# UWM Undergraduate Catalog 2015-2016

# College of Letters and Science Home Page

## MATH Mathematical Sciences

*This course information produced at 01:00 on 10/08/2015.*

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**90 Basic Mathematics. 0 cr. U.**Arithmetic operations involving whole numbers, integers, positive and negative rational numbers; decimals, percents; ratio, proportion; radicals; descriptive statistics; units of measure; geometry; introduction to algebra.|Fee for 3 cr assessed; credits count toward credit load for Financial Aid and enrollment verification only. Not open to students eligible for math courses that carry graduation credit. Prereq: none.

**92 Mathematical Literacy for College Students I. 0 cr. U.**Introduction to numeracy, proportional reasoning, algebraic reasoning, and functions. Emphasis on developing conceptual and procedural tools that support the use of key mathematical concepts in context.|Fee for 3 cr assessed; counts as 3 cr toward cr load for Fin Aid & enrollment verification. Prereq: Placement level E or D.

**94 Preparation for College Mathematics. 0 cr. U.**Combination of the content of Math 90 and Math 95. Use of computer adaptive testing software to provide individualized instruction in a laboratory setting.|Fee for 6 cr assessed; credits count toward credit load for Financial Aid and enrollment verification only. Math 94 counts as repeat of Math 90 & 95. Prereq: none.

**95 Essentials of Algebra. 0 cr. U.**Number systems; linear equations, inequalities; exponent notation, radicals; polynomials, operations, factoring, rational expressions; coordinate geometry; linear systems; quadratic equations.|Fee for 3 cr assessed; credits count toward credit load for Financial Aid and enrollment verification only. Prereq: math placement level D.

**98 Algebraic Literacy I. 0 cr. U.**Arithmetic number systems; linear equations, inequalities; exponent notation, radicals; polynomials, operations, factoring; modeling; coordinate geometry; linear systems; quadratic equations.|Fee for 3 cr assessed; counts as 3 cr toward credit load for Fin Aid & enrollment verification only. Prereq: math placement level D.

**102 Mathematical Literacy for College Students II. 3 cr. U.**Continuation of Math 92, with an integrated approach to numeracy, proportional reasoning, algebraic reasoning, and functions.|Prereq: C or better in Math 92.

**103 (106) Contemporary Applications of Mathematics. 3 cr. U.**Logical inference, probability and statistical inference, geometric growth, with selected topics such as linear programming, patterns, binary codes.|Prereq: math placement level C.

**105 Intermediate Algebra. 3 cr. U.**Algebraic techniques with polynomials, rational expressions, equations and inequalities, exponential and logarithmic functions, rational exponents, systems of linear equations.|Not open for cr to students who have cr in Math 108(ER). Prereq: math placement level C.

**108 Algebraic Literacy II. 3 cr. U.**Continuation of Math 98 in polynomials, equations, and inequalities; exponential, logarithmic, and periodic functions; rational expressions and exponents; and systems of linear equations.|Not open for credit for students who have cr in Math 105(ER). Prereq: C or better Math 98.

**116 College Algebra. 3 cr. U.**Function concepts. Polynomial, rational, exponential, and logarithmic functions. Systems of equations and inequalities. Matrices and determinants. Sequences and series. Analytic geometry and conic sections. Induction.|2 cr may be used to repeat 2 cr of Math 115. Prereq: math placement level B.

**117 Trigonometry. 2 cr. U.**Trigonometric functions; graphs, identities, equations, inequalities; inverse trigonometric functions; solutions of triangles with applications; complex numbers; polar coordinates.|Repeats 2 cr of Math 115. Prereq: math placement level B.

**175 Mathematical Explorations for Elementary Teachers I. 3 cr. U.**Theory of arithmetic of whole numbers, fractions, and decimals. Introduction to algebra, estimation and problem-solving strategies.|Prereq: Math placement level A, AB, B, or C & regis in elementary, early childhood, or exceptional educ curric.

**176 Mathematical Explorations for Elementary Teachers II. 3 cr. U.**A continuation of Math 175 in geometry, statistics, and probability.|Prereq: grade of C or better in Math 175(P).

**194 First-Year Seminar: (Subtitled). 3 cr. U.**Specific topics are announced in the Schedule of Classes each time the class is offered.|Open only to freshmen. Students may earn cr in just one L&S First-Year Sem (course numbers 192, 193, 194). Prereq: none.

**199 Independent Study. 1-3 cr. U.**Regularly offered courses may not be taken as Independent Study.|Retakable w/chg in topic to 9 cr max. Prereq: 2.5 gpa in all previous math courses; writ cons instr, dept chair & asst dean for SAS.

**205 Introductory Finite Mathematics. 3 cr. U.**Elements of mathematical logic, structures in sets; partitions and counting; probability theory, stochastic processes.|Prereq: math placement level B.

**211 Survey in Calculus and Analytic Geometry. 4 cr. U.**A one-semester survey with applications to business administration, economics, and non-physical sciences. Topics include coordinate systems, equations of curves, limits, differentiation, integration, applications.|May not be used as a prereq for Math 226, 227, 229, or 232. No cr for students with cr in Math 221, 226, 227, 228, or 231. Prereq: math placement level B.

**221 Honors Calculus I. 5H cr. U.**Calculus of functions of one and several variables; sequences, series, differentiation, integration; introduction to differential equations; vectors and vector functions; applications.|Max of 6 cr in combination of Math 221 & 222 may count toward Honors College reqs. Prereq: math placement level A+; cons instr or Honors College dir.

**222 Honors Calculus II. 5H cr. U.**Continuation of Math 221.|Max of 6 cr in combination of Math 221 & 222 may count toward Honors College reqs. Prereq: C or better in Math 221(P) or B or better in Math 232(P); cons instr.

**231 (effective 01/25/2016) Calculus and Analytic Geometry I. 4 cr. U.**Limits, derivatives, and graphs of algebraic, trigonometric, exponential, and logarithmic functions; antiderivatives, the definite integral, and the fundamental theorem of calculus, with applications.|Repeats Math 226, 227, & 228. Prereq: math placement level A.

**231 Calculus and Analytic Geometry. 4 cr. U.**Limits, derivatives, and graphs of algebraic, trigonometric, exponential, and logarithmic functions; antiderivatives, the definite integral, and the fundamental theorem of calculus, with applications.|Repeats Math 226, 227, & 228. Prereq: math placement level A.

**232 (effective 01/25/2016) Calculus and Analytic Geometry II. 4 cr. U.**Continuation of Math 231. Applications of integration, techniques of integration; infinite sequences and series; parametric equations, conic sections, and polar coordinates.|Prereq: grade of C or better in Math 226(P), 227(P), 228(P), 229(P), or 231(P).

**232 Calculus and Analytic Geometry. 4 cr. U.**Continuation of Math 231. Applications of integration, techniques of integration; infinite sequences and series; parametric equations, conic sections, and polar coordinates.|Prereq: grade of C or better in Math 226(P), 227(P), 228(P), 229(P), or 231(P).

**233 Calculus and Analytic Geometry. 4 cr. U.**Continuation of Math 232. Three-dimensional analytic geometry and vectors; partial derivatives; multiple integrals; vector calculus, with applications.|Counts as repeat of Math 229. Prereq: grade of C or better in Math 232(P).

**233 (effective 01/25/2016) Calculus and Analytic Geometry III. 4 cr. U.**Continuation of Math 232. Three-dimensional analytic geometry and vectors; partial derivatives; multiple integrals; vector calculus, with applications.|Counts as repeat of Math 229. Prereq: grade of C or better in Math 232(P).

**234 Linear Algebra and Differential Equations. 4 cr. U.**Elementary differential equations. Vectors; matrices; linear transformations; quadratic forms; eigenvalues; applications.|No cr for students with cr in Math 240. Prereq: grade of C or better in Math 232(P).

**240 Matrices and Applications. 3 cr. U.**Vectors, vector spaces, systems of linear equations, matrices, determinants, linear transformations, diagonalization, eigenvalues, eigenvectors; selected topics from quadratic forms, difference equations, numerical methods, and applications.|No cr for students with cr in Math 234. Prereq: Math placement level A or grade of C or better in a Math or MthStat course numbered 200 or higher.

**275 Problem Solving/Critical Thinking for Elementary Education Majors. 3 cr. U.**Course provides a strong foundation in the exploration, teaching and communication (oral and written) of mathematical concepts via problem-solving experiences and discussion.|Prereq: grade of C or better in Math 175(P) or cons instr.

**276 Algebraic Structures for Elementary Education Majors. 3 cr. U.**Topics for K-8 teachers. Basic patterns and rules that govern number systems, geometric transformations, and manipulation of algebraic expressions.|Counts as repeat of Math 299 w/same topic. Prereq: grade of C or better in Math 175(P) or cons instr.

**277 (effective 01/25/2016) Geometry for Elementary Education Majors. 3 cr. U.**Topics for K-8 teachers. Geometry as measuring tool-congruence, similarity, area, volume, and coordinates; geometry as axiomatic system-definitions, conjectures, proofs, counterexamples; rigid motions, symmetry.|Prereq: grade of C or better in Math 176(P) or cons instr.

**277 Geometry for Elementary Education Majors. 3 cr. U.**Topics for K-8 teachers. Geometry as measuring tool-congruence, similarity, area, volume, and coordinates; geometry as axiomatic system-definitions, conjectures, proofs, counterexamples; rigid motions, symmetry.|Prereq: Math 176(P) or cons instr.

**278 (effective 01/25/2016) Discrete Probability and Statistics for Elementary Education Majors. 3 cr. U.**Random experiments; histograms; sample spaces; equally likely outcomes for random experiments; permutations; combinations; binomial, geometric, hypergeometric distributions; expectation; conditional probabilities; max likelihood estimation & inference.|Math 278 and MthStat 278 are jointly offered; they count as repeats of one another. Prereq: grade of C or better in Math 176(P) or cons instr.

**278 Discrete Probability and Statistics for Elementary Education Majors. 3 cr. U.**Random experiments; histograms; sample spaces; equally likely outcomes for random experiments; permutations; combinations; binomial, geometric, hypergeometric distributions; expectation; conditional probabilities; max likelihood estimation & inference.|Math 278 and MthStat 278 are jointly offered; they count as repeats of one another. Prereq: Math 176(P) or cons instr.

**290 (effective 01/25/2016) Topics in Mathematics: (Subtitled). 3 cr. U.**Specific topics and any additional prerequisites announced in Schedule of Classes each time course is offered.|Retakable w/chg in topic to 9 cr max. Prereq: satisfaction of Quantitative Literacy Part A GER.

**297 Study Abroad: (Subtitled). 1-12 cr. U.**Designed to enroll students in UWM sponsored programs before course work level, content and credits are determined and/or in specially prepared program course work.|Retakable w/chg in topic. Prereq: acceptance for Study Abroad Prog.

**299 Ad Hoc: (Subtitled). 1-6 cr. U.**Course created expressly for offering in a specified enrollment period. Requires only dept & assoc dean approval. In exceptional circumstances, can be offered in one add'l sem.|Retakable w/chg in topic. Prereq: none; add'l prereqs may be assigned to specific topic.

**311 Theory of Interest. 4 cr. U/G.**Interest theory: annuities, ammortization schedules, portfolio immunization. Financial derivatives: bonds, forwards, options. Topics are consistent with syllabus of SOA Financial Mathematics Exam.|Does not carry grad cr in Math Dept. Prereq: jr st & Math 232(P); or grad st.

**311 (effective 01/25/2016) Theory of Interest. 4 cr. U.**Interest theory: annuities, amortization schedules, portfolio immunization. Financial derivatives: bonds, forwards, options. Topics consistent w/syllabus of SOA Financial Mathematics Exam.|Prereq: grade of C or better in Math 232(P).

**313 Linear Programming and Optimization. 3 cr. U/G.**Primal and dual formulations of linear programming problems; simplex and related methods of solution; algorithms for transportation; optimization.|Prereq: jr st; grade of C or better in Math/ElecEng 234(P) or Math 240(P); or grad st.

**315 (314) Mathematical Programming and Optimization. 3 cr. U/G.**Introduction to operations research. Network analysis; integer programming; game theory; nonlinear programming; dynamic programming.|Prereq: jr st, grade of C or better in either Math 234(P) or 240(P), & grade of C or better in either Math 211(P) or 233(P); or cons instr; or grad st.

**320 (effective 01/25/2016) Introduction to Differential Equations. 3 cr. U/G.**Elementary types and systems of differential equations, series solutions, numerical methods, Laplace transforms, selected applications.|No grad cr in Math Sci. Prereq: jr st, grade of C or better in both Math 232(P) & 240(P), or grade of C or better in Math/ElecEng 234(P); or grad st.

**320 Introduction to Differential Equations. 3 cr. U/G.**Elementary types and systems of differential equations, series solutions, numerical methods, Laplace transforms, selected applications.|No grad cr in Math Sci. Prereq: jr st, Math 233(P) & Math 234(P) or ElecEng 234(P); or grad st.

**321 Vector Analysis. 3 cr. U/G.**Topics selected from vector algebra; scalar and vector fields; line, surface, and volume integrals; theorems of Green, Gauss, and Stokes; vector differential calculus.|Prereq: jr st, grade of C or better in Math 233(P); or grad st.

**322 Introduction to Partial Differential Equations. 3 cr. U/G.**Partial differential equations of mathematical physics, boundary value problems in heat flow, vibrations, potentials, etc. Solved by Fourier series; Bessel functions and Legendre polynomials.|Prereq: jr st, Math 320(P), & grade of C or better in Math 233(P); or grad st.

**341 Seminar: Introduction to the Language and Practice of Mathematics. 3 cr. U.**Goal: facility with mathematical language and method of conjecture, proof and counter example, with emphasis on proofs. Topics: logic, sets, functions and others.|Prereq: math placement level A.

**371 Introduction to Stochastic Models in Finance. 3 cr. U/G.**Elementary modeling of financial instruments for students in mathematics, economics, business, etc. Statistical and stochastic tools leading to the Black-Scholes model. Real data parameter fitting.|Prereq: jr st & one of the following pairs; Econ 413(431)(P) & 506(P), Bus Adm 210(P) & 350(P), Bus Adm 701(P) & 702(P), or Math 234(P) & MthStat 361(P), or cons instr; or grad st.

**381 Honors Seminar: (Subtitled). 3H cr. U.**Significant topics to illustrate to non-mathematicians the characteristic features of mathematical thought. Only H.S. algebra and geometry assumed.|Retakable w/chg in topic to 9 cr max. Not open for cr toward a major in Math. Prereq: soph st; Honors 200(P); cons Honors College dir.

**405 (effective 01/25/2016) Mathematical Models and Applications. 3 cr. U/G.**Modeling techniques for analysis and decision-making in social and life sciences and industry. Deterministic and stochastic modeling. Topics may vary with instructors.| Prereq: jr st, grade of C or better in either Math 211(P) or Math 231(P), & grade of C or better in either Math/ElecEng 234(P) or Math 240(P); or grad st.

**405 Mathematical Models and Applications. 3 cr. U/G.**Construction and mathematical models with applications to the social and life sciences. Models may involve Markov chains, linear programming, game theory, graph theory and growth processes.|Prereq: jr st & either Math 234(P) or Math 205(P) & 211(P); or grad st.

**413 Introduction to Numerical Analysis. 3 cr. U/G.**Root finding and solution of nonlinear systems; direct solution of linear systems; interpolation & approximation of functions; least squares; fast Fourier transform; quadrature.|Prereq: jr st, Math 233(C), & Math 234(C) or ElecEng 234(C); or grad st.

**413 (effective 01/25/2016) Introduction to Numerical Analysis. 3 cr. U/G.**Root finding and solution of nonlinear systems; direct solution of linear systems; interpolation & approximation of functions; least squares; fast Fourier transform; quadrature.|Prereq: jr st, grade of C or better in Math 233(C), & grade of C or better in Math/ElecEng 234(C); or grad st.

**415 Introduction to Scientific Computing. 3 cr. U/G.**Nonlinear systems; iterative solution of linear systems; initial value problems in ordinary differential equations; boundary value problems in ordinary and partial differential equations.|Prereq: jr st, grade of C or better in Math 233(C), & grade of C or better in Math/ElecEng 234(C); or grad st.

**417 (416) Computational Linear Algebra. 3 cr. U/G.**Direct solution of linear systems; iterative solution of linear systems; least squares; eigenvalue problems.|Prereq: jr st & grade of C or better in Math/ElecEng 234(P) or Math 240(P); or grad st.

**431 Modern Algebra with Applications. 3 cr. U/G.**Groups, rings, fields, Boolean algebras with emphasis on their applications to computer science and other areas.|Does not carry grad cr in math sci. Prereq: jr st & Math 232(P); or grad st.

**431 (effective 01/25/2016) Modern Algebra with Applications. 3 cr. U/G.**Groups, rings, fields, Boolean algebras with emphasis on their applications to computer science and other areas.|Does not carry grad cr in math sci. Prereq: jr st & grade of C or better in Math 232(P); or grad st.

**451 Axiomatic Geometry. 3 cr. U/G.**An axiomatic approach to Euclidean and non-Euclidean geometry (historic role of the parallel postulate and models).|Dept permission necessary for grad cr in math sci. Prereq: jr st, Math 341(241)(P), & Math 232(C); or grad st.

**451 (effective 01/25/2016) Axiomatic Geometry. 3 cr. U/G.**An axiomatic approach to Euclidean and non-Euclidean geometry (historic role of the parallel postulate and models).|Dept cons req'd for grad cr in math sci. Prereq: jr st, grade of C or better in both Math 341(P) & Math 232(C); or grad st.

**453 Transformations in Geometry. 3 cr. U/G.**Selected topics from vector geometry and geometric transformations such as the study of invariants and conics.|Recom for secondary school teachers. Dept cons req'd for grad cr in math sci. Prereq: jr st, grade of C or better in both Math 341(P) & Math 232(C); or grad st.

**489 Internship in Mathematics, Upper Division. 1-6 cr. U.**Application of advanced principles of mathematics in a business, organizational, educational, governmental, or other appropriate setting.|One cr earned for academic work based on 40 hours in internship. Retakable to 6 cr max. Prereq: jr st; 6 cr 300-level or above in math or math stats; 2.5 gpa in the major; cons supervising faculty member.

**497 Study Abroad: (Subtitled). 1-12 cr. U/G.**Designed to enroll students in UWM sponsored programs before course work level, content and credits are determined and/or in specially prepared program course work.|Retakable w/chg in topic. Prereq: jr st; acceptance for Study Abroad Prog.

**499 Ad Hoc: (Subtitled). 1-6 cr. U.**Course created expressly for offering in a specified enrollment period. Requires only dept & assoc dean approval. In exceptional circumstances, can be offered in one add'l sem.|Retakable w/chg in topic. Prereq: jr st; add'l prereqs may be assigned to specific topic.

**511 Symbolic Logic. 3 cr. U/G.**First-order predicate calculus; formal properties of theoretical systems; chief results of modern mathematical logic; advanced topics such as completeness and computability.|CompSci 511, Math 511 & Philos 511 are jointly offered; they count as repeats of one another. Prereq: jr st & either Philos 212(P) or 6 cr in math at the 300-level or above.

**521 Advanced Calculus. 3 cr. U/G.**Fundamental notions of sets and functions; limits, continuity; Riemann integral, improper integral; infinite series; uniform convergence; power series; improper integrals with a parameter.|Prereq: jr st, Math 232(P), & 341(241)(P); or grad st. Math 233(R) & 234(R).

**522 Advanced Calculus. 3 cr. U/G.**Linear functions; differentiation of functions of several variables (implicit functions, Jacobians); change of variable in multiple integrals; integrals over curves, surfaces; Green, Gauss, Stokes theorems.|Prereq: jr st, Math 521(P), grade of C or better in Math 233(P), & grade of C or better in either Math 234(P) or 240(P); or grad st.

**531 Modern Algebra. 3 cr. U/G.**Integers; groups; rings; fields; emphasis on proofs.|Prereq: jr st; grade of C or better in Math 341(P).

**535 Linear Algebra. 3 cr. U/G.**Vector spaces; linear transformations and matrices; characteristic values and vectors; canonical forms; bilinear, quadratic, and Hermitian forms; selected applications.|Prereq: jr st, Math 234(P) or 240(P), & Math 341(241)(P); or grad st.

**535 (effective 01/25/2016) Linear Algebra. 3 cr. U/G.**Vector spaces; linear transformations and matrices; characteristic values and vectors; canonical forms; bilinear, quadratic, and Hermitian forms; selected applications.|Prereq: jr st, grade of C or better in either Math 234(P) or 240(P), & grade of C or better in Math 341(P); or grad st.

**537 Number Theory. 3 cr. U/G.**Number theoretic functions; distribution of primes; Diophantine approximation; partitions; additive number theory; quadratic reciprocity.|Prereq: jr st, grade of C or better in both Math 232(P) & 341(P); or grad st.

**551 Elementary Topology. 3 cr. U/G.**General theory of point sets in Euclidean spaces, with emphasis on topology of two-dimensional and three-dimensional spaces; elementary notions of metric spaces; applications.|Prereq: jr st & either Math 521(P) or 529(P); or grad st.

**553 Differential Geometry. 3 cr. U/G.**The theory of curves and surfaces by differential methods.|Prereq: jr st, grade of C or better in all of Math 233(P), 234(P) & 341(P); or grad st.

**571 (472) Introduction to Probability Models. 3 cr. U/G.**Probability review, Markov chains in discrete and continuous time. Random walks, branching processes, birth and death processes. Queuing theory. Applications to physical sciences, engineering, mathematics.|Prereq: jr st, Math 233(P), Math 234(P) or ElecEng 234(P), & one college-level course in statistics or probability; or grad st.

**571 (effective 01/25/2016) Introduction to Probability Models. 3 cr. U/G.**Probability review, Markov chains in discrete and continuous time. Random walks, branching processes, birth and death processes. Queuing theory. Applications to physical sciences, engineering, mathematics.|Prereq: jr st; grade of C or better in Math 233(P); grade of C or better in Math/ElecEng 234(P) or in both Math 240(P) & 320(P); & one calculus-based course in statistics or probability at the 300 level or above; or grad st.

**581 Introduction to the Theory of Chaotic Dynamical Systems. 3 cr. U/G.**Iterated mappings, one parameter families, attracting and repelling periodic orbits, topological transitivity, Sarkovski's theorem, chaos, bifurcation theory, period doubling route to chaos, horseshoe maps, attractors.|Prereq: jr st & Math 521(P), 529(P) or 621(P), or cons instr; or grad st.

**591 Undergraduate Seminar: (Subtitled). 1 cr. U.**Specific topics and any additional prerequisites announced in Schedule of Classes each time course is offered.|Retakable w/chg in topic to 4 cr max. Prereq: jr st; cons instr.

**599 Capstone Experience. 1 cr. U.**Student writes a paper under supervision of an advisor on an approved topic not covered in the student's regular course work.|Prereq: sr st; writ cons instr & dept chair.

**601 Advanced Engineering Mathematics I. 3 cr. U/G.**Sequences and series, elementary complex analysis; Fourier series; linear and nonlinear ordinary differential equations; matrix theory, elementary functional analysis; elementary solution of partial differential equations.|Prereq: jr st; grade of C or better in Math/ElecEng 234(P); 3 cr Math at 300-level or above; or cons instr; or grad st.

**602 Advanced Engineering Mathematics II. 3 cr. U/G.**Continuation of Math 601. Partial differential equations, Fourier and Laplace transforms, convolutions, special functions, mathematical modeling.|Prereq: jr st; Math 601(P).

**615 Numerical Solution of Partial Differential Equations. 3 cr. U/G.**Finite difference solution of elliptic boundary value problems and of evolution problems; solution of hyperbolic conservation laws; finite volume methods; finite element methods.|Prereq: jr st; Math 413(P), 415(414)(P), or 417(416)(P); Math 322(P) or 602(P); or cons instr.

**617 Optimization. 3 cr. U/G.**Unconstrained and constrained optimization: linear, nonlinear, and dynamic programming; barrier, penalty, and Lagrangian methods; Karush-Kuhn-Tucker theory, quadratic, and sequential quadratic programming; evolutionary algorithms.|Prereq: jr st; Math 321(P) or 602(P); or grad st or cons instr.

**621 Introduction to Analysis. 3 cr. U/G.**Topology of Euclidean space; continuity; differentiation of real and vector-valued functions; Riemann-Stieltjes integration.|Prereq: jr st, Math 233(P) 234(P), & 341(241)(P); or grad st.

**622 Introduction to Analysis. 3 cr. U/G.**Continues Math 621. Sequences and series of functions; uniform convergence; power series; functions of several variables; inverse and implicit function theorems; differential forms; Stokes' theorem.|Prereq: jr st & Math 621(P); or grad st.

**623 Complex Analysis. 3 cr. U/G.**Complex numbers; definition and properties of analytic functions of a complex variable; conformal mapping; calculus of residues; applications to mathematics and physics. See also Math 713.|Prereq: jr st & either Math 321(P) OR 521(P); or grad st.

**631 Modern Algebra. 3 cr. U/G.**Group theory, including normal subgroups, quotients, permutation groups, Sylow's theorems, Abelian groups; field theory; linear algebra over general fields.|Prereq: jr st, Math 531(P) or cons instr; or grad st.

**632 Modern Algebra. 3 cr. U/G.**Continuation of Math 631. Ring theory, including ideals, quotient rings, Euclidean rings, polynomial rings, unique factorization; modules, including vector spaces, linear transformations, canonical forms; bilinear forms.|Prereq: jr st & Math 631(P) or cons instr; or grad st.

**675 Topics in Modern Mathematics: (Subtitled). 3 cr. U/G.**Instructor and student presentations of modern topics. Specific topics and any additional prerequisites announced in Schedule of Classes each time course is offered.|Retakable w/chg in topic to 9 cr max. Prereq: jr st, Math 233(P) & cons instr; or grad st.

**699 Independent Reading. 1-3 cr. U.**See Advanced Independent Study. For further information, consult dept chair.|Retakable w/chg in topic. Prereq: jr st; 2.5 gpa, & writ cons instr, dept chair, & asst dean for SAS.

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