# Mathematics (MTH)

College of Science and Health

Department Chair: Bruce Riley

1034 Cowley Hall,

(608)785-8382

email:
riley.bruc@uwlax.edu

**Professors:** Hoar, Kelly, S., Riley, B.,
Senger, Skala;

**Associate Professors: **Baggett, Boyle, Elfessi,
Hulett, Kattchee, Koster, D., LeDocq, Matchett, Reineke, Will;

**Assistant Professors:** Bennie, Hasenbank, Kosiak,
Okamoto, P., Peirce, Toribio.

## Credit by Examination

The mathematics department awards credit by examination in MTH 150, College Algebra; MTH 151, Pre-calculus; MTH 207, Calculus I; and MTH 208, Calculus II. The exams may be taken by new freshmen and are scheduled for the second week of semester I. Students may not earn credit by examination in both MTH 150 and MTH 151. Credit for MTH 145 will be granted to entering freshmen with a College Board Advanced Placement Statistics Examination score of three or better. Credit for MTH 207 will be granted to entering freshmen with a College Board Advanced Placement AB exam score of three or better. Entering freshmen with a score of three or better on the BC form of this exam will receive credit for MTH 207 and should consult the mathematics department chair about additional credit and placement in mathematics courses.

## Retroactive Credits

Students taking MTH 207, 208, or 309, and earning a grade of "B" or better, will be given retroactive credit for MTH 151 provided that the student's transcript shows no record of prior or concurrent enrollment in MTH 151, 207, 208, or 309.

## Dual Degree Program in Mathematics and Engineering

Students with a math major who are also interested engineering are able to receive both a Bachelor of Science (mathematics major) from UW-La Crosse and a Bachelor of Science (engineering major) from UW-Madison. At UWL, students must complete a minimum of 85 credits, including the General Education requirements and specific math and science course in preparation for the engineering program. Students are recommended to include the following courses in their work at UWL: CHM 103,104; CS 120; ECO 110,120, 336; MTH 207, 208, 225, 309, 310, 341, 353, 371, 480 or 461; PHY 203, 204, 321. Students interested in Industrial Engineering should add ACC 221. Students should consult with the mathematics department chair for specific course and sequence advising for this agreement.

Students who express interest in the dual degree program will be selected for entrance into the UW-Madison portion of the program based on their GPA in all course work; their GPA in the chemistry, computer science, mathematics and physics course work required by the program; and the positive recommendation of the UWL mathematics department chair (or designee). Qualified UW-La Crosse applicants are assured admission in the College of Engineering at UW-Madison.

In order to receive the B.S. degree (mathematics major) from UWL, students have two options: 1) complete the selected engineering program at UW-Madison, or 2) complete applicable math courses at UW-Madison to satisfy the remaining math electives required for mathematics major at UWL. In either case the student would then transfer these credits to UW-La Crosse. For the typical student, the remaining credits must include 1013 credits at the 300 level or above. This transfer of credits and awarding of the B.S. degree (mathematics major) by UWL can take place as soon as the student demonstrates completion of math courses at UW-Madison to meet UWL mathematics electives requirements or the completion of the engineering sequence at UW-Madison.

## Mathematics Major

(All colleges, excluding Teacher Certification programs) - 38 credits - MTH 207, 208, 225, 309, 310, 407, 411; 9 credits from: 311, 317, 320, 331, 341, 353, 371, 408, 410, 412, 413, 441, 442, 461, 480, CS 453, PHY 470. In addition CS 120 must be taken. Mathematics-computer science double majors may count MTH 317, MTH 371, and CS 453 for credits in both majors. Mathematics-physics double majors may count MTH 461 and PHY 470 for credit in both majors.

## Mathematics Major with Applied Emphasis

(All colleges excluding Teacher Certificates Programs) - 39 credits - MTH 207, 208, 225, 309, 310, 353, 371; 461 or 480; 9 credits from: MTH 341, 407, 408, 410, 413, 441, 442, 448, 461, 480. Three of these credits may be met by completing one of the following courses: CHM 310, CS 453, PHY 470, PHY 474. In addition, CS 120 must be taken.

## Mathematics Major with Education Emphasis

(All colleges; student seeking teacher certification should complete the major through the School of Education) - 38 credits - MTH 207, 208, 225, 309, 310, 331, 341, 411; 6 credits from: MTH 151 (if taken for grade rather than retro-credit), 311, 317, 320, 353, 371, 407, 410, 413, 441, 442, 461, 480. In addition CS 120 must be taken (4 credits).

## Mathematics Major with Statistics Emphasis

(All colleges, excluding Teacher Certification programs) - 39 credits - MTH 207, 208, 309, 310, 341, 441, 442, 445, 446; six credits from MTH 305, 371, 407, 410, 444, 447, 448. In addition, CS 120 must be taken.

## Mathematics Minor

(All colleges, excluding Middle Childhood-Early Adolescence Level Education Program) - 22 credits - MTH 207, 208, 309; 9 credits from MTH 225, MTH courses numbered 310 and above. Credit for MTH 151 may be used to fulfill three of the nine additional credits. Computer science majors may count MTH317, 371 and CS 453 for credit in both the computer science major and the mathematics minor. Physics majors may count MTH 461 and PHY 470 for credit in both the physics major and the mathematics minor.

## Mathematics Minor

(Middle Childhood-Early Adolescence Education Program) - 22 credits - MTH 125, 126, 171, 280, 175 or 207; 4 credits from: MTH 145, 151, 208, MTH courses numbered 300 and above.

## Statistics Minor

(All colleges) - 20 credits - MTH 145, 175 or 207, 305; 9 credits from MTH 444, 445, 446, 447, 448. With the approval of the mathematics department chair, a research methods course from another department may be substituted for three of the nine additional credits. MTH 341 may be substituted for MTH 145. This minor is not available to mathematics majors.

## Mathematics Minor with Education Emphasis

(All colleges) - 28 credits - MTH 207, 208, 225, 309, 310, 331, 341.

**The mathematics department incorporates a significant
amount of writing through the required courses instead of
identifying particular courses as writing emphasis courses.
Students who complete the math major will fulfill the university
writing emphasis requirement. **

**+** next to a course number indicates a **General
Education** course.

### MTH 050 Cr.3 *transcript Basic Algebra

A review of beginning algebra. Topics include an elementary treatment of real numbers, polynomials, linear equations, inequalities, rational expressions, systems of linear equations, radicals, and quadratic equations. Letter grade, but only F calculated in GPA. Transcript credit only.

### MTH 051 Cr. 2 *transcript Topics in Intermediate Algebra

A course to enhance the student's skills in selected areas of intermediate algebra; areas covered include polynomials, rational expressions, exponents, equations, and inequalities. Prerequisite: MTH 050 or an appropriate placement test score. Letter grade, but only F calculated in GPA.

*Transcript credit does not count toward graduation.

### MTH 125 Cr. 4 Mathematics for Elementary Teachers

A study of the mathematical concepts and techniques that are fundamental to, and form the basis for, elementary school mathematics. Topics include problem solving, inductive and deductive reasoning, sets, number systems through the real numbers, number theory, measurement, and 2and 3dimensional geometry. Prerequisite: MTH 051 or satisfactory placement test score.

### + MTH 126 Cr. 4 Mathematics for Elementary Teachers II

Continued study of the mathematical concepts and techniques that are fundamental to, and form the basis for, elementary school mathematics. Topics include use of probability and statistics to explore real-world problems; representation and analysis of discrete mathematical problems using counting techniques, sequences, graph theory, arrays and networks; use of functions, algebra and the basic concepts underlying the calculus in real-world applications. Prerequisite: MTH 125 (and MTH 051 or satisfactory placement test score for transfer students)

### + MTH 145 Cr. 4 Elementary Statistics

An introductory course covering fundamentals of modern statistical methods. Topics include descriptive statistics, the binomial and normal distributions, estimation, and hypothesis testing. The z, t, F and chi-square test statistics are introduced. Instruction in computer use is included, and statistics software is used throughout the course for analyzing data files and carrying out statistical procedures. Prerequisite: MTH 050 or an appropriate placement test score.

### + MTH 150 Cr. 4 College Algebra

A college algebra course on the properties, graphs, and applications of elementary functions. Topics include the real and complex numbers, concepts from analytic geometry, solutions to equations and inequalities, the elementary algebraic functions, and the logarithmic and exponential functions. Prerequisite: MTH 051 or two years of high school algebra and an appropriate placement test score. (Successful completion of MTH 151, 175 or 207 precludes taking MTH 150 for credit.)

### + MTH 151 Cr. 4 Pre-calculus

A pre-calculus course on properties, graphs, and applications of elementary transcendental functions. Topics include concepts from analytic geometry; theory of equations; the logarithmic, exponential, trigonometric, and inverse trigonometric functions; and analytic trigonometry. Prerequisite: MTH 150 or two years of high school algebra and an appropriate placement test score. (Successful completion of MTH 151 precludes taking MTH 150 for credit. Successful completion of MTH 207 precludes taking MTH 151 for credit.)

### MTH 171 Cr. 3 Geometry for Elementary Teachers

Analysis and description of 2and 3dimension geometric concepts. Intuitive, direct, and indirect proofs and applications of geometric principles. Transformations, similarities and symmetry. Some topics from measurement. Prerequisite: MTH 125 and either MTH 150 or math placement above MTH 150. Offered Sem. I.

### + MTH 175 Cr. 4 Applied Calculus

Basic concepts and methods from differential, integral, and multivariate calculus. Logarithmic and exponential functions are included, but not trigonometric functions. Emphasis of the course is on models and applications in business and the social, life, and physical sciences. Prerequisite: MTH 150 or two years of high school algebra and an appropriate placement test score. (Successful completion of MTH 175 precludes taking MTH 150 for credit. Successful completion of MTH 207 precludes taking MTH 175 for credit.)

### + MTH 207 Cr. 5 Calculus I

A rigorous introduction to calculus. Topics include limits, rules for differentiation, derivatives of trigonometric, logarithmic and exponential functions, the Mean Value Theorem, integration, and the Fundamental Theorem of Calculus. In the area of applications, the course covers problems on related rates, extrema, areas, volumes, and Newton's Second Law. Prerequisite: MTH 151 or four years of high school mathematics, including trigonometry, and appropriate placement score. (Successful completion of MTH 207 precludes taking MTH 150, 151, or 175 for credit.)

### + MTH 208 Cr. 4 Calculus II

A continuation of Calculus I with a rigorous introduction to sequences and series. Topics include techniques of integration and indeterminate forms, improper integrals, applications of integrals to the physical sciences, tests for the convergence of series, absolute convergence, power series, and Taylor's Theorem with Remainder. First order linear differential equations are explored, as well as the geometry of space. Prerequisite: MTH 207.

### MTH 225 Cr. 4 Logic and Discrete Mathematics

An introduction to mathematical reasoning. Mathematical logic, including quantification and the predicate calculus is introduced and used to discuss set theory, relations, functions, counting, graphs, and algorithms. Elementary proofs, including proofs by induction are stressed. Prerequisite: MTH 175 or 207.

### MTH 280 Cr. 3 Problem Solving for Elementary Teachers

A high activity course designed to enhance skills in problem solving. Includes methods of representing problems, general strategies for solving problems, creative problem posing and ways to evaluate progress in problem solving skills. Examples taken from the elementary school curriculum. Prerequisite: MTH 125 and either MTH 150 or math placement above MTH 150.Offered Sem. II.

### MTH 305/505 Cr. 3 Statistical Methods

A survey of statistical methods from the point of view of how these methods are implemented with a standard statistics software package. Topics include descriptive statistics, graphical methods, tests of location, goodness of fit, simple and multiple regression, design of experiments, ANOVA, multiple comparisons, chi-square tests. Both parametric and nonparametric methods are treated. Computer use is an integral part of the course. Prerequisite: MTH 145 or 341. Offered Sem. I.

### MTH 309 Cr. 4 Linear Algebra with Differential Equations

A systematic study of linear algebra, and its interactions with differential equations. Topics include: vectors, matrices, systems of linear equations, determinants, vector spaces, subspaces, basis and dimension, linear transformations and their matrix representations, similar matrices and diagonalization, systems of first order linear differential equations, and higher order linear differential equations. Prerequisite: MTH 208.

### MTH 310 Cr. 4 Calculus III: Multivariate Calculus

A course in higher dimensional calculus, partial derivatives, and multiple integrals. Topics include parametric curves, polar (and other) coordinate systems, vector fields, scalar fields, the gradient vector, chain rule, Jacobian, Green's Theorem, Stokes' Theorem, and the Divergence Theorem. Prerequisite: MTH 208.

### MTH 311 Cr. 3 Number Theory

Topics are selected from such areas as: divisibility and factorization, congruence, distribution of prime numbers, and Diophantine equations. Problem-solving strategies and unsolved problems are stressed. Applications are developed in areas such as primality testing and public key cryptography. Prerequisite: MTH 225 and 309 or concurrent enrollment. Offered Sem. I, even-numbered years.

### MTH 317 Cr. 3 Graph Theory

An introduction to graph theory-emphasizing algorithms. Topics include graphs and sub graphs, isomorphism, degree sequences, digraphs, networks, algorithm complexity and NP-completeness, trees, Euler circuits and Hamilton cycles, planarity and graph coloring. Prerequisite: MTH 225, 309 and CS 120. Offered Sem. I, odd numbered years.

### MTH 320 Cr. 3 History of Mathematics

A study of the evolution of mathematics. Discussion and evaluation of major periods of development including the lives and works of preeminent mathematicians. A sampling of problem solving methods from various historical periods. Emphasis is on Western mathematics from earliest recorded history through the initial developments of calculus and modern mathematics. Prerequisite: MTH 309 or concurrent enrollment. Offered Sem. II, odd-numbered years.

### MTH 331 Cr. 3 Introduction to Modern Geometry

A thorough discussion of transformations and their use in proving congruence of geometric figures; selected theorems concerning the triangle and circle, and constructions possible given different parts of a triangle. Prerequisite: MTH 225 and 309 or concurrent enrollment. Offered Sem. II.

### MTH 341 Cr. 4 Probability and Statistics

An initial course in probability and statistics for students strong in mathematics. Probability topics include sample spaces, random variables, independence, and the binomial, Poisson, normal, and exponential distributions and their applications. Calculus-based methods will be used for analyzing continuous distributions. Statistics topics include descriptive statistics, sampling distributions, confidence intervals, hypothesis testing, regression, and ANOVA. Prerequisite: MTH 208.

### MTH 353 Cr. 3 Differential Equations

Fundamental existence and uniqueness theory, linear independence and the Wronskian, series solutions near regular singular points, Laplace transforms and systems of first order linear equations. Fourier Series and the method of separations of variables will be applied to the heat equation, wave equation and Laplace's equation. Prerequisite: MTH 309 and 310. Offered Sem. I.

### MTH 371 Cr. 3 Introduction to Numerical Methods

Techniques devised for use with computing machinery are applied to problems such as: solving nonlinear equations and linear systems, curve-fitting and function approximation, numerical integration. Prerequisite: MTH 309 and CS 120. Offered Sem. II.

### MTH 395 Cr. 13 Special Topics in Mathematics

Special topics in mathematics not covered by regular courses taught in this department. The particular topic is decided mutually by the student and instructor. Prerequisite: written consent of department chair. Repeatable for credit - maximum 6.

### MTH 407 Cr. 4 Real Analysis I

This course covers the basic theory underlying the differential and integral calculus. Convergence of sequences and series is examined. Theoretical concepts of calculus are examined and particular attention is given to writing proofs. Prerequisite: MTH 225, 309, 310. Offered Sem. I.

### MTH 408 Cr. 3 Real Analysis II

Further study of real analysis via selected topics such as the theory of convergence, Lebesque-Stieltjes integration, Fourier Analysis, probability theory, approximation theory and metric spaces. Prerequisite: MTH 407. Offered Sem. II, even-numbered years.

### MTH 410 Cr. 3 Complex Analysis

Introduction to complex numbers and complex functions. Analytical functions are developed via power series and Cauchy's Theorem. Topics include Cauchy-Riemann conditions, integration theorems, residues, conformal mapping and applications. Prerequisite: MTH 309 and 310. Offered Sem. II, even-numbered years.

### MTH 411 Cr. 4 Abstract Algebra I

A rigorous approach to algebraic systems including the study of groups, rings, integral domains and fields with application to polynomials. Prerequisite: MTH 225 and 309. Offered Sem. I.

### MTH 412 Cr. 3 Abstract Algebra II

Continuation of MTH 411. Further study of rings, integral domains and fields. Prerequisite: MTH 411. Offered Sem. II, even-numbered years.

### MTH 413 Cr. 3 Topics in Linear Algebra

A thorough study of selected topics such as abstract vector spaces, linear transformations and their matrix representations, inner product spaces, canonical forms; selected applications. Prerequisite: MTH 225 and 309. Offered Sem. II, odd-numbered years.

### MTH 440 Cr. 1 Statistical Consulting

Experiences will include interpersonal written and oral communication and interdisciplinary exposure as well as opportunities to apply statistical knowledge in a broad variety of situations. Students will take part in consultations (i.e. extracting information, listening, asking appropriate questions), apply knowledge in experimental design, data modeling, use of statistical software, and/or sampling; diagnose and conduct appropriate statistical procedures and interpret and communicate results. Reading past and present literature on statistical consulting also will be required. Prerequisites: MTH 305 or MTH 341 and written consent of the Statistical Consulting Center director. Repeatable for credit maximum - 3.

### MTH 441 Cr. 3 Mathematical Statistics I

Review of discrete and continuous random variables. Moment generating functions, multivariate probability distributions, marginal and conditional probability distributions, functions of random variables, order statistics, Central Limit Theorem, point estimation and confidence intervals. Prerequisite: MTH 310 and 341. Offered Sem. I.

### MTH 442 Cr. 3 Mathematical Statistics II

Methods of estimating, including method of moments and maximum likelihood. Sufficient statistics, hypothesis testing, power of tests, likelihood ratio tests and introduction to regression and analysis of variance. Prerequisite: MTH 441. Offered Sem. II, even-numbered years.

### MTH 444/544 Cr. 3 Introduction to Sampling

Methodology of simple random sampling, stratified sampling, systematic sampling, cluster sampling, and multistage sampling is developed and applications related to physical, social, and biological sciences are given. Single and two variable estimation techniques are discussed. Prerequisite: MTH 305 or 341. Offered Sem. I, odd-numbered years.

### MTH 445/545 Cr. 3 Correlation and Regression Analysis

An introduction to simple linear regression, multiple regression, polynomial regression. Inferences, appropriateness of model, model diagnostics/adequacy, difficulties in the application of models are discussed. A computer package will be used. Course participants will be involved with hands-on statistical applications and consulting. Prerequisite: MTH 305 or 341. Offered Sem. I, even-numbered years.

### MTH 446/546 Cr. 3 Analysis of Variance and Design of Experiments

An introduction to single factor, multiple factor, and randomized block designs in analysis of variance. Inferences, appropriateness of model, model diagnostics/adequacy, difficulties in the application of models are discussed. Design or structure of an experiment will be discussed. A computer package will be used. Course participants will be involved with hands-on statistical applications and consulting. Prerequisite: MTH 305 or 341. Offered Sem. II, odd-numbered years.

### MTH 447/547 Cr. 3 Nonparametric Statistics

An introductory course presenting the theory and procedures for using distribution-free methods in data analysis. Standard procedures, such as the Wilcoxon tests, Kruskal-Wallis, Kolmogorov-Smirnov, nonparametric confidence intervals, regression analysis, and powers of the tests will be included. Computer programs will be used when appropriate. Prerequisite: MTH 305 or 341. Offered Sem. II, even-numbered years.

### MTH 448 Cr. 3 Operations Research

An introductory course which applies mathematics/statistics to management decision making. Included are methods of optimizing systems, inventory and production control, scheduling, game theory bidding, queuing, quality control, reliability and time series. Various programming, analysis and Monte Carlo techniques are introduced with the computer used as a tool where appropriate. Prerequisite: MTH 305 or 341. Offered Sem. II, odd-numbered years.

### MTH/PHY 461 Cr. 3 Mathematical Physics

In depth study of topics from vector analysis, Fourier analysis and special functions with emphasis on modeling physical phenomena involving conservative fields, fluid flow, heat conduction, and wave motion. Prerequisite: MTH 353. (Cross-listed with PHY; may only earn credit in MTH or PHY.) MTH 461 may be counted towards both a MTH and PHY major. Offered Sem. II, odd numbered years.

### MTH 480 Cr. 3 Studies in Applied Mathematics

Advanced studies of applications of mathematics and computation to solve problems and understand processes from a variety of fields (for example, industry, medicine and the physical and life sciences.) Requirements include an application/ modeling project with a written report and class presentation. Prerequisite: MTH 353. Offered Sem. II.

### MTH 495/595 Cr. 1-3 Special Topics in Mathematics

Special topics in mathematics not covered by regular courses taught in this department, such as topology, set theory and advanced numerical analysis. The particular topic is decided mutually by the students and the instructor. Prerequisite: written consent of the department chair. Repeatable for credit - maximum 6.

### MTH 496/596 Cr. 1-3 Special Topics in Statistics

Special topics in statistics not covered by regular courses taught in this department. The particular topic is decided by the instructor. Prerequisite: Written consent of the department chair. Repeatable for credits- maximum 6.

### MTH 498 Cr. 1-3 Independent Study

Directed readings or presentation of material not available in formal departmental courses under the supervision of a faculty member. Prerequisite: written consent of the supervising faculty member and the department chair. Repeatable for credit - maximum 6.

### MTH 499 Cr. 1-3 Research Topics

An opportunity to pursue individual research topics under the direction of a faculty member. Depending on the nature of the research project, study is expected to involve substantial computational or theoretical work in addition to literature review and instruction. In addition to a written report to the supervising faculty member, expected outcomes may include: software, papers and presentations to the department and regional meetings. Prerequisite: written consent of the supervising faculty member and the department chair. Not applicable to a mathematics major or minor. Repeatable for credit - maximum 6.