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2005-2007 Academic Catalog

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2005-2007 Academic Catalog
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Mathematics

**Department Chair:** Gary Brown

**Faculty:** Marc Brodie, Gary Brown, Philip Byrne, Robert Dumonceaux, Jennifer Galovich, Michael Gass, David Hartz, Robert Hesse, James Johnson, Kristen Nairn, Thomas Sibley, 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. 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 a number of mathematics majors are invited by the department to serve as teaching assistants paid on an hourly basis. Some mathematics majors are also employed to teach labs which accompany the pre-calculus and calculus courses. Those labs, which meet regularly, provide students with additional opportunities to discuss course material and to practice problem-solving skills.

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, as a part of the core curriculum, all students will be required to take and pass one course which satisfies the core requirement in mathematics. While different courses cover different topics, all mathematics core courses stress mathematics as a conceptual discipline, and address issues in the history, philosophy and contemporary role of mathematics. Students enrolled in core 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.

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.

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 119 or 123, 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.

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 credits)**Required Courses:

119, 120, 239, 241, 331, 343 plus 16 addition upper-division credits in mathematics.

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

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

Suggestions:

At least 1 credit in 105-108 or 300-303 (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.

Required Courses:

119, 120, 239; plus either 12 additional upper-division credits in mathematics, or 241 plus 8 additional upper-division credits in mathematics.

105, 106, 107, 108 History of Mathematics. (1 credit each)

Independent guided readings, discussion and written projects on the history of topics covered in mathematics courses in the curriculum. Prerequisite: instructor's consent.

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, math proficiency.

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 and mathematics proficiency. Does not satisfy Mathematics Core Requirement.

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 satisfactory performance on a calculus readiness exam.

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.

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 and mathematics proficiency.

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, math proficiency.

123 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 calculus readiness exam.

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, math proficiency.

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)

Nature and construction of proofs. Matrices and matrix operations, vector spaces and subspaces, complex numbers, linear transformations and linear systems, determinants, eigenvalues and eigenvectors, applications. Prerequisites: 120 or 123 and consent of instructor.

240 Discrete Mathematics. (4)

Discrete mathematics is the study of mathematical objects and structures which occur in discrete states rather than continuously. All digital computation is based on discrete mathematics. Course topics include sets, functions, relation and partial order, logic, counting techniques, recurrence relations, asymptotic analysis, sequences and series, proof (including induction) and graphs. Prerequisites: 119 or 123, and 120 or 124.

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. Time permitting, additional topics from discrete structures and formal languages may be included. 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, 301, 302, 303 History of Mathematics. (1 credit each)

Advanced level independent guided readings, discussion and written projects on the history of mathematics. Prerequisite: instructor’s consent.

305 Multivariable Calculus. (4)

Topics selected from Geometry of Rn, differentiation in Rn, 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. 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 selected from the following: 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. Fall in odd years.

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. Spring in even years and every 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, complex analysis, 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: 305 or permission of instructor. 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.

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.