# Department of Mathematics

262-472-1313

## Mathematics

Mathematics has traditionally been the language of science and now plays a vital role in the social sciences and business as well. Every field in which quantitative activity is found requires that its workers have good backgrounds in mathematics. A student well-versed in mathematics will find their skills of use in almost any profession.

Although mathematics is studied for its applications, it also attracts many who are interested in its aesthetic and creative side. A study of mathematics improves one's ability to think about abstract concepts, to conceptualize about hypothetical structures, to reach logical conclusions, and to consider many solution techniques when problem solving.

### Majors/Emphases

Students may elect one of five emphases in the mathematics major. Emphases consist of the CORE^{1} courses and an additional 4 to 5 required courses in their emphasis. In addition, each major must complete a writing requirement (ENGLISH 370 or ENGLISH 372), a computer science requirement (courses are listed below for all emphases except the Secondary Education Emphasis which does not have a computer science requirement), and the milestones (Major Field Test and Math Exit Interview). An approved minor is also required with each emphasis.

^{1} | The one exception is the Applied Mathematics Emphasis for which the CORE course MATH 452 is not required. |

For more information on the different emphases including 4-year plans and Math Major FAQ, please visit the department's website: http://www.uww.edu/cls/departments/math.

### CORE courses in the Mathematics Major - 25 units

Code | Title | Units |
---|---|---|

MATH 253 | CALCULUS AND ANALYTIC GEOMETRY I (GM) | 5 |

MATH 254 | CALCULUS AND ANALYTIC GEOMETRY II | 5 |

MATH 255 | CALCULUS AND ANALYTIC GEOMETRY III | 3 |

MATH 280 | DISCRETE MATHEMATICS | 3 |

MATH 301 | INTRODUCTION TO ANALYSIS | 3 |

MATH 355 | MATRICES AND LINEAR ALGEBRA | 3 |

MATH 452 | INTRODUCTION TO ABSTRACT ALGEBRA ^{1} | 3 |

Total Units | 25 |

^{1} | The one exception is the Applied Mathematics Emphasis for which the CORE course MATH 452 is not required. |

### Courses

MATH 40 PRE-ALGEBRA 3 Units

A course for students who need a review of basic mathematics or who lack the computational skills required for success in algebra and other University courses. Topics include fractions, decimals, percent, descriptive statistics, English and metric units of measure, and measures of geometric figures. Emphasis is on applications. A brief introduction to algebra is included at the end of the course. This course does count toward the semester credit load and will be computed into the grade point average. It will not be included in the 120 credits required for graduation. It may be taken for a conventional grade or on a satisfactory/no credit basis. Not available to students who have satisfied the University Proficiency requirement in mathematics. ACT Math subscore 14 or below (SAT 340 or below) Arithmetic skills test required.

MATH 41 BEGINNING ALGEBRA 4 Units

A course for those who have a sound background in basic arithmetic, but who have not been exposed to algebra, or who need to strengthen their basic algebra skills. Topics include properties of the real numbers, linear and quadratic equations, linear inequalities, exponents, polynomials, rational expressions, the straight line, and systems of linear equations. The course counts towards the semester credit load and will be computed into the grade point average. It will not, however, be included in the credits necessary for graduation. It may be taken for a conventional grade or on a satisfactory/no credit basis.
Prereq: MATH 040 or equivalent demonstration of capability. Students cannot receive credit for MATH 041 if they have been waived from the Mathematics Proficiency Requirement. Not available to students who have satisfied the University Proficiency requirement in mathematics.

PREREQ: MATH 40 OR ITS EQUIVALENT

MATH 49 WORKSHOP *Repeatable* 1-3 Units

Variable credit course offering with a defined topic. Repeatable with a change of topic.

MATH 140 MATHEMATICAL IDEAS 3 Units

Designed to give students a broad understanding and appreciation of mathematics. Includes topics not usually covered in a traditional algebra course. Topics encompass some algebra, problem solving, counting principles, probability, statistics, and consumer mathematics. This course is designed to meet the University Proficiency Requirement for students who do not wish to take any course having MATH 141 as a prerequisite.

PREREQ: MATH 41 WITH A GRADE OF C OR BETTER OR WAIVER

CROSS-LISTED: MATH 140, MATH 140W

MATH 141 FUNDAMENTALS OF COLLEGE ALGEBRA 4 Units

A functional approach to algebra with emphasis on applications to different disciplines. Topics include linear, exponential, logarithmic, quadratic, polynomial and rational equations and functions, systems of linear equations, linear inequalities, radicals and rational exponents, complex numbers, variation. Properties of exponents, factoring, and solving linear equations are reviewed.

PREREQ: MATH 41 WITH A GRADE OF C OR BETTER OR WAIVER

MATH 143 FINITE MATHEMATICS FOR BUSINESS AND SOCIAL SCIENCES (GM) 3 Units

Mathematical preparation for the understanding of various quantitative methods in modern management and social sciences. Topics included are sets, relations, linear functions, interest, annuities, matrix theory, the solution of linear systems by the graphical, algebraic, Gauss-Jordan, and inverse methods, linear programming by graphical and simplex methods, counting and probability, and decision theory. College of Business and Economics majors must take this course on a conventional grade basis.

PREREQ: MATH 141 WITH A GRADE OF C OR BETTER OR WAIVER.

MATH 148 MATHEMATICS FOR THE ELEMENTARY TEACHER I (GM) 3 Units

A study of sets, whole numbers, fractions, integers, decimals and real numbers, basic arithmetic operations and their properties, standard and alternative algorithms and estimations strategies; problem-solving, proportional reasoning and algebraic thinking. Manipulatives and cooperative learning activities are used throughout the course. For elementary education majors.

PREREQ: A GRADE OF C OR BETTER IN MATH 141 OR WAIVER

MATH 149 MATHEMATICS FOR THE ELEMENTARY TEACHER II 3 Units

Topics in probability and statistics, with emphasis on descriptive techniques. Investigations in geometric figures, measurement, construction, transformations, congruent and similar geometric figures. Problem solving strategies, manipulatives, and cooperative learning activities are emphasized throughout the course.

PREREQ: MATH 147 WITH A GRADE OF C OR BETTER OR MATH 148 WITH A GRADE OF C OR BETTER

MATH 152 ELEMENTARY FUNCTIONS (GM) 5 Units

Review of algebraic functions, inequalities, mathematical induction, theory of equations, exponential and logarithmic functions, circular functions, trigonometric identities and equations, inverse trigonometric functions, solution of triangles.

PREREQ: MATH 141 WITH A GRADE OF C OR BETTER OR WAIVER.

MATH 177 THE LOGIC OF CHESS 1 Units

A study of logic particularly as it is used in the game of chess and, most particularly, in chess strategy and the end game of chess. The rules are taught to those who are not already acquainted with the game.

PREREQ: MATH 141 OR MATH 140

MATH 230 INTRODUCTION TO STATISTICAL REASONING AND ANALYSIS (GM) 3 Units

A course on the principles, procedures and concepts surrounding the production, summarization and analysis of data. Emphasis on critical reasoning and interpretation of statistical results. Content includes: probability, sampling, and research design; statistical inference, modeling and computing; practical application culminating in a research project. Unreq: ECON 245, PSYCH 215, SOCIOLGY 295

PREREQ: GRADE OF C OR BETTER IN MATH 141 OR CONSENT OF INSTRUCTOR

MATH 243 SHORT CALCULUS FOR BUSINESS AND SOCIAL SCIENCES (GM) 3 Units

A general survey of the calculus. Topics covered include limits, differentiation, max-min theory, exponential and logarithmic functions, and integration. Business and social science applications are stressed.

PREREQ: MATH 143 OR MATH 152 WITH A GRADE OF C OR BETTER

UNREQ: MATH 243, MATH 250, MATH 253

MATH 250 APPLIED CALCULUS SURVEY FOR BUSINESS AND SOCIAL SCIENCES (GM) 5 Units

An applied calculus course covering elementary analytic geometry, limits, differentiation, max-min theory, exponential and logarithmic functions, integration, functions of several variables, and elementary differential equations. Some computer topics may be included. A student may earn credit for only one of MATH 243, MATH 250, and MATH 253.

PREREQ: MATH 143 WITH A C OR BETTER OR MATH 152 WITH A C OR BETTER

UNREQ: MATH 243, MATH 250, MATH 253

MATH 253 CALCULUS AND ANALYTIC GEOMETRY I (GM) 5 Units

Review of algebraic and trigonometric functions, transcendental functions, limits, study of the derivative, techniques of differentiation, continuity, applications of the derivative, L' Hopital's Rule and indeterminate forms, the Riemann integral, Fundamental Theorem of Calculus, and substitution rule.

PREREQ: MATH 152 WITH A GRADE OF C OR BETTER OR EQUIVALENT HIGH SCHOOL PREPARATION AS DETERMINED BY THE MATHEMATICS DEPARTMENT.

UNREQ: MATH 243, MATH 250, MATH 253

MATH 254 CALCULUS AND ANALYTIC GEOMETRY II 5 Units

Techniques of integration, applications of the integral, introduction to differential equations, polar coordinates and conic sections, infinite sequences and series. This course includes a writing component.

PREREQ: MATH 250 WITH A GRADE OF B OR BETTER OR MATH 253 WITH A GRADE OF C OR BETTER

MATH 255 CALCULUS AND ANALYTIC GEOMETRY III 3 Units

Solid analytic geometry, vectors and vector functions, functions of several variables, multiple integrals and their applications.

PREREQ: MATH 254 WITH A C OR BETTER

MATH 280 DISCRETE MATHEMATICS 3 Units

This course will supply a thorough grounding in the mathematical topics which are central to the study of computer science, and which form the basis for many modern applications of mathematics to the social sciences. Topics covered will include sets, logic, Boolean algebra and switching circuits, combinatorics, probability, graphs, trees, recursion, and algorithm analysis. Expressing mathematical ideas and writing proofs will be emphasized.

PREREQ: MATH 250 WITH A GRADE OF B OR BETTER OR MATH 253 WITH A GRADE OF C OR BETTER

MATH 301 INTRODUCTION TO ANALYSIS 3 Units

A first course in real analysis. Topics include properties of the real numbers, convergence of sequences, monotone and Cauchy sequences, continuity, differentiation, the Mean Value Theorem, and the Riemann integral. Emphasis is placed on proof-writing and communicating mathematics.

PREREQ: MATH 255 AND MATH 280

MATH 342 APPLIED STATISTICS 3 Units

This course will cover the basics of statistical testing, regression analysis, experimental design, analysis of variance, and the use of computers to analyze statistical problems. This course contains a writing component.

PREREQ: MATH 253 OR MATH 250 OR CONSENT OF INSTRUCTOR

MATH 346 THEORY OF INTEREST 3 Units

This course will cover the topics of interest theory listed in the Society of Actuaries/Casualty Actuarial Society syllabus for Exam FM/2. Topics include the time value of money, annuities, loans, bonds, general cash flows and portfolios, and immunization schedules.

PREREQ: MATH 254 WITH A C OR BETTER

MATH 352 INFINITE PROCESSES FOR THE ELEMENTARY TEACHER 3 Units

This course is primarily for pre-service elementary and middle school teachers. Students will be introduced to the concepts of calculus, which include infinite precesses, limits, and continuity. In addition, dirivatives and integrals, and their relationship to area and change will be covered.

PREREQ: MATH 152 WITH C OR BETTER

MATH 353 COLLEGE GEOMETRY 5 Units

The topics included in this course are foundations of Euclidean geometry, Euclidean transformational geometry, modern synthetic geometry that builds on Euclidean geometry, selected finite geometries, and an introduction to non-Euclidean and projective geometry, including their relationship to Euclidean geometry. Although the course is adapted to the prospective teacher of geometry, it will also meet the needs of those in other majors needing a background in geometry. Standards and guidelines of appropriate national and local bodies will be implemented.

PREREQ: MATH 253 AND MATH 280

MATH 355 MATRICES AND LINEAR ALGEBRA 3 Units

Systems of linear equations, matrices and determinants, finite dimensional vector spaces, linear dependence, bases, dimension, linear mappings, orthogonal bases, and eigenvector theory. Applications stressed throughout.

PREREQ: MATH 250 WITH A GRADE OF B OR BETTER OR MATH 253 WITH A GRADE OF C OR BETTER

MATH 359 PROBABILITY & STATISTICS FOR TEACHERS 3 Units

An introduction to probability and statistics for teachers. Topics covered include counting techniques, basic probability theory, exploratory data analysis, simulation, randomization, and statistical inference. This course contains a writing component.

PREREQ: MATH 250 WITH A GRADE OF B OR BETTER OR MATH 253 WITH A GRADE OF C OR BETTER

MATH 361 DIFFERENTIAL EQUATIONS 3 Units

Ordinary differential equations: general theory of linear equations, special methods for nonlinear equations including qualitative analysis and stability, power series and numerical methods, and systems of equations. Additional topics may include transformation methods and boundary value problems. Applications stressed throughout.

PREREQ: MATH 254 WITH A C OR BETTER

MATH 370 PROBLEM SOLVING FOR THE ELEMENTARY TEACHER 3 Units

This course is primarily for pre-service elementary and middle school teachers. Students will learn a variety of problem solving strategies applicable in elementary and middle school. The applications will cover many different areas of mathematics.

PREREQ: MATH 149 WITH C OR BETTER

MATH 375 DEVELOPMENT OF MATHEMATICS 3 Units

A study of the development of mathematical notation and ideas from prehistoric times to the present. Periods and topics will be chosen corresponding to the backgrounds and interests of the students.

PREREQ: MATH 152 WITH A GRADE OF C OR BETTER, OR CONSENT OF INSTRUCTOR

MATH 381 MATHEMATICAL MODELING AND SIMULATION 3 Units

Modeling involving formulation of deterministic, stochastic and rule-based models and computer simulation in order to make predictions. Topics may include unconstrained and constrained growth models, equilibrium and stability, force and motion, predator-prey model, enzyme kinetics, data-driven models, probability distributions, Monte Carlo simulations, random walk, diffusion, cellular automaton simulations, and high performance computing.

PREREQ: MATH 254 WITH A GRADE OF C OR BETTER AND MATH 355

MATH 415 MODERN ALGEBRA AND NUMBER THEORY FOR THE ELEMENTARY TEACHER 3 Units

An introduction to modern algebra with special emphasis on the number systems and algorithms which underlie the mathematics curriculum of the elementary school. Topics from logic, sets, algebraic structures, and number theory.

PREREQ: MATH 149 WITH C OR BETTER AND MATH 152 WITH C OR BETTER

UNREQ: MATH 415 AND MATH 452

MATH 416 GEOMETRY FOR THE ELEMENTARY TEACHER 3 Units

A study of the intuitive, informal geometry of sets of points in space. Topics include elementary constructions, coordinates and graphs, tessellations, transformations, problem solving, symmetries of polygons and polyhedra, and use of geometry computer software.

PREREQ: MATH 149 WITH C OR BETTER

MATH 417 THEORY OF NUMBERS 3 Units

A study of the properties of integers, representation of integers in a given base, properties of primes, arithmetic functions, module arithmetic. Diophantine equations and quadratic residues. Consideration is also given to some famous problems in number theory.

PREREQ: MATH 280 OR MATH 415 OR CONSENT OF INSTRUCTOR

MATH 420 APPLIED REGRESSION ANALYSIS 3 Units

This is a second course in regression analysis and its applications. Topics include correlation, simple and multiple linear regression, model assumptions, inference of regression parameters, regression diagnostics and remedial measures, categorical predictors, multicollinearity,and model selection. Real data re emphasized and analyzed using statistical software such as R or SAS.

PREREQ: MATH 342 OR CONSENT OF INSTRUCTOR

MATH 421 MATHEMATICS FOR HIGH SCHOOL TEACHERS I 3 Units

The course revisits the high school curriculum from an advanced perspective. The focus is on deepening understanding of concepts, highlighting connections and solving challenging problems. The mathematical content includes number systems, functions, equations, integers, and polynomials. Connections to geometry are emphasized throughout the course.

PREREQ: MATH 280, MATH 301 AND AT LEAST AN ADDITIONAL 3 CREDITS IN UPPER LEVEL MATH

MATH 422 MATHEMATICS FOR HIGH SCHOOL TEACHERS II 3 Units

The course continues the exploration of the high school curriculum from an advanced perspective that was started in MATH 421. The focus is on deepening understanding of concepts, highlighting connections and solving challenging problems. The mathematical content includes congruence, distance, similarity, trigonometry, area, and volume. Connections to algebra are emphasized throughout the course.

PREREQ: MATH 353 AND MATH 421

MATH 431 TOPOLOGY 3 Units

An introduction to point-set topology, including such topics as topological spaces, mappings, connectedness, compactness, separation axioms, metric spaces, complete spaces, product spaces and function spaces.

PREREQ: MATH 255 AND EITHER MATH 280 OR CONSENT OF INSTUCTOR

MATH 441 PROBABILITY THEORY 3 Units

Probability spaces, discrete and continuous random variables, mathematical exceptation, discrete and continuous distributions.

PREREQ: MATH 255 WITH A GRADE OF C OR BETTER

MATH 442 MATHEMATICAL STATISTICS 4 Units

This course will cover moment generating functions, moments of linear combinations of random variables, conditional expection, functions of random variables, sampling distributions, the theory of estimation, Bayesian estimation, hypothesis testing, nonparametric tests, and linear models.

PREREQ: MATH 441 WITH GRADE OF C OR BETTER

MATH 448 ACTUARIAL EXAMINATION PREPARATION FOR EXAM P/1 1 Units

The course is designed to prepare students for Exam P/1, the first actuarial exam which tests students' knowledge of and ability to use and apply fundamental probability tools in assessing risk. Basic concepts from risk theory are introduced, probability theory is reviewed, and sample questions from previous exams are discussed.

COREQ: MATH 442

MATH 449 ACTUARIAL EXAMINATION PREPARATION *Repeatable* 1 Units

This course is designed to prepare students for Exam FM/2, the second acturial exam which tests students' knowledge and understanding of the fundamental concepts of financial mathematics. Derivatives are introduced, interest theory is reviewed, and sample questions from previous exams and practice exams from other sources are discussed.

PREREQ: MATH 346

MATH 450 GRAPH THEORY 3 Units

This course will examine basic concepts and applications of graph theory. Topics covered will be selected from trees, connectivity, paths and cycles, coloring, matching and covering problems, digraphs, and network flows.

PREREQ: MATH 280 OR (COMPSCI 215 AND (MATH 253 WITH C OR BETTER OR MATH 250 WITH B OR BETTER))

MATH 452 INTRODUCTION TO ABSTRACT ALGEBRA 3 Units

An introductory survey of abstract algebra and number theory with emphasis on the development and study of the number systems of integers, integers mod n, rationals, reals, and complex numbers. These offer examples of and motivation for the study of the classical algebraic structures of groups, rings integral domains and fields. Applications to algebraic coding theory and crystallography will be developed if time allows.

PREREQ: MATH 280 OR CONSENT OF INSTUCTOR

UNREQ: MATH 415 AND MATH 452

MATH 453 ABSTRACT ALGEBRA 3 Units

This course is a continuation of MATH 452 with emphasis on ring and field theory. Topics include a review of group theory, polynomial rings, divisibility in integral domains, vector spaces, extension fields, algebraic extension fields, finite fields, etc.

PREREQ: MATH 355 AND MATH 452

MATH 458 APPLIED MATHEMATICAL ANALYSIS 3 Units

Selected topics in ordinary differential equations: series solutions, stability, transform methods, special functions, numerical methods, vector differential calculus, line and surface integrals.

PREREQ: MATH 361

MATH 459 PARTIAL DIFFERENTIAL EQUATIONS 3 Units

Fourier analysis, partial differential equations and boundary value problems, complex variables, and potential theory.

PREREQ: MATH 361

MATH 463 COMPLEX VARIABLES 3 Units

This course is a study of the algebra and geometry of complex numbers, the properties of analytic functions, contour integration, the calculus of residues and the properties of power series.

PREREQ: MATH 255

MATH 464 ADVANCED CALCULUS 3 Units

This course presents a rigorous treatment of the differential and integral calculus of single variable functions, convergence theory of numerical sequences and series, uniform convergence theory of sequences and series of functions, metric spaces, functions of several real variables, and the inverse function theorem. This course contains a writing component.

PREREQ: MATH 301

MATH 471 NUMERICAL ANALYSIS 3 Units

Emphasis on numerical algebra. The problems of linear systems, matrix inversion, the complete and special eigenvalue problems, solutions by exact and iterative methods, orthogonalization, gradient methods. Consideration of stability and elementary error analysis. Extensive use of microcomputers and programs using a high level language. This course contains a writing component.

PREREQ: MATH 355 AND (COMPSCI 170 OR COMPSCI 171 OR COMPSCI 172 OR COMPSCI 174 OR COMPSCI 220 OR COMPSCI 222 OR COMPSCI 347)

MATH 474 DYNAMICAL SYSTEMS & CHAOS 3 Units

An analytic, geometric, and intuitive study of continuous and discrete low-dimensional nonlinear dynamical systems. The basic notions of stability, bifurcations, chaotic systems, strange attractors, and fractals are examined. Specific applications will be taken from diverse fields such as Biology, Chemistry, Economics, Engineering, and Physics.

PREREQ: MATH 361 WITH A C OR BETTER

MATH 490 WORKSHOP 1-3 Units

Variable topics. Group activity oriented presentations emphasizing `hands on` and participatory instructional techniques. Repeatable. Prereq: Consent of instructor

MATH 492 FIELD STUDY *Repeatable* 1-3 Units

A study for which data is obtained or observations are made outside the regular classroom. Repeatable. Instructor Consent required.

MATH 494 SEMINAR *Repeatable* 2 Units

Variable topics. Group activity. An advanced course of study in a defined subject matter area emphasizing a small group in intense study with a faculty member. Repeatable. Instructor Consent required.

MATH 496 SPECIAL STUDIES *Repeatable* 1-3 Units

Variable topics. Group activity. Not offered regularly in the curriculum but offered on topics selected on the basis of timeliness, need, and interest, and generally in the format of regularly scheduled Catalog offerings. Repeatable three times maximum in 6 years. Instructor Consent required.

MATH 497 EXCHANGE STUDY *Repeatable* 1-12 Units

Variable topics

MATH 498 INDEPENDENT STUDY *Repeatable* 1-5 Units

Study of a selected topic or topics under the direction of a faculty member. Repeatable. Department Consent required.

MATH 498R INDEPENDENT STUDY - UNDERGRADUATE RESEARCH *Repeatable* 1-3 Units

Study of a selected topic or topics under the direction of a faculty member. Repeatable. Department Consent required.

MATH 499 PROJECT FOR MAJORS 1 Units

This course is designed to give students experience and to improve their skill in reading, writing, and understanding mathematics by requiring them to research one or more mathematical topics and then write a report about their activities and discoveries. The focus is on the learning and communication of mathematics: how to read with understanding, write with clarity and precision, and in the process discover how writing can aid in understanding.

PREREQ: JUNIOR/SENIOR STATUS OR CONSENT OF INSTRUCTOR