Mathematics and Computer Science

Professors: DAVID F. APPLEYARD, JACK GOLDFEATHER, ROGER B. KIRCHNER, MARK KRUSEMEYER, RICHARD W. NAU, Chair

Associate Professors: GAIL S. NELSON, SAMUEL E. PATTERSON

Assistant Professors: LYNNE ADRIENNE BAUR, AMY L. BIESTERFELD, DEANNA BETH HAUNSPERGER, STEPHEN F. KENNEDY, JEFFREY R. ONDICH

Senior Lecturer: CRIS T. ROOSENRAAD

Requirements for a Mathematics Major:

The course requirements are Mathematics 110 or 111, 121, 211, 232, 236 and six advanced courses from among: Mathematics courses numbered above 236 and Computer Science 227, 237. Potential majors with especially strong preparation may petition the department for exemption from the Mathematics 232 and/or 236 requirement(s). Mathematics majors are strongly encouraged to take Computer Science 117, preferably during their first two years. Concepts and skills from Computer Science 117 can be particularly valuable in advanced mathematics courses.

At least three of the following five areas of mathematics must be represented by the six advanced courses.

Algebra: Mathematics 312, 332, 342, 352

Analysis: Mathematics 251, 311, 321, 331, 351

Applied Mathematics: Mathematics 241, 315, 325, 341

Discrete Structures: Mathematics 333, Computer Science 227, 237

Geometry and Topology: Mathematics 244, 344, 354

In addition, each senior major must complete an integrative exercise which consists of a senior lecture and a written comprehensive examination (see 400-01 and 400- 02), and attend a total of twelve senior lectures during the junior and senior years.

There are many patterns of courses for the major depending upon a student's mathematical interests and career goals. A handbook for majors, which supplies information about suitable patterns of courses, is available on the web at http://www.mathcs.carleton.edu. Those planning to attend graduate school should acquire a reading knowledge of at least one of the following languages: French, German or Russian.

In order to meet State of Minnesota certification requirements, prospective secondary school teachers must take 342, 315, 244 and 349, and a computer science course is strongly recommended.

Mathematics Major under Combined Plan (see Combined Plan, Engineering in index):

Required courses: Mathematics 110 or 111, 121, 211, 232, 241, 341; Physics 122, 128, 231, and 340; Chemistry 123, 230; and one additional course chosen from Mathematics 244, 251, 311, 312, 315, 321, 332, 333, 342, 344, 351, 352 and 354.

Mathematics Skills Center:

This Center offers extra assistance to students in lower-level mathematics courses.

Requirements for a Computer Science Major:

The course requirements are Mathematics 110 or 111, 121; Computer Science 117, 223 (or Mathematics 236), 127, 207, 217, 227, 237; and two additional courses from among: Computer Science courses numbered 240 or above, Mathematics 311. Additional courses which are often recommended are Mathematics 232 and a probability and statistics course. In addition, each senior major must complete an integrative exercise which consists of a senior lecture and a written comprehensive examination (see 400- 01 and 400-02), and attend a total of twelve senior lectures during the junior and senior years. Potential majors should take Computer Science 127 before the end of the sophomore year.

Students contemplating graduate study in computer science should consider taking additional courses in both computer science and mathematics. Those interested in computer engineering should consider taking physics courses through Electricity and Magnetism and Electronics.

A handbook for majors is available on the web at http://www.mathcs.carleton.edu. Those planning to attend graduate school should acquire a reading knowledge of at least one of the following languages: French, German, Russian.

Mathematics Courses

106. Introduction to Mathematics

This course is designed to provide students with an understanding of fundamental concepts and applications of mathematics. It attempts to provide insights into the nature of mathematics and its relation to other branches of knowledge, and helps the student develop skill in mathematical reasoning. No prerequisites. 6 credits, MS

Spring -- C. Roosenraad

109. Calculus I with Review, Part 1

Mathematics 109 and 110 cover in two terms the same material covered in Mathematics 111. In addition, topics from precalculus mathematics are reviewed and practiced as needed. Precalculus topics include: algebra and analytic geometry; linear, quadratic, polynomial and rational functions; and trigonometric functions. Mathematics 109 and 110 are intended for students who wish to take Calculus I but who have not taken a precalculus course or who were advised to take Math 109-110 on the basis of the department's Diagnostic Examination. The two-term Math 109-110 sequence serves as an alternate prerequisite for all college courses requiring Math 111. 6 credits, S/CR/NC, ND

Winter -- L. Baur

110. Calculus I with Review, Part 2

This course continues the study of calculus begun in Mathematics 109. Review of precalculus mathematics continues as needed. Prerequisite is Mathematics 109 or permission of the instructor. 6 credits, MS

Spring -- L. Baur

111. Calculus I

An introduction to the differential and integral calculus. Derivatives, antiderivatives, the definite integral, applications, and the fundamental theorem of calculus. Prerequisite is one of the following: admission by way of the department's decision tree to be found in the New Student Course Description booklet, or a satisfactory grade on the Diagnostic Examination. Not open to students who have received credit for Mathematics 110. 6 credits, MS

Fall, Winter and Spring -- Staff

115. Introduction to Statistics

An introduction to data analysis and statistical inference intended primarily for students in the social sciences who have relatively little background in mathematics. Use will be made of statistical packages for computing, but no previous computer experience is required. Prerequisite: 3 years of high school mathematics or a satisfactory grade on the Department's Diagnostic Examination or Mathematics 109. Students who have taken Mathematics 111 should consider taking Mathematics 215 instead of 115. Not open to students who have received credit for Mathematics 121. Students may not receive credit for both Mathematics 115 and either Mathematics 215 or Psychology 124. 6 credits, MS

Fall, Winter and Spring -- Staff

116. Highlights in the Development of Mathematics

The multiple roles of mathematics in the history of culture. Mathematics as a language, a tool for science and technology, a model for deductive systems, and as an artistic pursuit. Topics will include the developments of combinatorial mathematics, number systems and algebra, various geometries, logic and others. Attention will be given to historical roots and applications of these developments. No prerequisite. Cross-listed with IGS 116. 6 credits, MS

Spring -- S. Schuster

121. Calculus II

Integration techniques, improper integrals, the calculus of the inverse-trigonometric, exponential, and logarithmic functions, applications, indeterminate forms, Taylor polynomials, infinite series. Prerequisite: Mathematics 110 or 111 or permission of the department. 6 credits, MS

Fall, Winter and Spring -- Staff

205. Environmental Decision-Making

Computer modeling of interacting species and population dynamics. Dominant paradigms (economic, political, ecological) for making "rational" policy choices. How to include uncertainty, risk and environmental ethics. Common formal decision-aiding methods: computer simulation, mathematical optimization, benefit-cost analysis, game theory. Cross-listed with Environmental and Technology Studies 205. 6 credits, MS

Winter -- L. Baur

206. A Tour of Mathematics

A series of eight lectures intended for students considering a Mathematics major. The emphasis will be on presenting various striking ideas, concepts and results in modern mathematics, rather than on developing extensive knowledge or techniques in any particular subject area. 1 credit, S/CR/NC, MS

Winter and alternate years -- Staff

211. Calculus III

Introduction to multivariable calculus: vectors, curves, partial derivatives, gradient, multiple and iterated integrals, line integrals, Green's theorem. Prerequisite: Mathematics 121 or permission of the department. 6 credits, MS

Fall, Winter and Spring -- Staff

215. Introduction to Probability and Statistics

Frequency distributions, one- and two- way tables, sampling, normal distributions and approximations, hypothesis testing, exploratory data analysis, linear regression and an introduction to multiple regression and analysis of variance. Use will be made of statistical packages for computing but no previous computer experience is required. Students who have taken or plan to take Mathematics 211 and 232 should consider taking Mathematics 315 instead of 215. Prerequisite: Mathematics 110 or 111 or consent of the instructor. Students may not receive credit for both Mathematics 115 and 215. Mathematics 215 does not count toward the Mathematics major. 6 credits, MS

Fall, Winter and Spring -- L. Baur, D. Haunsperger, Staff

216. Seminar on the History of Mathematics

This seminar will focus on selected episodes in the history of mathematics from the 17th century to the present. Each participant will give one public presentation, which will be followed by discussion. Some weekly preparatory reading, often on the life and work of a prominent mathematician, will be required. Prerequisite: Mathematics 211 or concurrent registration with Mathematics 211 or consent of the instructor. 2 credits, MS

Given in alternate years; not offered in 1996-1997.

223. Discrete Mathematics

Elements of logic; methods of proof; sets, relations, and functions; counting techniques; and simple finite probability. Additional topics may include recurrence relations, trees and graphs, finite-state machines, applications of linear transformations, and group theory. Prerequisite: Mathematics 110 or 111 and Computer Science 117 or Mathematics 110 or 111 and concurrent registration in Computer Science 117. Cross- listed with Computer Science 223. Students may not receive credit for both Mathematics 223 and 236. 6 credits, MS

Spring -- R. Kirchner

232. Linear Algebra

Vector spaces, linear transformations, determinants, inner products and orthogonality, eigenvectors and eigenvalues; connections with multivariable calculus. Prerequisite: Mathematics 211. 6 credits, MS

Fall, Winter and Spring -- J. Goldfeather, M. Krusemeyer, G. Nelson

234. Philosophical Issues in the History of Mathematics

Before 1800, the theorems of mathematics were generally regarded as paradigms of absolute truth, and philosophers (Plato, Kant) were happy to construct their theories on the firm bedrock of mathematics. In the nineteenth century this foundation collapsed as new discoveries (non-Euclidean geometry, non-commutative algebras, continuous non-differentiable functions) forced a critical re-examination of the foundations of mathematics. We will study some of these discoveries, and in light of this knowledge ask ourselves philosophical questions such as: In what sense do mathematical objects (triangles, the number 42) exist? In what sense are mathematical truths true? Why does mathematics seemingly describe the real world? Cross- listed with Integrated General Studies 234. 6 credits, 3 HU and 3 MS

Spring -- S. Kennedy

236. Introduction to Mathematical Structures

Basic concepts and techniques used throughout mathematics. Topics include logic, mathematical induction and other methods of proof, problem solving, sets, cardinality, equivalence relations, functions and relations, and foundations of the real number system. Other topics may include: algebraic structures such as semigroups, groups and rings; basic combinatorics. Prerequisite: Mathematics 232 or consent of the instructor. Students may not receive credit for both Mathematics 223 and 236. 6 credits, MS

Fall, Winter and Spring -- S. Patterson, D. Haunsperger, M. Krusemeyer

241. Ordinary Differential Equations

An introduction to the theory and methods of solution of ordinary differential equations. Prerequisites: Mathematics 232 or consent of the instructor. 6 credits, MS

Winter -- J. Goldfeather, D. Appleyard

244. Modern Geometries

Euclidean geometry from an advanced perspective; projective, hyperbolic, inversive, and/or other geometries. In addition to foundations, various topics such as transformation and convexity will be treated. Recommended for prospective high school teachers. Prerequisite: Mathematics 236. 6 credits, MS

Given in alternate years; not offered in 1996-1997.

251. Chaotic Dynamics

An exploration of the behavior of non- linear dynamical systems. Topics include one-dimensional dynamics, Feigenbaum's universality, Sarkovskii's Theorem, chaos, symbolic dynamics, fractals, structural stability, Smale's horseshoe map, strange attractors and bifurcation theory. Some point-set topology will be developed as needed. Prerequisite: Mathematics 232. 6 credits, MS

Winter and in alternate years -- S. Kennedy

291. Independent Study

Credit by Arrangement -- Staff

311. Topics in Numerical Analysis

Topics chosen from the following: the numerical solution of algebraic, differential, and difference equations; integration; functional approximation; treatment of empirical data; computational linear algebra; error analysis. Prerequisite: Mathematics 232 and Computer Science 117 or consent of the instructor. 6 credits, MS

Not offered in 1996-1997.

312. Elementary Theory of Numbers

Properties of the whole numbers. Topics included are the Euclidean algorithm, classical unsolved problems in number theory, prime factorization, Diophantine equations, congruences, divisibility, Euler's phi function and other multiplicative functions, primitive roots, and quadratic reciprocity. Other topics may include integers as sums of squares, continued fractions, distribution of primes, integers in extension fields, p-adic numbers. Prerequisite: Mathematics 236 or consent of the instructor. 6 credits, MS

Fall and in alternate years -- M. Krusemeyer

315. Probability and Statistics I

Introduction to the axioms of probability, random variables, joint and conditional distributions, and expectation. Discussion of the law of large numbers, the central limit theorem, and distributions derived from the normal. Applications include survey sampling. A computer package will be used for simulating random phenomena. Prerequisite: Mathematics 232. 6 credits, MS

Fall -- A. Biesterfeld

321. Real Analysis I

A systematic study of concepts basic to calculus, such as topology of Rⁿ, limits, differentiation, integration, convergence of sequences, series of functions, and the implicit function theorem. Prerequisite: Mathematics 236 or consent of the instructor. 6 credits, MS

Fall -- G. Nelson

325. Probability and Statistics II

Introduction to descriptive statistics, parameter estimation, hypothesis testing and experimental design, regression, analysis of variance, decision theory and Bayesian inference. A computer package will be used to analyze real data sets. Prerequisite: Mathematics 315. 6 credits, MS

Winter -- A. Biesterfeld

331. Real Analysis II

Further topics in analysis such as measure theory, Lebesgue integration or Banach and Hilbert spaces. Prerequisite: Mathematics 321 or consent of the instructor. 6 credits, MS

Winter and in alternate years -- G. Nelson

332. Advanced Linear Algebra

Advanced topics in linear algebra. Prerequisite: Mathematics 232 or consent of the instructor. 6 credits, MS

Given in alternate years; not offered in 1996-1997.

333. Combinatorial Theory

Study of existence and enumeration of specified arrangements of elements of a set. Enumeration techniques include the inclusion-exclusion principle, recurrence relations and generating functions; existence criteria include the pigeonhole principle, marriage theorem and Ramsey's theorem. Some graph theory. Applications, as time permits, to probability, geometry and computer science. Prerequisite: Mathematics 236 or consent of the instructor. 6 credits, MS

Spring and in alternate years -- M. Krusemeyer

341. Partial Differential Equations

Methods for solving parabolic, hyperbolic, and elliptic equations and boundary value problems arising in physical situations. Topics will include Fourier series, Fourier transforms, and Laplace transforms. Prerequisite: Mathematics 241. 6 credits, MS

Spring -- S. Patterson

342. Abstract Algebra I

Introduction to algebraic structures, including groups, rings, and fields. Homomorphisms and quotient structures, polynomials, unique factorization. Other topics may include applications such as Burnside's counting theorem, symmetry groups, polynomial equations, geometric constructions, algebraic coding and Boolean algebras. Prerequisite: Mathematics 236 or consent of the instructor. 6 credits, MS

Spring -- J. Goldfeather

344. Differential Geometry

Local and global theory of curves, Frenet formulas. Local theory of surfaces, normal curvature, geodesics, Gaussian and Mean curvatures, Theorema Egregium. Riemannian geometry. Prerequisite: Mathematics 236 or consent of the instructor. 6 credits, MS

Winter and in alternate years -- S. Patterson

349. Methods of Teaching Mathematics

Methods of and materials and technology for teaching mathematics in secondary school. Issues in contemporary mathematics education. Regular visits to secondary school classrooms and teaching a class are required. Prerequisite: Consent of the instructor. Normally taken during the senior year. 6 credits, ND

Winter -- C. Roosenraad

351. Functions of a Complex Variable

Algebra and geometry of complex numbers, analytic functions, complex integration, series, residues, applications. Prerequisite: Mathematics 211. 6 credits, MS

Given in alternate years; not offered in 1996-1997.

352. Abstract Algebra II

An intensive study of one or more of the types of algebraic systems studied in Mathematics 342. Prerequisite: Mathematics 342 or consent of the instructor. 6 credits, MS

Given in alternate years; not offered in 1996-1997.

354. Topology

An introduction to the topological point of view in mathematics. Topics include continuous transformations, compactness, connectedness, simplicial complexes and surfaces. Prerequisite: Mathematics 236 or consent of the instructor. 6 credits, MS

Given in alternate years; not offered in 1996-1997.

391. Independent Study

Credit by Arrangement -- Staff

392. Independent Research

Credit by Arrangement -- Staff

395. Set Theory

Introduction to set-theoretic foundations of mathematics The axiom system of Zermelo-Fraenkel, cardinal and ordinal numbers and the Axiom of Choice. As time permits, additional topics may include construction of the real numbers, transfinite induction, partition calculus, trees, the constructible universe or consistency/independence proofs, according to the interests of the students. 6 credits, MS

Spring -- L. Baur

400-01. Integrative Exercise (Senior Lecture)

A mathematical talk on an assigned topic presented by the registered senior mathematics major. Required of all senior majors. Prerequisite: Mathematics 236 and successful completion of three courses from among: Mathematics courses numbered above 236, Computer Science 227, Computer Science 237. 3 credits, S/NC, ND

Fall, Winter and Spring -- Staff

400-02. Integrative Exercise (Senior Examination)

A three-hour written test on material from Mathematics 110 or 111, 121, 211, 232 and 236. Required of all senior majors. 3 credits, S/NC, ND

Spring -- Staff

Computer Science Courses

117. Introduction to Computer Science

Elements of procedure-oriented languages. Problem solving and implementation of algorithms. Program design and documentation. Measures of efficiency and complexity. Iterative and recursive techniques. Non-numerical and numerical applications. Files, pointers and linked structures. 6 credits, MS

Fall, Winter and Spring -- R. Nau, J. Ondich

127. Data Structures

An introduction to abstract data types, recursion, searching, sorting, stacks, queues, linked lists, trees, graphs and hash tables. Prerequisite: Computer Science 117 or consent of the instructor. 6 credits, MS

Winter, Spring -- J. Goldfeather, J. Ondich

207. Computer Organization and Computer Systems

The design and organization of hardware and software. Topics include: internal data representations, digital logic, micro-processor architecture and the history of computer architecture, micro-programming, assemblers and assembly language programming, memory organization, caches, RISC architectures. Prerequisite: Computer Science 127 or consent of the instructor. 6 credits, MS

Fall -- J. Ondich

217. Programming Languages

Design principles for high-level programming languages. Syntax and semantics. Assignment, control structures, data types, procedures, nesting and scope. Mechanisms for recursion, parameter-passing, functional and object-oriented programming. Realizations through interpreters and/or selected languages. Prerequisite: Computer Science 127 or consent of the instructor. 6 credits, MS

Spring -- R. Kirchner

223. Discrete Mathematics

Elements of logic; methods of proof; sets, relations, and functions; counting techniques; and simple finite probability. Additional topics may include recurrence relations, trees and graphs, finite-state machines, applications of linear transformations, and group theory. Prerequisite: Mathematics 110 or 111 and Computer Science 117 or Mathematics 110 or 111 and concurrent registration in Computer Science 117. Cross-listed with Mathematics 223. Students may not receive credit for both CS 223 and Mathematics 236. 6 credits, MS

Spring -- R. Kirchner

227. Computer Algorithms

Design and analysis of algorithms. Divide and conquer, dynamic programming, greedy method, backtracking, and inductive approaches. Recurrence relations, lower-bound methods, NP-completeness. Prerequisites: Computer Science 223 and 127 and Mathematics 121. 6 credits, MS

Winter -- J. Ondich

237. Theory of Computation

Abstract automata, especially finite state machines, push-down automata, and Turing machines. Formal languages, especially context-free languages. The relationship between automata and languages. Computability and solvability. Prerequisites: Computer Science 127; Computer Science 223 or Mathematics 236. 6 credits, MS

Fall -- D. Appleyard

247. Digital Electronics

A study of the digital electronics involved in computers, ranging from basic logic circuits to microprocessors. Weekly lab. Each student will complete a term paper that will involve projections about future developments in computer electronics, and a lab project that will involve circuit design. Prerequisite: Computer Science 207. Cross-listed with Physics 247. 6 credits, MS

Given in alternate years; not offered in 1996-1997.

257. Object-Oriented Design

Object-oriented programming including objects, methods, polymorphisms, inheritance and exceptions. The software design life cycle. Program specification, design, implementation, testing, measuring, verification, and documentation. Readings/discussions on object-oriented languages and software development. Introductory examples of object-oriented programming. Labs. Group projects. Individual project. Prerequisite: Computer Science 127. 6 credits, MS

Spring and in alternate years -- R. Nau

291. Independent Study

Credit by Arrangement -- Staff

307. Operating Systems

Introduction to the design and construction of operating systems. Sequential and concurrent processes, synchronization and mutual exclusion, memory management techniques, file systems design, security and protection systems, CPU scheduling, input/output device handling, and distributed operating systems. Prerequisite: Computer Science 207 or consent of instructor. 6 credits, MS

Given in alternate years; not offered in 1996-1997.

317. Computer Graphics

The raster graphics representation of 2- and 3-dimensional images. Topics include frame buffers, data structures for image storage, geometric transformations, hidden surface algorithms, splines, and lighting models. Prerequisites: Computer Science 127, Mathematics 121 and 232. 6 credits, MS

Given in alternate years; not offered in 1996-1997.

327. Artificial Intelligence

Heuristic search; knowledge representation using logic, frames and rules; expert systems. LISP, Prolog, and expert system shells used for implementation. Prerequisite: Computer Science 127 or consent of the instructor. 6 credits, MS

Winter and in alternate years -- R. Kirchner

391. Independent Study

Credit by Arrangement -- Staff

392. Independent Research

Credit by Arrangement -- Staff

395. Seminar

Not offered in 1996-1997.

400-01. Integrative Exercise (Senior Lecture)

A talk on an assigned topic presented by the registered senior computer science major. Required of all senior majors. Prerequisite: Mathematics 110 or 111, 121, Computer Science 117, 127, 223, (or Mathematics 236); three courses from among Computer Science 207, 217, 227, 237; one course from among Computer Science courses numbered 240 or above or Mathematics 311. 3 credits, S/NC, ND

Fall, Winter, Spring -- Staff

400-02. Integrative Exercise (Senior Examination)

A three-hour written test on material from Computer Science 117, 127, 207, 217, 223, 227, and 237. Required of all senior majors. 3 credits, S/NC, ND

Spring -- Staff