## Mathematics Major/Minor

Mathematics is "the universal language." The mathematics major at Saint Mary's offers a flexible path that fits a variety of interests and opens the door to a wide range of careers.

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**Career Options**

A degree in mathematics is excellent preparation not only for a career in mathematics or elementary, middle school, or high school teaching, but also in the fields of actuarial science, computer science, finance, research analysis, or statistics.

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**High School Preparation**

High school coursework that will support a student in his or her pursuit of a mathematics degree includes exposure to Calculus, Computer Science, Discrete Math, Statistics, and Trigonometry.

#### Enhance Your Experience

Additional areas of study that go well with the mathematics major include minors in computer science, geographic information science, business, statistics, physics, and scientific computing.

#### Mathematics Minor

Mathematics minors find it it supplements most any major in the liberal arts. A minor in mathematics will provide foundational knowledge helpful to any student's post-graduate career.

#### Degree Requirements

**A. All of the following**

Please note: Students may take either M148 and M149, or M151

M148 Calculus I with Precalculus (part 1) (4 cr.)

This course, followed by M151 or courses equivalent to college algebra and college trigonometry.

M149 Calculus I with Precalculus (part 2) (4 cr.)

This course completes the two-semester sequence that begins with M151.

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 M149.

This course is a continuation 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.

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

**B. All of the following:**

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.

M301 Foundations of Advanced Mathematics (2 cr.)

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 quantifiers 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.

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' Theorem, random variables, discrete and continuous probability distributions, and expected value.

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

M411 Introduction to Analysis (3 cr.)

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

This course consists of student presentations from mathematics, mathematical modeling, mathematics education, or statistics. Each student chooses a topic in consultation with the instructor, does appropriate background reading, and prepares an oral presentation and written paper on the topic. A senior assessment test is administered as part of this course.

ST232 Introduction to Statistics (2 cr.)

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.

**C. Four of the following courses**

Four of the following courses; or three of the following courses and one course in another field using mathematics with the approval of the major advisor and the department chair.

M310 Combinatorics and Graph Theory (3 cr.)

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.

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.

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.

M341 Differential Equations with Applications (3 cr.)

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.

M342 Numerical Analysis (3 cr.)

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.

M344 Applied Mathematics (3 cr.)

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.

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's Theorem and Cauchy's Integral Formula, Laurent series, and residues and poles.

M356 Introduction to Scientific Computing (3 cr.)

This course is 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.

M361 Operations Research (3 cr.)

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.

M380-389 Special Topics (3 cr.)

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

M496/497 Mathematics Internship (1–17 cr.)

This opportunity provides the student with experience in mathematical research or applications. The internship must be approved by the department chair 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 cr.)

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

ST371 Applied Regression Analysis (3 cr.)

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.

ST373 Design of Experiments (3 cr.)

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.

ST431 Mathematical Statistics (3 cr.)

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.

ST496/497 Statistics Internship (1–17 cr.)

This opportunity provides the student with experience and training in statistical techniques. The internship must be approved by the department chair 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.

**D. Either CS106 or CS110 & CS111:**

CS106 Introduction to Programming for Sciences (3 cr.)

This course teaches introductory programming within a problem solving framework applicable to the sciences. The course emphasizes technical programming, introductory data storage techniques, and the processing of scientific data. There is an emphasis on designing and writing correct code using an easy to learn scientific programming language such as Python. Advanced excel spreadsheet concepts will be taught and utilized during the programming process. Credit is not granted for this course and CS101.

CS110 Computer Science I: Introduction to Programming (3 cr.)

This course introduces students to the practice of software development. Students learn the fundamentals of programming, algorithm development, and basic design principles.

CS111 Computer Science I Laboratory (1 cr.)

The laboratory course complements CS110 lectures.

**A. All of the following**

Please note: Students may take either M148 and M149, or M151

M148 Calculus I with Precalculus (part 1) (4 cr.)

This course, followed by M151 or courses equivalent to college algebra and college trigonometry.

M149 Calculus I with Precalculus (part 2) (4 cr.)

This course completes the two-semester sequence that begins with M151.

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 M149.

This course is a continuation 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.

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

**B. Both of the following:**

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.

ST232 Introduction to Statistics (2 cr.)

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.

**C. Two of the following courses**

M301 Foundations of Advanced Mathematics (2 cr.)

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 quantifiers 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.

M310 Combinatorics and Graph Theory (3 cr.)

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.

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.

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.

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' Theorem, random variables, discrete and continuous probability distributions, and expected value.

M341 Differential Equations with Applications (3 cr.)

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.

M342 Numerical Analysis (3 cr.)

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.

M344 Applied Mathematics (3 cr.)

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.

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's Theorem and Cauchy's Integral Formula, Laurent series, and residues and poles.

M356 Introduction to Scientific Computing (3 cr.)

This course is 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.

M361 Operations Research (3 cr.)

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.

M380-389 Special Topics (3 cr.)

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

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

M411 Introduction to Analysis (3 cr.)

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

M496/497 Mathematics Internship (1–17 cr.)

This opportunity provides the student with experience in mathematical research or applications. The internship must be approved by the department chair 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 cr.)

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

ST371 Applied Regression Analysis (3 cr.)

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.

ST373 Design of Experiments (3 cr.)

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.

ST431 Mathematical Statistics (3 cr.)

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.

ST496/497 Statistics Internship (1–17 cr.)

This opportunity provides the student with experience and training in statistical techniques. The internship must be approved by the department chair 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.