1981-1982 Senate Document #24 APC Document #21 Redefinition of Major/Change in Major Requirements/Renumbering of Courses/Additional Courses Delete: All Catalog material describing the mathematics majors and mathematics courses found on pages 114-117. Add: The text on the following pages of this memorandum Rationale: 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 MATH 131 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. MATH 132 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. MATH 151 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. MATH 161 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. MATH 163 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. MATH 164 Trigonometry (3) Circular functions, identities, equations, complex numbers, polar coordinates, and applications. Prerequisite: MATH 163 or satisfactory performance on the departmental placement exam. MATH 191 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. MATH 192 Calculus II (4) Page 4 The calculus of exponential, logarithmic, and trigonometric functions, the Mean Value Theorem, indeterminate forms, improper integrals, and infinite series. Prerequisite: MATH 191. MATH 266 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. MATH 280 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. MATH 291 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. MATH 332 Geometry (3) Euclidean geometry and the fifth postulate. Hyperbolic and elliptic geometries, the consistency of non-Euclidean geometries, and projective geometry. Prerequisite: MATH 291. MATH 361 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. MATH 362 Modern Algebra II (3) A continuation of the study of topics in MATH 361. Prerequisite: MATH 361. MATH 365 Linear Algebra I (3) Vector spaces matrices and linear transformations. Prerequisites: MATH 280 and MATH 291 or permission of the department. MATH 366 Linear Algebra II (3) A continuation of the study of topics in MATH 365. Prerequisite: MATH 365. MATH 391 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. MATH 394 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. MATH 392 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. MATH 431 Topology I (3) Metric spaces, topological spaces, separation axioms, connectedness and compactness. Prerequisite: MATH 280 and MATH 291 or permission of department. MATH 432 Topology II (3) A continuation of the study of topics in MATH 431. Prerequisite: MATH 431. MATH 452 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. MATH 462 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. MATH 491 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. MATH 492 Analysis II (3) A continuation of the study of topics in MATH 491. Prerequisite: MATH 491. Page 6 STAT 125 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. STAT 225 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. STAT 228 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. STAT 325 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. STAT 327 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. STAT 425 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. STAT 426 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 -   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.