# Mathematics Minor

23 credits

**Why mathematics?**

A mathematics minor will benefit students majoring in subjects that involves critical thinking and/or problem solving. Students who are majoring in a field that makes significant use of mathematics (e.g., Biology, Business, Chemistry, Computer Science, or Physics) are especially encouraged to consider a Mathematics minor.

**Goes great with**:

- Accounting
- Biochemistry
- Biology
- Biology Clinical Laboratory Science/Medical Technology
- Biology Cytogenetic Technology
- Biology Cytotechnology
- Biology Nuclear Medicine Technology
- Biology Pre-Physical Therapy
- Chemistry
- Entreprenuership
- Environmental Biology
- Human Resource Management
- Marketing
- Physics
- Physics with Engineering Emphasis
- Sport Management

(From the 2013-15 Catalog)

A. **Calculus Core**

All of the following:

(Either M148 and M149 or M151)

M148 - Calculus with Precalculus I (4 credits)

This course, followed by M149, provides a two-semester sequence that covers the material of a traditional Calculus I course along with built-in coverage of precalculus topics. Topics in M148 include: solving equations, functions, classes of functions (polynomial, rational, algebraic, exponential, logarithmic), right triangle trigonometry, angle measure, limits and continuity, derivatives, rules for derivatives. Credit is not granted for this course and M151 or courses equivalent to college algebra and college trigonometry.

Prerequisite: mathematics competency satisfied.

M149 - Calculus with Precalculus II (4 credits)

This course completes the two-semester sequence that begins with M148, and together with M148 provides a two-semester sequence that covers the material of a traditional Calculus I course along with built-in coverage of precalculus topics. Topics in M149 include: trigonometric and inverse trigonometric functions, rules for derivatives, applications of derivatives, and definite and indefinite integrals. Credit is not granted for this course and M151.

Prerequisite: M148.

This course provides an introduction to the differential and integral calculus. Topics include: the concepts of function, limit, continuity, derivative, definite and indefinite integrals, and an introduction to transcendental functions. Credit is not granted for this course and M148 and M149.

Prerequisites: departmental placement or courses equivalent to college algebra and college trigonometry.

M152 - Calculus II (4 credits)

This course is a continuation of M151. Some of the topics of M151 are revisited at a higher mathematical level. Topics include: applications of the definite integral, techniques of integration, improper integrals, introduction to differential equations, numerical methods for integration and approximation, curves in the plane given parametrically, polar coordinates, and vectors in 2-space and 3-space.

Prerequisite: Minimum grade of C in either M149 or M151 or departmental placement.

M251 - Calculus III (4 credits)

This course continues the development of Calculus from M151 and M152. Topics include: sequences and series, and differentiation and integration of vector-valued functions and functions of several variables.

Offered fall semester. Prerequisite: Minimum grade of C in M152.

B. Both of the following:

M252 - Linear Algebra (4 credits)

This course provides an introduction to techniques and applications of linear algebra. Topics include: systems of linear equations, matrices, determinants, Euclidean n-space, real vector spaces, basis and dimension, linear transformations, inner products, and eigenvalues and eigenvectors.

Offered spring semester. Prerequisite: Minimum grade of C in M152.

ST232 - Introduction to Statistics (2 credits)

This course is designed to provide the basic ideas and techniques of statistics. Topics include: descriptive and inferential statistics, an intuitive introduction to probability, estimation, hypothesis testing, chi-square tests, regression and correlation. This course makes significant use of appropriate technology. Topics in this course are treated at a higher mathematical level than they are treated in ST132. Credit is not granted for this course and any of the following: BU215, B392 or ST132.

Offered fall semester. Prerequisite: departmental placement or minimum C grade in M148.

C. Two of the following courses (M332 is recommended).

M301 - Foundations of Advanced Mathematics (2 credits)

This course looks at topics central to further study in mathematics. Topics include symbolic logic, especially as it applies to mathematical proof; methods of mathematical proof such as direct proof, indirect proof, proof by induction; use and meaning of mathematical quantifies and predicates; sets; relations; equivalence relations and partitions; order relations; functions and their properties; and complex numbers. A junior assessment test is administered as part of this course.

Offered fall semester. Prerequisite: M251 (may be concurrent).

M310 - Combinatorics and Graph Theory (3 credits)

This course provides an introduction to combinatorial and graph theoretical techniques in mathematics. It is also designed for students in computer science. Topics include: sets, functions, combinatorial techniques, graph theory, searching algorithms, and trees.

Prerequisites: CS106 or CS110/111, and M152.

M315 - Number Theory (3 credits)

This course provides an introduction to elementary number theory. Topics include: divisibility, prime and composite numbers, congruences, arithmetical functions, primality testing, factorization techniques, and applications to cryptography.

Prerequisite: M152.

M321 - Modern Geometry (3 credits)

This course is required for the Mathematics Education major. The course is designed to be an introduction to the foundations of geometry. Topics include: Euclidean geometry, non-Euclidean geometry, projective geometry, and geometric transformations.

Offered in alternate spring semesters. Prerequisite: M152.

M332 - Probability (2 credits)

This calculus-based course is designed to provide mathematics majors and minors with an introduction to the mathematical underpinnings of statistics. Topics include: probability axioms, probability, Bayesﾒ Theorem, random variables, discrete and continuous probability distributions, and expected value.

Offered spring semester. Prerequisite: M152.

M341 - Differential Equations with Applications (3 credits)

This course provides an introduction to the theory, methods, and applications of ordinary differential equations. Topics include: first order differential equations, linear differential equations with constant coefficients, and systems of differential equations.

Prerequisites: M251 and M252.

M342 - Numerical Analysis (3 credits)

This course provides an introduction to the theory and methods of numerical analysis. Topics include: numerical methods for solving linear and nonlinear equations, polynomial approximation of functions, numerical integration and differentiation, numerical approximation to solutions of differential equations, direct and iterative methods for solving systems of equations.

Prerequisites: either CS106 or CS110/111, and M251, M252.

M344 - Applied Mathematics (3 credits)

This course serves physics majors as well as those mathematics majors whose area of interest is analysis. Topics include: Fourier series, the complex numbers, analytic functions, and derivatives and integrals of complex functions. Other topics may include Laurent series and residues, partial differential equations and boundary value problems.

Prerequisites: M251 and M252.

M348 - Complex Analysis (3 credits)

This course provides an introduction to the theory of functions of one complex variable. Topics include: the complex numbers, the complex derivative, analytic functions, power series, complex integration, Cauchyﾒs Theorem and Cauchyﾒs Integral Formula, Laurent series, and residues and poles.

Prerequisite: M251.

M356 - Introduction to Scientific Computing (3 credits)

A course designed to provide undergraduates students with the basic computational tools and techniques needed for their study in science and mathematics. Students learn by doing projects that solve problems in physical sciences and mathematics using symbolic and compiled languages with visualization. By use of the Sage problem-solving environment and the Python programming language, the students learn programming and numerical analysis in parallel with scientific problem solving.

Also offered as CS356 and P356. Prerequisites: CS106, M251, M252, and ST232.

M361 - Operations Research (3 credits)

This course is required for the mathematics education major, providing an introduction to techniques and applications of operations research. Topics include: linear programming, game theory, queuing theory, Markovian decision processes, and decision theory.

Offered in alternate spring semesters. Prerequisites: M252 and M332.

M380-389 - Special Topics (3 credits)

Special topics in mathematics may be offered depending on student interest.

M401 - Abstract Algebra (3 credits)

This course provides an introduction to algebraic structures. Topics include: groups, subgroups, quotient groups, group homomorphisms, rings, ideals, and fields.

Offered in alternate spring semesters. Prerequisites: M251, M252, and M301.

M411 - Introduction to Analysis (3 credits)

This course provides a rigorous treatment of topics in calculus. Topics include: sequences, functions, limits, continuity, derivatives, and integration.

Offered in alternate spring semesters. Prerequisites: M251, M252, and M301.

M496/497 - Mathematics Internship (1-17 credits)

This opportunity provides the student with experience in mathematical research or applications. The internship must be approved by the department and, depending on the nature of the internship, may be counted towards the major. Students generally are expected to give a presentation following the internship.

ST350-359 - Special Topics (3 credits)

Selected topics in statistics may be offered depending on student interest.

ST371 - Applied Regression Analysis (3 credits)

This course provides students with an introduction to linear and non-linear models in statistics. Topics include: linear regression, multiple regression, one-, two-, and higher-way analysis of variance, and popular experimental designs. Real-world problems are analyzed using appropriate technology.

Prerequisites: M251, M332, and ST232.

ST373 - Design of Experiments (3 credits)

This course provides an introduction to the principles of the design of experiments from a statistical perspective. Topics include: Analysis of variance, covariance, randomization, completely randomized, randomized block, Latin-square, factorial, response surface methods and other designs.

Prerequisites: M152 and ST232.

ST431 - Mathematical Statistics I (3 credits)

This course provides a mathematical treatment of probability and statistics. Topics include: several descriptions of the concept of probability, univariate and bivariate probability distributions, joint and marginal probability distributions, covariance, hypothesis testing, estimation, data analysis, and sampling distributions.

Prerequisites: M332 (may be concurrent) and ST232.

ST496/497 - Statistics Internship (1-17 credits)

This opportunity provides the student with experience and training in statistical techniques. The internship must be approved by the department and, depending on the nature of the internship, may be counted towards the major. Students usually are expected to give a presentation following the experience.

Kevin Dennis, Ph.D.

Chair, Mathematics & Statistics Department

Saint Mary's University of Minnesota

700 Terrace Heights #59

Winona, MN 55987-1399

(800) 635-5987, Ext. 6650

kdennis@smumn.edu