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Mathematics (MATH)

Chair: Professor Samuel E. Patterson

Professors: David F. Appleyard, Jack Goldfeather, Stephen F. Kennedy, Mark Krusemeyer, Richard W. Nau, Gail S. Nelson, Jeffrey R. Ondich, Samuel E. Patterson

Associate Professors: Laura M. Chihara, Robert P. Dobrow, Deanna B. Haunsperger

Assistant Professor: Eric S. Egge

Senior Lecturer: Cris T. Roosenraad

Mathematics is an art, a pure science, a language, and an analytical tool for the natural and social sciences, a means of exploring philosophical questions, and a beautiful edifice that is a tribute to human creativity. The mathematic curriculum is designed to provide essential skills for students in a variety of disciplines and to provide mathematics majors with a deep understanding of mathematics as it has evolved over the past two thousand years and how it is practiced today.

Requirements for a Major:

The course requirements are Mathematics 110 or 111, 121, 211, 232, 236 and six advanced courses from among: Mathematics courses numbered above 236 and Computer Science 227, 237. Potential majors with especially strong preparation may petition the department for exemption from the Mathematics 232 and/or 236 requirement(s). Mathematics majors are strongly encouraged to take Computer Science 117, preferably during their first two years. Concepts and skills from Computer Science 117 can be particularly valuable in advanced mathematics courses.

At least three of the following five areas of mathematics must be represented by the six advanced courses.

Algebra: Mathematics 312, 332, 342, 352

Analysis: Mathematics 251, 311, 321, 331, 351

Applied Mathematics: Mathematics 241, 265, 275, 341

Discrete Structures: Mathematics 333, Computer Science 227, 237

Geometry and Topology: Mathematics 244, 344, 354

In addition, each senior major must complete an integrative exercise which consists either of a senior lecture and a written comprehensive examination, or a group research project. Majors must attend a total of twelve other senior lectures during the junior and senior years.

There are many patterns of courses for the major depending upon a student's mathematical interests and career goals. A guide for majors, which supplies information about suitable patterns of courses, is available on the web at Those planning to attend graduate school should consider acquiring a reading knowledge of at least one of the following languages: French, German or Russian.

In order to meet State of Minnesota certification requirements, prospective secondary school teachers must take Mathematics 265, 275, 342, 244 (recommended) or 344, and 349. A computer science course is also strongly recommended.

Major under Combined Plan in Engineering (see Engineering in index):

In addition to completing requirements for the mathematics major listed above including Mathematics 241 and 341, the student should take the following courses required for admission to engineering schools: Physics 113 or 114, 115, 128, Chemistry 123, 230, and Computer Science 117.

Mathematics Skills Center:

This Center offers extra assistance to students in lower-level mathematics courses and other courses requiring basic mathematical skills.

Mathematics Courses:

MATH 100. Chance in the News

Probability is not just a key branch of mathematics, on a deeper level it is our attempt to make sense of an uncertain world. It's hard to read a newspaper without confronting arguments involving chance and randomness, whether it's the "increased risk" of a new drug or the "likelihood of guilt" in a courtroom drama. Wall Street debates the "random walk" hypothesis for stocks and sportscasters make predictions using the "law of averages." In this class we will read the newspaper as "probability detectives," searching out stories, claims and controversies involving chance. With current events as our backdrop we'll study the basic principles and paradoxes of probability. 6 cr., S/CR/NC, MS, FallR. Dobrow

MATH 101. Calculus with Problem Solving Topics will be chosen to provide an introduction to the central ideas of calculus and to provide review and practice of those skills needed for the continued study of calculus. Problem solving strategies will be emphasized. 6 cr., MS, FallD. Haunsberger

MATH 106. Introduction to Mathematics This course is designed to provide students with an understanding of fundamental concepts and applications of mathematics. It attempts to provide insights into the nature of mathematics and its relation to other branches of knowledge, and helps students develop skill in mathematical reasoning. No prerequisites. 6 cr., MS, SpringM. Krusemeyer

MATH 111. Calculus I An introduction to the differential and integral calculus. Derivatives, antiderivatives, the definite integral, applications, and the fundamental theorem of calculus. Prerequisite is one of the following: admission by way of the department's decision tree to be found in the New Student Course Description booklet, or a satisfactory grade on the Diagnostic Examination. Not open to students who have received credit for Mathematics 110. 6 cr., MS, Fall,Winter,SpringStaff

MATH 115. Statistics: Concepts and Applications Introduction to statistical concepts with emphasis on understanding and interpretation of statistical information, especially in the context of media reports and scholarly articles. Examples taken from a wide-range of areas such as public policy, health and medicine, and the social and natural sciences. Computationally less intensive than Math 215. Students will learn how to use statistical software. Topics include: Uncertainty and variability, statistical graphs, types of studies, correlation and linear regression, two-way tables, and inference. Not open to students who have already received credit for Math 211, Math 215 or Psychology 124. 6 cr., MS, Fall,SpringL. Chihara, R. Dobrow

MATH 121. Calculus II Integration techniques, improper integrals, the calculus of the exponential, logarithmic, and inverse trigonometric functions, applications, indeterminate forms, Taylor polynomials, infinite series. Prerequisite: Mathematics 110 or 111 or placement by or consent of the department. 6 cr., MS, Fall,Winter,SpringStaff

MATH 206. A Tour of Mathematics A series of eight lectures intended for students considering a Mathematics major. The emphasis will be on presenting various striking ideas, concepts and results in modern mathematics, rather than on developing extensive knowledge or techniques in any particular subject area. 1 cr., S/CR/NC, MS, WinterStaff

MATH 211. Calculus III Introduction to multivariable calculus: vectors, curves, partial derivatives, gradient, multiple and iterated integrals, line integrals, Green's theorem. Prerequisite: Mathematics 121 or placement by or consent of the department. 6 cr., MS, Fall,Winter,SpringStaff

MATH 215. Introduction to Statistics Introduction to statistics and data analysis. Practical aspects of statistics, including extensive use of statistical software, interpretation and communication of results, will be emphasized. Topics include: exploratory data analysis, correlation and linear regression, design of experiments, basic probability, the normal distribution, sampling distributions, estimation, hypothesis testing, and two-way tables. Not open to students who have already received credit for Math 115 or Math 275. Students who have received MS credit for Psychology 124-126 cannot receive MS credit for Math 215. Students who have taken Math 211 are encouraged to consider the more advanced Math 265-275 probability-statistics sequence. 6 cr., MS, Fall,Winter,SpringStaff

MATH 216. Seminar: History of Mathematics This seminar will focus on selected episodes in the history of mathematics from the seventeenth century to the present. Each participant will give at least one public presentation, which will be followed by discussion. Some weekly preparatory reading, often on the life and work of a prominent mathematician, will be required. Prerequisite: Mathematics 211 or concurrent registration with Mathematics 211 or consent of the instructor. 2 cr., MS, Offered in alternate years. Not offered in 2006-2007.

MATH 232. Linear Algebra Vector spaces, linear transformations, determinants, inner products and orthogonality, eigenvectors and eigenvalues; connections with multivariable calculus. Prerequisite: Mathematics 211. 6 cr., MS, Fall,Winter,SpringR. Dobrow, E. Egge, J. Goldfeather, S. Kennedy

MATH 236. Mathematical Structures Basic concepts and techniques used throughout mathematics. Topics include logic, mathematical induction and other methods of proof, problem solving, sets, cardinality, equivalence relations, functions and relations, and the axiom of choice. Other topics may include: algebraic structures such as groups and rings and basic combinatorics. Prerequisite: Mathematics 232 or consent of the instructor. Students may not receive credit for both Computer Science 177 and Mathematics 236. 6 cr., MS, Fall,Winter,SpringD. Appleyard, D. Haunsperger, M. Krusemeyer

MATH 241. Ordinary Differential Equations An introduction to the theory and methods of solution of ordinary differential equations. Prerequisites: Mathematics 232 or consent of the instructor. 6 cr., MS, WinterL. Chihara

MATH 244. Geometries Euclidean geometry from an advanced perspective; projective, hyperbolic, inversive, and/or other geometries. In addition to foundations, various topics such as transformation and convexity will be treated. Recommended for prospective secondary school teachers. Prerequisite: Mathematics 236. 6 cr., MS, Offered in alternate years. Not offered in 2006-2007.

MATH 245. Applied Regression Analysis A second course in statistics covering simple linear regression, multiple regression and ANOVA, and logistic regression. Exploratory graphical methods, model building and model checking techniques will be emphasized with extensive use of software to analyze real-life data. Prerequisites: Mathematics 215 (or equivalent) or 275. 6 cr., MS, SpringL. Chihara

MATH 251. Chaotic Dynamics An exploration of the behavior of non-linear dynamical systems. Topics include one-dimensional dynamics, Feigenbaum's universality, Sarkovskii's Theorem, chaos, symbolic dynamics, fractals, structural stability, Smale's horseshoe map, strange attractors and bifurcation theory. Some point-set topology will be developed as needed. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Offered in alternate years. WinterD. Haunsperger

MATH 265. Probability Introduction to probability and its applications. Topics include: combinatorial analysis used in computing probabilities, random variables, independence, joint and conditional distributions, expectation, law of large numbers and properties of the most common probability distributions. Prerequisite: Mathematics 211. 6 cr., MS, FallL. Chihara

MATH 275. Mathematical Statistics Introduction to mathematical statistics. The mathematics underlying fundamental statistical concepts will be covered as well as applications of these ideas to real-life data. Topics include: confidence intervals, hypothesis testing, parameter estimation, maximum likelihood, goodness of fit tests and regressions. A statistical software package will be used to analyze data sets. Prerequisite: Mathematics 265. 6 cr., MS, WinterR. Dobrow

MATH 285. Topics in Probability and Statistics Topic to be determined based on students' and instructor's interests. 6 cr., MS, Offered in alternate years. Not offered in 2006-2007.

MATH 295. Computational Number Theory Modern computational techniques for primality testing and factoring. Pseudo primes; Carmichael numbers; the Lucas-Lehmer test; probabilistic, quadratic sieve, and number sieve methods. Extensive use of mathematics. 6 cr., MS, SpringS. Kennedy

MATH 311. Topics in Numerical Analysis Topics chosen from: the numerical solution of algebraic, differential, and difference equations; integration; functional approximation; treatment of empirical data; computational linear algebra; computational geometry; computational number theory; error analysis. Prerequisite: Mathematics 232. No specific programming prerequisite. 6 cr., MS, Not offered in 2006-2007.

MATH 312. Elementary Theory of Numbers Properties of the integers. Topics include the Euclidean algorithm, classical unsolved problems in number theory, prime factorization, Diophantine equations, congruences, divisibility, Euler's phi function and other multiplicative functions, primitive roots, and quadratic reciprocity. Other topics may include integers as sums of squares, continued fractions, distribution of primes, integers in extension fields, p-adic numbers. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Offered in alternate years. FallJ. Goldfeather

MATH 321. Real Analysis I A systematic study of concepts basic to calculus, such as topology of the real numbers, limits, differentiation, integration, convergence of sequences, and series of functions. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, FallC. Roosenraad

MATH 331. Real Analysis II Further topics in analysis such as measure theory, Lebesgue integration or Banach and Hilbert spaces. Prerequisite: Mathematics 321 or consent of the instructor. 6 cr., MS, Offered in alternate years. WinterG. Nelson

MATH 332. Advanced Linear Algebra Vector spaces, linear operators, canonical forms, inner-product spaces. Emphasis on the interplay of theory and applications. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Offered in alternate years. Not offered in 2006-2007.

MATH 333. Combinatorial Theory Deciding on the existence of, finding, and/or counting arrangements, functions, and other desired structures involving finite sets. Some graph and network theory. Counting techniques include the inclusion-exclusion principle, generating functions, and recurrence relations. Existence criteria include the pigeonhole principle, Ramsey's theorem, and Hall's ("marriage") theorem. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Offered in alternate years. WinterE. Egge

MATH 341. Partial Differential Equations Fourier series and their applications to boundary value problems in partial differential equations. Topics include separation of variables, orthogonal sets of functions, representations of functions in series of orthogonal functions, Fourier transforms, and uniqueness of solutions. Prerequisite: Mathematics 241. 6 cr., MS, SpringS. Patterson

MATH 342. Abstract Algebra I Introduction to algebraic structures, including groups, rings, and fields. Homomorphisms and quotient structures, polynomials, unique factorization. Other topics may include applications such as Burnside's counting theorem, symmetry groups, polynomial equations, or geometric constructions. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, SpringD. Haunsperger

MATH 344. Differential Geometry Local and global theory of curves, Frenet formulas. Local theory of surfaces, normal curvature, geodesics, Gaussian and mean curvatures, Theorema Egregium. Riemannian geometry. Prerequisite: Mathematics 236 or consent of the instructor. 6 cr., MS, Offered in alternate years. FallG. Nelson

MATH 349. Methods of Teaching Mathematics Cross-listed with EDUC 350. Methods of teaching mathematics in grades 7-12. Issues in contemporary mathematics education. Regular visits to school classrooms and teaching a class are required. Prerequisite: Senior standing and permission of the instructor. 6 cr., ND, SpringC. Roosenraad

MATH 351. Functions of a Complex Variable Algebra and geometry of complex numbers, analytic functions, complex integration, series, residues, applications. Prerequisite: Mathematics 211. 6 cr., MS, Offered in alternate years. Not offered in 2006-2007.

MATH 352. Abstract Algebra II An intensive study of one or more of the types of algebraic systems studied in Mathematics 342. Prerequisite: Mathematics 342 or consent of the instructor. 6 cr., MS, Offered in alternate years. Not offered in 2006-2007.

MATH 354. Topology An introduction to the topology of surfaces. We will cover basic point-set, geometric and algebraic topology. Topics include continuity, connectedness and compactness; triangulations and classification of surfaces; topological invariants (Euler characteristic); homology. Prerequisite: Mathematics 236. 6 cr., MS, Offered in alternate years. Not offered in 2006-2007.

MATH 395. Senior Seminar 6 cr., MS, SpringM. Krusemeyer

MATH 400. Integrative Exercise (Senior Lecture) A mathematical talk on an assigned topic, presented by the registered senior mathematics major. Required of all senior majors. Prerequisite: Mathematics 236 and successful completion of three courses from among: Mathematics courses numbered above 236, Computer Science 227, Computer Science 237. 3 cr., S/NC, ND, Fall,Winter,SpringStaff

MATH 400. Integrative Exercise (Senior Examination) A three-hour written test on material from Mathematics 110 or 111, 121, 211, 232 and 236. Required of all senior majors. 3 cr., S/NC, ND, SpringStaff