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# Mathematics

**Department Chair:** Robert Hesse

**Faculty:**Bret Benesh, Philip Byrne, Robert Campbell, Sunil Chetty, Robert Dumonceaux, Jennifer Galovich, Michael Gass, David Hartz, Robert Hesse, Kristen Nairn, Thomas Sibley, Anne Sinko, Michael Tangredi.

**Math Skills Center** **Director:** Marilyn Creed

The mathematics department offers courses to fit the needs of a wide variety of students: the student majoring in mathematics, the student majoring in another field who needs or chooses supporting courses in mathematics and the general liberal arts student.

Since a knowledge of mathematics can be useful in disciplines as diverse as biology, philosophy and economics, the mathematics department offers a number of options to students. The major offerings are flexible enough to prepare students to apply for further study in graduate school, for a career in secondary education or as a mathematician or statistician in business or industry. It is also possible for a student to arrange for an individualized major in mathematics and another discipline. This should be done in careful consultation with a member of the mathematics department and a member of the student's major department. A student majoring in another discipline may choose to minor in mathematics. A major in elementary education may choose a minor in mathematics or the concentration designed especially for elementary teachers. (See the education department listing for more information.)

In addition to the formal courses described below, there are many other opportunities available for students interested in mathematics. An individual learning project on a topic of mutual interest can be designed with the assistance of a faculty member. The department supports students to engage in summer research in mathematics, mathematical biology or biostatistics through a generous stipend program. Opportunities are available to combine the summer research with an honors thesis. An active student math club and a local chapter of Pi Mu Epsilon (a national honor society for students of mathematics) cooperate with the mathematics department to offer a rich program of seminars, films, visiting speakers, career information and social activities. Each spring the department hosts a regional Pi Mu Epsilon conference at which students and faculty from several colleges gather at Saint Benedict's and Saint John's for two days of presentations by students and invited speakers.

Each semester the mathematics department employs students paid on an hourly basis as calculus teaching assistants, course assistants, and tutors. Calculus teaching assistants grade papers and, in consultation with the course instructor, supervise the calculus labs. Those labs, which meet regularly, provide students with additional opportunities to discuss course material and to practice problem-solving skills. Course assistants grade papers for lower division classes other than calculus I and II. Tutors give individual help to students at the Math Skills Center.

## Mathematics in the Common Curriculum

Mathematics as a skill and as a theoretical structure has played a crucial role in modern civilization as well as in the everyday lives of individuals. Therefore, all students will be required to take and pass one course which satisfies the common curriculum requirement in mathematics. While different courses cover different topics, all courses meeting the requirement stress mathematics as a conceptual discipline, and address its contemporary role. These courses will also enable students to understand and appreciate the power and limitations when using mathematical reasoning, its language and notation to solve a variety of problems from other disciplines and from everyday life. Students enrolled in common curriculum courses are actively involved in doing mathematics.

The director of the Math Skills Center will provide assistance for students who have not fulfilled this requirement.

Certain mathematics common curriculum courses (MATH 114, 121, 122, 124) have as prerequisite satisfactory performance on the Quantitative Skills Inventory Test. Students who have an ACT-Math score of 21 or greater or SAT-Math score of 530 or greater will be granted satisfactory performance status without taking the examination. Otherwise, the examination will be administered by appointment with the Mathematics Skills Center. All students enrolled in MATH 118 or 119 will be asked to take a calculus readiness exam during the first week of classes.

## Major

The mathematics department offers concentrations in mathematics and mathematics/secondary education; it also offers a major in numerical computation jointly with the computer science department. Information about the numerical computation major is in a separate section for that major. Students may not earn majors in both mathematics and numerical computation. Students may not earn a minor in mathematics with a major in numerical computation.

Special Requirements:

Students anticipating a major in mathematics and/or the natural sciences ordinarily begin their study of mathematics with 119. However, a student needing further preparation before beginning calculus, either 118 or 119, should enroll in 115. Students interested in advanced placement should contact the department chair.

Admission to the major requires a grade of C or higher in MATH 119, 120 and MATH 239 or 241.

Before admission to the major (ordinarily in the sophomore year), prospective majors must consult with their advisors in the mathematics department to plan their mathematics courses. Students should choose their courses and non-curricular activities with regard to their goals for careers and graduate school. Students should be aware of which semesters upper-division mathematics courses will be offered.

Senior majors are required to take a comprehensive exam in mathematics (the Major Field Test).

Suggestions:

Prospective majors should have familiarity with computer programming before taking upper-division mathematics courses. Students preparing for graduate school in mathematics should include 332 and 344 or 348.

**Concentration in Mathematics (40-42 credits)**Required Courses:

119, 120, 239, 241, 331, 343, 395, 16 additional upper-division credits in. 395 may be waived for students who complete an undergraduate research project in mathematics. See department chair for details.

**Concentration in Mathematics/Secondary Education (40-42 credits)**Required Courses:

Same as concentration in mathematics, but include 333, 345.

Suggestions:

At least 2 credits 300 (History of Mathematics) is also recommended. Check with the chairs of the education department and the mathematics department for requirements for certification by the Minnesota Department of Education. See the education department listing for minor requirements.

## Minor (24 credits)

Required Courses:

119, 120, 239; plus either 12 additional upper-division credits in mathematics, or 241 plus 8 additional upper-division credits in mathematics. Note: students may not earn a minor in mathematics with a major in numerical computation.

## Courses (MATH)

The minimum prerequisite for any mathematics course is: Math ACT subscore of 21 or above OR Math SAT subscore of 530 or above OR satisfactory performance on the CSB/SJU Quantitative Skills Inventory. Other prerequisites may also apply, as noted in the course descriptions.

114 Mathematics Exploration. (4)

A course to enrich the students' liberal arts education by presenting the spirit and some insights of mathematics. The course will emphasize understanding over techniques. Topics will illustrate the nature of contemporary mathematics and the relationship between mathematics and our cultural heritage. Some possible topics include: algorithms, exotic geometries, finance, map coloring, graphs, groups and mathematical modeling. Prerequisites: three years of college preparatory mathematics or permission of instructor.

115 Pre-Calculus Mathematics. (2)

Properties of polynomial, trigonometric, exponential functions. For the student who needs further preparation for calculus. Prerequisites: three years of college preparatory mathematics. Does not satisfy Mathematics Common Curriculum Requirement.

118 Essential Calculus. (4)

Preliminary concepts; derivatives, integrals and the concept of limit; application of differentiation and integration; calculus of several variables; exponentials, logarithms and growth problems. Other topics may include differential equations and probability theory. Prerequisites: 115 or four years of college preparatory mathematics or permission from the chair of the mathematics department.

119 Calculus I. (4)

Definition and nature of limits, continuity, derivatives of polynomial, algebraic and trigonometric functions and applications. Definite integrals and application. Prerequisites: 115 or four years of college preparatory mathematics or permission from the chair of the mathematics department. Note: Credit will be awarded for MATH 119 upon completion of MATH 120 with a grade of C or higher.

120 Calculus II. (4)

Continuation of applications of the integral. Infinite series, Taylor's theorem, methods of integration, introduction to functions of several variables. Additional topics may include complex numbers, polar coordinates, parametric equations, approximation methods, differential equations. Prerequisite: 119 or permission of the chair of the mathematics department. Note: Credit will be awarded for MATH 120 upon completion of MATH 305 with a grade of C or higher.

121 Fundamentals of Mathematics. (4)

Basic concepts of sets, numeration, structure of number systems, arithmetic and algebraic operations, problem solving, and other topics to prepare students for elementary school mathematics teaching. Prerequisites: three years of college preparatory mathematics.

122 Finite Mathematics. (4)

Mathematics for students in the life, social and management sciences. Topics chosen from symbolic logic, set theory, combinatorial analysis, probability, linear equations, vectors, matrices, mathematics of finance, linear programming, Markov chains and matrix games. Prerequisites: three years of college preparatory mathematics.

124 Probability and Statistical Inference. (4)

Graphs and charts, mean, median and other measures of location. Terminology and rules of elementary probability; normal distribution, random sampling, estimation of mean, standard deviation and proportions, correlation and regression, confidence intervals, tests of hypotheses. Prerequisites: three years of college preparatory mathematics.

127 Number Systems (1)

Topics to include sets, functions, other number bases, elementary number theory, rational and irrational numbers and problem solving strategies related to these topics. Prerequisite: Elementary Education majors who have completed Math 118, Math 119 or the equivalent, with a grade of C or better.

180 Fundamentals of Mathematics II. (4)

Continuation of 121. Probability and statistics, geometry, discrete mathematics including combinatorics and graph theory, and other topics to prepare students for middle school mathematics teaching. Prerequisite: 121.

239 Linear Algebra. (4)

Systems of linear equations, matrices and matrix operations, vector spaces, subspaces, linear independence, basis and dimension, linear transformations, eigenvalues and eigenvectors, inner products, elementary proofs appropriate to the course content, and selected applications. Prerequisite: 120 or permission of the chair of the mathematics department.

241 Foundations and Structures of Mathematics. (4)

The basic theme of this course is mathematical thinking and writing. Emphasis will be placed on formulating and writing proofs. The course will cover topics in the following areas: logic, sets, relations, functions, counting, graph theory, infinite sets, algebraic structures and the real number system. Additional topics as time permits. Prerequisite: 120.

271 Individual Learning Project. (1-4)

Supervised reading or research at the lower-division level. Permission of department chair required. Consult department for applicability towards major requirements. Not available to first-year students.

300 History of Mathematics. (2 credits)

Advanced level independent guided readings, discussion and written projects on the history of mathematics. May be repeated for credit when topics vary. Prerequisite: instructor’s consent.

305 Multivariable Calculus. (4)

Topics selected from Geometry of R^{n}, differentiation in R^{n}, vector-valued functions, optimization, multiple integrals, line and surface integrals, vector analysis and introduction to differential forms. Prerequisite: 239. Fall.

315 Operations Research. (4)

Topics selected from: linear programming, duality theory, dynamic and integer programming, graph-theoretic methods, stochastic processes, queuing theory, simulation, non-linear programming, PERT/CPM. Applications to social and natural sciences and business. Prerequisite: 239. Fall in even years.

318 Applied Statistical Models. (4)

The relationships among variables in real data sets will be explored through the theory and application of linear models. The focus of the course will be on building such models, assessing their adequacy, and drawing conclusions. Statistical computing programs will be used to analyze the data. Prerequisite: 239. Spring in even years.

322 Combinatorics and Graph Theory. (4)

Basic enumerative combinatorics and graph theory including counting principles, generating functions, recurrences, trees, planarity and vertex colorings. Additional topics at the discretion of the instructor. Prerequisite: 239. 241 recommended. Spring in odd years.

331 Algebraic Structures I. (4)

Definitions and basic properties of sets and relations, groups, rings, ideals, integral domains, fields, algebras and applications. Prerequisites: 239, 241. Spring and fall in even years.

332 Algebraic Structures II. (4)

Continuation of 331, additional topics in Algebra such as: Sylow theorems, coding theory, free groups, Euclidean rings, extension fields, Galois theory, categories, functors, tensor products. Prerequisite: 331. Spring in odd years.

333 Geometry I. (4)

Foundations of geometry, study of axiom systems for finite geometries and Euclidean geometry, topics in synthetic geometry; introduction to hyperbolic and other geometries. Geometric transformation theory and classification of geometries by transformation groups. Prerequisite: 239. Fall in odd years.

337 Differential Equations. (4)

The concept of a solution, tangent fields, the existence and uniqueness theorem and its implications, elementary solution techniques, series and numerical solutions, linear equations and systems, Laplace transforms, applications. Prerequisite: 239. Spring.

338 Numerical Analysis. (4)

Numerical algorithms and error estimations, solutions of linear and nonlinear equations and systems, numerical solutions of differential equations, numerical integration, interpolation and approximation techniques, matrix methods and power series calculations. Prerequisite: 239 and familiarity with computer programming. Spring in even years.

339 Mathematical Modeling. (4)

Mathematical modeling is the art of finding mathematical dexcriptions of real-world phenomena, with the goal of attaining a deeper understanding of those phenomena. The mathematical tools will vary according to the application. This course will cover both continuous and discrete mathematical models. Applications will be drawn from a variety of fields, such as population dynamics, economics, and physical sciences. Prerequisite: Math 239.

340 Topics in Advanced Mathematics. (4)

Content varies from semester to semester. Topics will be chosen from both pure and applied mathematics and may include algebraic coding theory, cryptology, number theory, mathematical modeling, mathematical logic, complex analysis, topology, dynamical systems, applications to computer science. May be repeated for credit when topics vary. Prerequisite: 239. Additional prerequisites possible depending on the topic. Fall.

341 Fourier Series and Boundary Value Problems. (4)

Separable partial differential equations from theoretical physics. Fourier series, convergence, orthogonal systems. Fourier integrals. Sturm-Liouville theory, solutions to boundary value problems. Applications. Prerequisite: 239. Spring in odd years.

343 Analysis I. (4)

Set theory, real numbers, topology of Cartesian spaces, Heine-Borel Theorem, sequences, series, convergence, continuity, differentiation, integration. Prerequisites: 239, 241. Spring and fall in odd years.

344 Analysis II. (4)

Topics selected from the following: mapping theorems and extremum problems, Riemann-Stielties integral, main theorems of integral calculus, point set topology, Lebesque integral, functions defined by integrals, convergence theorems. Prerequisite: 343.

345 Mathematical Statistics I. (4)

Probability spaces, random variables, statistics and sampling distributions, statistical hypotheses and decision theory, statistical inference, estimation. Prerequisite: 239. Spring and fall in even years.

346 Mathematical Statistics II. (4)

Topics selected from the following: sampling, order statistics, Monte Carlo methods, asymptotic efficiencies, maximum likelihood techniques, inference, multivariate normal, analysis of variance, regression, correlation. Prerequisite: 345. Spring in odd years.

348 Complex Analysis. (4)

Topics will generally include properties of complex numbers; complex functions and their derivatives; analyticity; Cauchy’s Theorem and related results; series representations of functions; contour integration and the theory of residues. Additional topics at the discretion of the instructor. Prerequisite: 343 or 305. Spring in even years.

371 Individual Learning Project. (1-4)

Supervised reading or research at the upper-division level. Permission of department chair and completion and/or concurrent registration of 12 credits within the department required. Consult department for applicability towards major requirements. Not available to first-year students.

395 Mathematics Capstone. (2)

Critical analysis of readings or topics and/or an in-depth investigation leading to a project. The course will be structured as a seminar. The instructor will select the subject matter. Students will present and discuss the material of the course, and complete regular assignments (short papers or problem sets). Prerequisite: Senior standing, 241 and completion of at least two 300 level mathematics courses.

398 Honors Senior Essay, Research or Creative Project. (4)

Required for graduation with "Distinction in Mathematics." Prerequisite: HONR 396 and approval of the department chair and director of the Honors Thesis program. For further information see HONR 398.