regression. Prerequisite: MATH 192.
[4mSTAT 228[0m Exploratory Data Analysis (3)
Box plots, stem-and-leaf dislays, re-expressions, median
polish, smoothing, and robust regression. Data analysis will
entail the use of interactive APL programs. Prerequisite: STAT
125 or STAT 225.
[4mSTAT 325[0m Introduction to Linear Models (3)
The mathematics of linear relationships, least squares and
maximum likelihood estimation procedures, analysis of residuals,
the general linear model, dummy variables, and the problems
associated with heteroskedasticity, multicollinearity, and
autocorrelation. Prerequisite: Either STAT 125 or STAT 225, and
MATH 266.
[4mSTAT 327[0m Applied Multivariate Analysis (3)
Methods of multivariate analysis, including canonial
correlation, clustering, discriminant analysis, factor analysis
multivariate analysis of variance, multiple regression, and
principle components analysis. Prerequisite: Either STAT 125 or
STAT 225, and MATH 266.
[4mSTAT 425[0m Introduction to Probability Theory (3)
Various formulations of probability, the structure of
probability spaces, combinatorial analysis, discrete and continuous
random variables, joint distributions, the Central Limit Theorem,
monent generating functions, and characteristic functions.
Prerequisite: STAT 225 and MATH 291.
[4mSTAT 426[0m Introduction to Mathematical Statistics (3)
Page 7
Sampling distributions of statistics, properties of
statistics, general principles of statistical inference, linear
statistical models, some nonparametric statistics, Bayesian
statistics, and an introduction to statistical decision theory.
Prerequisite: MATH 425.
Page 8
SUMMARY
New Listing Old Listing Change(s) Notes
MATH 131 MATH 131 - -
MATH 132 MATH 132 - -
MATH 151 MATH 121 Course Number -
MATH 161 MATH 173 Course Number -
MATH 163 MATH 181 Course Number 1
MATH 164 MATH 182 Course Number 1
MATH 191 MATH 191 - -
MATH 192 MATH 192 - -
MATH 266 - New Course 2
MATH 280 MATH 273 Course Number 3
MATH 291 MATH 201 Course Number -
MATH 332 MATH 351 Course Number -
MATH 361 MATH 361 - -
MATH 362 - New Course 4
MATH 365 MATH 366 Course Number -
MATH 366 - New Course 4
MATH 391 MATH 301 Course Number -
MATH 394 MATH 305 Course Number -
MATH 398 MATH 302 Course Number -
MATH 431 MATH 391 Course Number -
MATH 432 MATH 392 Course Number -
MATH 452 - New Course 5
MATH 462 - New Course 5
MATH 491 MATH 381 Course Number -
MATH 492 MATH 382 Course Number -
[4m [0m
STAT 125 MATH 125 MATH to STAT
Cr Hrs 3 to 4 6
STAT 225 - New Course 7
STAT 228 - New Course 8
STAT 325 MATH 228 MATH to STAT -
STAT 327 MATH 225 MATH to STAT -
STAT 425 MATH 421 MATH to STAT -
STAT 426 MATH 426 MATH to STAT -
Page 9
NOTES
1. This course is being added in order to support both Computer
Science and Statistics courses. It is felt that there is a need
for a Linear Algebra course which does not require calculus as a
prerequisite.
2. This course has been offered on a trial basis (MATH 273 in the
Spring of 1980) and the department feels strongly that this will
serve as a bridge to help students in going from the calculus
sequence to the upper-level, abstract mathematics courses.
3. Hitherto, the department has offered only one semester each of
Abstract Algebra and Linear Algebra. Whle this has served to give
students the "bare necessities" in a mathematics major, true
justice can only be done to these topics in a two-semester sequence
(as is done in Analysis, Topology, and Probability and Statistics).
4. These are courses offered at most all institutions of higher
education. While we do not feel that they are courses that should
be required of every major, nor do we anticipate that they would be
offered every year, we do strongly feel that they should be a part
of the curriculum.
5. MATH 125 has been, in the main, a service course for other
departments - e.g. Management and Economics as such, there is a
great deal of material which must be covered. The department feels
that to do justice to this amount of material requires 4 hours per
week rather than 3.
6. This course, also, is offered in support of the anticipated
Computer Science program. It amounts to little more than
partitioning the students interested in a statistics course into
various classes on the bases of their interests and mathematical
backgrounds. For example, students who are encouraged to take
calculus early in their programs (e.g. majors in chemistry,
computer science, economics, engineering, mathematics, physics)
would be encouraged to take STAT 225. Those without such direction
(e.g. majors in biology, management, the social sciences) would
take STAT 125.
7. Exploratory data analysis is clearly the most important
innovation in statistics in the past decade, and, rather than being
a short-lived fad, it is destined to become one of the important
"subfields" of statistics.