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| ADMISSIONS | BURSAR | FINANCIAL AID | GRADUATE SCHOOL | RESIDENTIAL LIFE | STUDENT ADMINISTRATION | UNDERGRADUATE CATALOG | |
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Mathematics (MATH)Head of Department: Professor Michael Neumann For major requirements, see the College of Liberal Arts and Sciences section of this Catalog. 101. Basic Algebra with Applications Either semester. Three credits. Not open to students who have passed any Q-course. Strongly recommended as preparation for Q-courses for students with a weak algebra background. This course does not count toward the minimum credit requirement for graduation. Polynomials, exponents, Cartesian coordinate system, linear and quadratic equations, inequalities. Either semester. Three credits. Recommended preparation: MATH 101, 104 or the equivalent. Not eligible for course credit by examination. Not open for credit to students who have passed any mathematics course other than MATH 101, 103, 104, 105, 107, 108 or 109. Vinsonhaler An introduction to the techniques used by mathematicians to solve problems. Skills such as Externalization (pictures and charts), Visualization (associated mental images), Simplification, Trial and Error, and Lateral Thinking learned through the study of mathematical problems. Problems drawn from combinatorics, probability, optimization, cryptology, graph theory, and fractals. Students will be encouraged to work cooperatively and to think independently. 103Q. Elementary Discrete Mathematics Either semester. Three credits. Recommended preparation: MATH 101, 104 or the equivalent. Not open for credit to students who have passed any mathematics course other than MATH 101, 102, 104, 105, 107, 108 or 109. Problem solving strategies, solutions of simultaneous linear equations, sequences, counting and probability, graph theory, deductive reasoning, the axiomatic method and finite geometries, number systems. 104Q. Introductory College Algebra and Mathematical Modeling Both semesters. Three credits. Five class periods. Not open for credit to students who have passed MATH 101, or any Q course. Strongly recommended as preparation for Q courses for students whose high school algebra needs reinforcement. The course emphasizes two components necessary for success in 100-level courses which employ mathematics. The first component consists of basic algebraic notions and their manipulations. The second component consists of the practice of solving multi-step problems from other disciplines, called mathematical modeling. The topics include: lines, systems of equations, polynomials, rational expressions, exponential and logarithmic functions. Students will engage in group projects in mathematical modeling. 105Q. Mathematics for Business and Economics Either semester. Three credits. Recommended preparation: MATH 101,104 or the equivalent. Linear equations and inequalities, exponents and logarithms, matrices and determinants, linear programming. Applications. 106Q. Calculus for Business and Economics Either semester. Three credits. (One credit for students who have passed MATH 113, 115, 120, or 135. ) Recommended preparation: MATH 101,104 or the equivalent and MATH 105. Not open for credit to students who have passed MATH 118. Derivatives and integrals of algebraic, exponential and logarithmic functions. Functions of several variables. Applications. 107Q. Elementary Mathematical Modeling Either semester. Three credits. Recommended preparation: MATH 101, 104 or the equivalent. Not open to students who have passed any mathematics course other than MATH 101, 102, 103, 104, 105, or 108. This course and MATH 109 cannot both be taken for credit. This course should not be considered as adequate preparation for MATH 106, 112, 115, 120 or 135. Use of algebraic and trigonometric functions with technology to analyze quantitative relationships and illustrate the role of mathematics in modern life; graphical numerical and symbolic methods. Most sections require a graphing calculator; some require work with a computer spreadsheet. 108QC. Mathematical Modeling in the Environment Either semester. Three credits. Recommended preparation: MATH 101, 104 or the equivalent. A solid background and good performance in high school algebra are highly recommended. An interdisciplinary approach to environmental issues, such as: ground water contamination, air pollution, and hazardous materials handling. Emphasis on mathematical models, social and ethical implications, and physical and chemical principles. Includes a spread sheet program for water and air pollution data; a computer modeling package to analyze hazardous materials emergencies; creative use of the internet and field research. CA 3. Either semester. Three credits. Recommended preparation: MATH 101, 104 or the equivalent. Not open for credit to students who have passed MATH 112, 115, or 120. Students may not recieve credit for this course and MATH 107. Preparation for calculus. Review of algebra. Functions and their applications; in particular, polynomials, rational functions, exponentials, logarithms, and the trigonometric functions. Either semester. Four credits. Four class periods. Recommended preparation: MATH 101, 104 or the equivalent. Students cannot receive credit for MATH 112 and either MATH 115, 120, or 135. Students who have not passed the Calculus Placement Survey take this course rather than MATH 115 or 120. Limits, derivatives, and extreme values of algebraic functions, with supporting algebraic topics. Either semester. Four credits. Four class periods. Prerequisite: MATH 112. Recommended preparation: A grade of C- or better in MATH 112. Students cannot receive credit for MATH 113 and either MATH 115, 120, or 135. May be used in place of MATH 115 or 120 to fulfill any requirement satisfied by MATH 115 or 120. Limits, derivatives, and extreme values of trigonometric functions, with supporting trigonometric topics; anti-derivatives of algebraic and trigonometric functions; the definite integral and applications. Either semester. Four credits. Four class periods. Prerequisite: MATH 113. Recommended preparation: A grade of C- or better in MATH 113. Note: MATH 115 is not adequate preparation for MATH 114. Not open for credit to students who have passed MATH 116, 121, or 136. The transcendental functions, formal integration, polar coordinates, infinite sequences and series, lines and planes in three dimensions, vector algebra. Either semester. Four credits. Four class periods. Prerequisite: Passing score on the Calculus Placement Survey. Students cannot receive credit for MATH 115 and either MATH 112, 113, 120, or 135. Suitable for students with some prior calculus experience. May be used in place of MATH 112, 120, or 135 to fulfill any requirement satisfied by MATH 112, 120, or 135. Limits, continuity, differentiation, antidifferentiation, definite integrals, with applications to the physical and engineering sciences. Sections with QC credit integrate computer-laboratory activity. Either semester. Four credits. Four class periods. Prerequisite: MATH 113, 115, 120, or 135, or advanced placement credit for calculus (a score of 4 or 5 on the Calculus AB exam or a score of 3 on the Calculus BC exam). Recommended preparation: A grade of C- or better in MATH 113 and 115. Not open to students who have passed MATH 114, 121, or 136. Substitutes for MATH 114 or 121 as a requirement. Transcendental functions, formal integration, polar coordinates, infinite sequences and series, vector algebra and geometry, with applications to the physical sciences and engineering. Sections with QC credit integrate computer-laboratory activity. 118Q. A Survey of Calculus with Applications I Either semester. Three credits. Recommended preparation: MATH 101, 104, or the equivalent. Not open for credit to students who have passed MATH 106, 113, 115, or 120, or 135. Derivatives and integrals of elementary functions including the exponential and logarithm functions; applications include optimization, marginal functions, exponential growth and decay, compound interest. (Formerly offered as MATH 120Q.) First semester. Four credits. Prerequisite: Passing score on the Calculus Placement Survey. Students cannot receive credit for MATH 135 and either MATH 113, 115, or 120. May be used in place of MATH 115 to fulfill any requirement satisfied by MATH 115. The subject matter of MATH 115 in greater depth, with emphasis on the underlying mathematical concepts. (Formerly offered as MATH 121Q.) Both semesters. Four credits. Prerequisite: MATH 135 or advanced placement credit for calculus (a score of 4 or 5 on the calculus AB examination or a score of 3 on the Calculus BC examination) or consent of instructor. Students cannot receive credit for MATH 136 and either MATH 114, 116, or 121. May be used in place of MATH 116 to fulfill any requirement satisfied by MATH 116. The subject matter of MATH 116 in greater depth, with emphasis on the underlying mathematical concepts. Either or both semesters. Credits and hours by arrangement. May be repeated for credit (to a maximum of 15 for MATH 193 and 293 together). Consent of the Department Head or Undergraduate Coordinator required, normally before the student's departure. Either semester. Credits, prerequisites, and hours as determined by the Senate Curricula and Courses Committee. Recommended preparation: MATH 101 or equivalent. May be repeated for credit with a change in topic. Either semester. One credit. Prerequisite: Either MATH 210, 220, 230, or 245; and either MATH 211, 221, or 246. Open to sophomores or higher. The student will attend 6-8 seminars per semester, and choose one mathematical topic to investigate in detail. The student will write a well-revised, comprehensive paper on this topic, including a literature review, description of technical details, and a summary and discussion. 201W. Undergraduate Seminar II Either semester. One credit. Prerequisite: MATH 200; ENGL 110 or 111 or 250. Open to sophomores or higher. The student will attend 6-8 seminars per semester, and choose one mathematical topic to investigate in detail. The student will write a well-revised, comprehensive paper on this topic, including a literature review, description of technical details, and a summary and discussion, building upon the writing experience in MATH 200. Either semester. One credit. Corequisite: MATH 210. Prerequisite: ENGL 110 or 111 or 250. Open to sophomores or higher . Weekly seminars and short essays reflecting on the learning experiences and content of MATH 210. 204. Introduction to Mathematical Modeling Either semester. Three credits. Prerequisite: MATH 221 or 246; or MATH 211 and 227. Knowledge of a programming language is strongly recommended. Not open for credit to students who have passed MATH 304 or 305, CHEM 305, or PHYS 305. Construction of mathematical models in the social, physical, life and management sciences. Linear programming, simplex algorithm, duality. Graphical and probabilistic modeling. Stochastic processes, Markov chains and matrices. Basic differential equations and modeling. Either semester. Four credits. Four class periods. Prerequisite: MATH 116, 121, or 136 or a score of 4 or 5 on the Advanced Placement Calculus BC exam. Recommended preparation: A grade of C- or better in MATH 116. Not open for credit to students who have passed MATH 220 or 230 or 245. Open to sophomores or higher. Two- and three-dimensional vector algebra, calculus of functions of several variables, vector differential calculus, line and surface integrals. 211Q. Elementary Differential Equations Either semester. Three credits. Prerequisite: MATH 116, 121, 136 or 244. Recommended preparation: A grade of C- or better in MATH 116; and MATH 210, 220 or 230. Not open for credit to students who have passed MATH 221 or 246. Open to sophomores or higher. Introduction to ordinary differential equations and their applications, linear differential equations, systems of first order linear equations, numerical methods. 213. Transition to Advanced Mathematics Either semester. Three credits. Prerequisite: MATH 210 or 220 or instructor consent. Open to sophomores or higher. Not open for credit to students who have passed MATH 214 or CSE 254. Students intending to major in mathematics should ordinarily take this course during the third or fourth semester. Basic concepts, principles, and techniques of mathematical proof common to higher mathematics. Logic, set theory, counting principles, mathematical induction, relations, functions. Concepts from abstract algebra and analysis. Either semester. Alternate years. Three credits. Prerequisite: MATH 227 or 246 and MATH 213. Vector spaces and linear transformations over fields. Either semester. Three credits. Prerequisite: MATH 213 or 244. Recommended preparation: MATH 215 or 227 or 246. The fundamental topics of modern algebra including elementary number theory, groups, rings, polynomials and fields. Either semester. Three credits. Prerequisite: MATH 216. Topic chosen from modules, linear algebra, geometric algebra, extension fields, algebraic coding, algebraic combinatorics. 221Q. Honors Differential Equations Either semester. Three credits. Prerequisite: MATH 136 or instructor consent. Not open to students who have passed MATH 211 or 246. Open to sophomores or higher. MATH 221 satisfies any requirement met by MATH 211, and provides superior preparation for prospective mathematics, science, and engineering majors. The subject matter of MATH 211 in greater depth, with emphasis on the underlying mathematical concepts. Either semester. Three credits. Prerequisite: MATH 113, 115, 120, 135, or 244. MATH 113 may be taken concurrently. Open to sophomores or higher. Deductive reasoning and the axiomatic method, Euclidean geometry, parallelism, hyperbolic and other non-Euclidean geometries, geometric transformations. Either semester, alternate years. Three credits. Prerequisite: Either (i) MATH 210 or 230, and 211, and MATH 213, or 244, or (ii) MATH 246. The in-depth study of curves and surfaces in space. Either semester. Three credits. Prerequisite: MATH 116, 121, 136 or 244. Recommended preparation: A grade of C- or better in MATH 116. Not open for credit to students who have passed MATH 215 or 246 . Open to sophomores or higher. Systems of equations, matrices, determinants, linear transformations on vector spaces, characteristic values and vectors, from a computational point of view. The course is an introduction to the techniques of linear algebra with elementary applications. 230Q. Honors Multivariable Calculus (Formerly offered as MATH 220Q.) Both semesters. Four credits. Prerequisite: MATH 136 or advanced placement credit for one year of calculus (a score of 4 or 5 on the Calculus BC examination) or consent of instructor. Open to sophomores or higher. Not open to students who have passed MATH 210 or 245. May be used in place of MATH 210 to fulfill any requirement satisfied by MATH 210. The subject matter of MATH 210 in greater depth, with emphasis on the underlying mathematical concepts. Either semester. Three credits. Prerequisite: MATH 210, 220, 230 or 245 which may be taken concurrently with the consent of the instructor. Introduction to the theory of probability. Discussion of some of the probability problems encountered in scientific and business fields. 232. Elementary Stochastic Processes (Also offered as STAT 235.) Either semester. Three credits. Prerequisite: STAT 220 or 224 or 230 or MATH 231. Conditional distributions, discrete and continuous time Markov chains, limit theorems for Markov chains, random walks, Poisson processes, compound and marked Poisson processes, and Brownian motion. Selected applications from actuarial science, biology, engineering, or finance. 235. Introduction to Mathematical Logic Either semester, alternate years. Three credits. Prerequisite: MATH 213 or 244 or CSE 207. PHIL 211 is recommended. Formalization of mathematical theories, elementary model theory with applications to algebra, number theory, and non-standard analysis. Additional topics: Elementary recursion theory and axiomatic set theory. Emphasis on the applications of logic to mathematics rather than the philosophical foundations of logic. 236. Introduction to Actuarial Science Both semesters. Three credits. Prerequisite: Consent of instructor. An introduction to actuarial science, covering many of the topics in the first Foundations of Actuarial Practice module, Role of the Actuary, of the Society of Actuaries. Topics include: what an actuary is and does; external forces that influence actuarial work; and the framework and processes actuaries use to perform actuarial work using Microsoft Excel. Open to sophomores or higher. Either semester, alternate years. Three credits. Prerequisite: MATH 213 or 244 or CSE 254. Finite automata and regular languages, pushdown automata and context-free languages and grammars. Turing machines, recursively enumerable sets and grammars, Church's thesis, the halting problem, and other undecidable problems. Computational complexity and NP-completeness. First semester. Three credits. Prerequisite: MATH 231 and STAT 230. Regression and time series applied to actuarial science. Covers the learning objectives established by the Society of Actuaries for Validation by Educational Experience in Applied Statistics. Either semester, alternate years. Three credits. Prerequisite: Either (i) MATH 210 or 230, and 211, or (ii) 221 or 246 ;and ENGL 110 or 111 or 250. This course may not be counted in any of the major groups described in the Mathematics Departmental listing. Open to sophomores or higher. A historical study of the growth of the various fields of mathematics. 243Q-244Q. Advanced Calculus I, II Both semesters. 4 credits each semester. May be taken for honors credit but open to any qualified student. Open to sophomores or higher. Prerequisite: A year of calculus (that may include high school) and instructor consent. MATH 243Q may be used in place of MATH 115 or 135 to fulfill any requirement satisfied by MATH 115 or 135. MATH 244Q may be used in place of MATH 116 or 136 to fulfill any requirement satisfied by MATH 116 or 136. A rigorous treatment of the mathematics underlying the main results of one-variable calculus. Intended for students with strong interest and ability in mathematics who are already familiar with the computational aspects of basic calculus. 245Q-246Q. Advanced Calculus III, IV Both semesters. 4 credits each semester. May be taken for honors credit but open to any qualified student. Open to sophomores or higher. Prerequisite: MATH 244Q or consent of instructor. MATH 245 may be used in place of MATH 210 to fulfill any requirement satisfied by MATH 210. MATH 246 may be used in place of MATH 211 to fulfill any requirement satisfied by MATH 211. A rigorous treatment of more advanced topics, including vector spaces and their application to multivariable calculus and first-order, second-order and systems of differential equations. 247Q-248Q. Fundamentals of Algebra and Geometry Either semester. Three credits each semester. Prerequisite: PSYC 132 and three credits of Mathematics other than MATH 101. Not open for credit to students who have passed MATH 210, 211, 220, 230, or 245. This course may not be counted in any of the major groups described in the Mathematics Departmental listing. The development of the number system with applications to elementary number theory and analytic geometry. This course is recommended for students in elementary education. Open to sophomores or higher. Either semester, alternate years. Three credits. Prerequisite: MATH 213 or 244 . Metric spaces, topological spaces and functions, topological properties, surfaces, elementary topics in geometric topology. Either semester. Three credits. Prerequisite: MATH 213 or 244. Analysis of combinatorial problems and solution methods. Topics include: Enumeration, generating functions, bijective proofs, sieve methods, recurrence relations, graphs, partially ordered sets, and extremal combinatorics. 252. Introduction to Complex Variables (Also offered as MATH 352.) Either semester. Three credits. Prerequisite: MATH 210 and 211, or 221 or 246 . Not open for credit to students who have passed MATH 352. Functions of a complex variable, integration in the complex plane, conformal mappings. . Representation of two- and three-dimensional data, internal representation of data structures, transformations, mapping of functions to graphics screen, graphics hardware. Programming projects assigned. 258. Introduction to Number Theory Either semester, alternate years. Three credits. Prerequisite: MATH 213 or 244. Congruences, unique factorization, primitive roots, numerical functions, quadratic reciprocity and other selected topics, with emphasis on problem solving. 261. Introduction to Modern Analysis (Also offered as MATH 301.) First semester. Three credits. Prerequisite: Consent of instructor. Not open for credit to students who have passed MATH 301. Metric spaces, sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, functions of several variables. (Also offered as MATH 315.) First semester. Three credits. Prerequisite: Consent of instructor. Not open for credit to students who have passed MATH 315. A study of the fundamental concepts of modern algebra: groups, rings, fields. Also selected topics in linear algebra. 267. Introduction to Geometry and Topology (Also offered as MATH 307.) First semester. Three credits. Prerequisite: Consent of instructor. Not open for credit to students who have passed MATH 307. Topological spaces, connectedness, compactness, separation axioms, Tychonoff theorem, compact-open topology, fundamental group, covering spaces, simplicial complexes, differentiable manifolds, homology theory and the De Rham theory, intrinsic Riemannian geometry of surfaces. 272. Differential Equations for Applications Either semester. Three credits. Prerequisite: MATH 210 and 211, or 221 or 246. Not open for credit to students who have passed MATH 279. Series solutions of differential equations, Bessel functions, Fourier series, partial differential equations and boundary value problems, nonlinear differential equations. Either semester. Three credits each semester. Prerequisite: MATH 213 or 244 and 211, 221 or 246. Introduction to the theory of functions of one and several real variables. Either semester. Three credits. Prerequisite: MATH 231 or STAT 220 or 230; and MATH 285. Introduction to the design of computerized simulations for analyzing and interpreting actuarial and financial problems. This course, together with MATH 392, 393, and 395, helps the student prepare for the actuarial examination on the construction and evaluation of risk models
(Also offered as Mathematics 377.) Either semester. Three credits. Prerequisite: MATH 272. Offered in alternate years. Not open for credit to students who have passed MATH 377. Convergence of Fourier Series, Legendre and Hermite polynomials, existence and uniqueness theorems, two point boundary value problems, and Green's functions. 278. Partial Differential Equations (Also offered as Mathematics 378.) Either semester, alternate years. Three credits. Prerequisite: MATH 272 or its equivalent. Not open for credit to students who have passed MATH 378. Solution of first and second order partial differential equations with applications to engineering and the sciences. 279. Introduction to Field Theory Either semester. Three credits. Prerequisite: Either (i) MATH 210 or 230, and 211 or 221 or (ii) MATH 246. Not open for credit to students who have passed MATH 272. Vector analysis in rectangular, circular-cylindrical and spherical coordinates, postulational derivation of the partial differential equations of classical physics, Fourier series, Bessel and Legendre functions, solutions of Laplace, Poisson, diffusion and scalar and vector wave equations. 280. Financial Mathematics Problems Both semesters. One credit. Two class periods. Prerequisite: MATH 285 and 284, which may be taken concurrently. Preparation for the financial mathematics actuarial examinaton, which tests a student's knowledge of the theory of interest and financial economies at an introductory level. Either semester. Three credits. Prerequisite: Either (i) MATH 210 or 230, 211, and either 215 or 227 or (ii) MATH 246; and knowledge of at least one programming language. Analysis of numerical methods associated with linear systems, eigenvalues, inverses of matrices, zeros of non-linear functions and polynomials. Roundoff error and computational speed. Either semester. Three credits. Prerequisite: MATH 281. Approximate integration, difference equations, solution of ordinary and partial differential equations. Either semester. One credit. Two class periods. Prerequisite: MATH 210, 230 or 245; and MATH 231. Preparation through problem solving for the probability actuarial examination, which tests a student's knowledge of the fundamental probability tools for quantitatively assessing risk. Recommended prior knowledge: a thorough command of probability, as well as basic concepts in insurance and risk management. 284. Advanced Financial Mathematics Either semester. Three credits. Prerequisite: MATH 285 and 231. Advanced topics in financial mathematics such as single period, multi-period and continuous time financial models; Black-Scholes formula; interest rate models; and immunization theory. (Also offered as MATH 365.) First semester. Three credits. Prerequisite: MATH 116, 136, or 244. Open to sophomores or higher. Fundamental concepts of financial mathematics, with applications in calculating present and accumulated values for various streams of cash flows as a basis for future use in: reserving, valuation, pricing, duration calculation, asset/liability management, investment income, capital budgeting and valuing contingent cash flows. 286. Introduction to Operations Research (Also offered as STAT 286 and STAT 356.) Either semester. Three credits. Prerequisite: MATH 231 or STAT 220 or 230. Not open for credit to students who have passed Math 366, STAT 286 or 356. Introduction to the use of mathematical and statistical techniques to solve a wide variety of organizational problems. Topics include linear programming, network analysis, queueing theory, decision analysis. 287-288. Actuarial Mathematics (Also offered as MATH 387-388.) Either semester. Three credits each semester . Prerequisite: MATH 231 or STAT 230; and MATH 285 , which may be taken concurrently . MATH 287 is not open to students who have passed MATH 387. MATH 288 is not open to students who have passed MATH 388. Survival distributions, claim frequency and severity distributions, life tables, life insurance, life annuities, net premiums, net premium reserves, multiple life functions, and multiple decrement models. Either semester. Three credits. Prerequisite: MATH 285 and ACCT 131, which may be taken concurrently. Not open for credit to students who have passed MATH 369. The continuation of MATH 285. Measurement of financial risk, the mathematics of capital budgeting, mathematical analysis of financial decisions and capital structure, and option pricing theory. Either or both semesters. One to three credits. May be repeated for credit (to a maximum of 6 credits). Consent of the Department Head, Director of the Actuarial Program, or the Undergraduate Coordinator required. Prerequisite: Completion of Freshman - Sophomore level requisite courses in the major. Students taking this course will be assigned a final grade of S (satisfactory) or U (unsatisfactory). 291W. Technical Writing for Actuaries Second semester. Three credits. Prerequisite: ENGL 110 or 111 or 250. Consent of Director of Actuarial Science required. Students will write a technical report on an advanced topic in actuarial science. 292W. Senior Thesis in Mathematics Either semester. Three credits.Prerequisite: ENGL 110 or 111 or 250. Open only by consent of Department Head or Departmental Honors Committee. The student should define a general subject area for the thesis before choosing a thesis advisor and seeking consent at the time of registration. The student should submit a written proposal for the senior thesis to the advisor by the end of the semester preceding enrollment for thesis credit. Either or both semesters. Credit and hours by arrangement. May be repeated for credit (to a maximum of 15 for MATH 193 and 293 together). Consent of the Department Head or Undergraduate Coordinator required, normally before the student's departure. May count toward the major with consent of the Advisor and either the Department Head or Undergraduate Coordinator. Either semester. Three credits. With a change in topic, may be repeated for credit. Prerequisites and recommended preparation vary. Either semester. One credit. One class period. Prerequisite: MATH 114, 116, 121, or 136 . Open to sophomores or higher. This course, with a change of topic, may be repeated for credit. Problem sequences selected from algebra, geometry, calculus, combinatorics, and other branches of mathematics, designed to introduce mathematical concepts and to give experience in problem solving. Either semester. Three credits. Open only with consent of instructor. This course, with a change of topic, may be repeated for credit. Either semester. Credits and hours by arrangement. With a change in content, may be repeated for credit. Prerequisites and recommended preparation vary. Either semester. Credits and hours by arrangement. Open only with consent of instructor. This course, with a change of topic, may be repeated for credit.
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