Mathematics and Computer Science

Chair: Professor Gail S. Nelson

Professors: David F. Appleyard, Jack Goldfeather, Mark Krusemeyer, Richard W. Nau, Gail S. Nelson, Samuel E. Patterson

Associate Professors: Laura M. Chihara, Deanna Beth Haunsperger, Stephen F. Kennedy, Jeffrey R. Ondich

Visiting Associate Professor: Karen Saxe

Assistant Professors: Robert P. Dobrow, David R. Musicant

Visiting Assistant Professors: Kristina C. Garrett, David Romano

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, 265 (formerly 315), 275 (formerly 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, and majors must attend a total of twelve other 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 guide 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 consider acquiring 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 Mathematics 265, 275, 342, 244 (recommended) or 344, and 349, and a computer science course is strongly recommended.

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

In addition to completing requirements for the mathematics major listed above including Mathematics 241 and 341, the student should take the following courses required for admission to engineering schools: Physics 113 or 114, 115, 128, Chemistry 123, 230, and Computer Science 117.

Requirements for a Computer Science Major:

The course requirements are Mathematics 110 or 111, 121; Computer Science 117, 177 or 223 (or Mathematics 236), 127, 207, 217, 227, 237; and two additional courses from among: Computer Science courses numbered 240 or above, Mathematics 311, Physics 247 or 343. Additional courses that are often recommended are Mathematics 232 and a probability and statistics course. In addition, each senior major must complete an integrative exercise that consists of a senior lecture and a written comprehensive examination, and majors must attend a total of twelve other 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 guide for majors is available on the web at http://www.mathcs.carleton.edu.

Mathematics Skills Center:

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

Mathematics Courses (MATH)

MATH 100. CryptographyAttempts to create "codes" are as ancient as written languages and such secret writings have had major impacts on many events in history and have been the focus of much controversy. From the beginning, the desire to "break" such hidden messages has been a source of some very original thinking. In this course we will examine the history and practice of cryptography and cryptanalysis from the Egyptians to computers, with a focus on the twentieth century and World War II in particular. No knowledge of mathematics beyond algebra and geometry will be assumed6 credits cr., S/CR/NC, MS, Spring—C. Roosenraad

MATH 106. Introduction to MathematicsThis 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 students develop skill in mathematical reasoning. No prerequisites.6 credits cr., MS, Spring—D. Appleyard

MATH 109. Calculus I with Review, Part 1Mathematics 109 and 110 cover in two terms the 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. Enrollment is restricted to those 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 Mathematics 111.6 credits cr., S/CR/NC, ND, Winter—K. Garrett

MATH 110. Calculus I with Review, Part 2This course continues the study of calculus begun in Mathematics 109. Review of precalculus mathematics continues as needed. Prerequisite: Mathematics 109 or consent of the instructor.6 credits cr., MS, Spring—K. Garrett

MATH 111. Calculus IAn 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 cr., MS, Fall,Winter,Spring—Staff

MATH 115. Statistics: Concepts and ApplicationsIntroduction to statistical concepts and their applications. Emphasis will be placed on interpretation and understanding of statistical results with examples taken from scholarly journals, newspapers or magazines. Topics include: data collection, exploratory data analysis and graphs, correlation and linear regression, hypothesis testing, and two-way tables. Students will learn how to use statistical software. Not open to students who have already received credit for Mathematcs 121, 215 or Psychology 124. Students who have taken Mathematics 111 should consider taking Mathematics 215 instead of Mathematics 115.6 credits cr., MS, Fall,Spring—Staff

MATH 121. Calculus IIIntegration techniques, improper integrals, the calculus of the exponential, logarithmic, and inverse trigonometric functions, applications, indeterminate forms, Taylor polynomials, infinite series. Prerequisite: Mathematics 110 or 111 or placement by or consent of the department.6 credits cr., MS, Fall,Winter,Spring—Staff

MATH 206. A Tour of MathematicsA 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 cr., S/CR/NC, MS, Winter—Staff

MATH 211. Calculus IIIIntroduction to multivariable calculus: vectors, curves, partial derivatives, gradient, multiple and iterated integrals, line integrals, Green's theorem. Prerequisite: Mathematics 121 or placement by or consent of the department.6 credits cr., MS, Fall,Winter,Spring—Staff

MATH 215. Introduction to StatisticsIntroduction to statistics and data analysis. Topics include: graphs, exploratory data analysis, correlation and linear regression, design of experiments, introduction to probability, normal distribution and approximations, sampling distribution, estimation, hypothesis testing, two-way tables, and introduction to analysis of variance. Emphasis will be placed on real-life uses of statistics from a wide range of applications. Students will learn how to use statistical software. Students who have taken or plan to take Mathematics 211 and 232 should consider taking Mathematics 265 instead of 215. Not open to students who have already received credit for Mathematics 115, Mathematics 275 or Psychology 124. Mathematics 215 does not count toward the Mathematics major. Prerequisite: Mathematics 111 or consent of the instructor.6 credits cr., MS, Fall,Winter,Spring—Staff

MATH 216. Seminar: History of MathematicsThis seminar will focus on selected episodes in the history of mathematics from the seventeenth century to the present. Each participant will give at least 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 cr., MS, Spring—Staff

MATH 232. Linear AlgebraVector spaces, linear transformations, determinants, inner products and orthogonality, eigenvectors and eigenvalues; connections with multivariable calculus. Prerequisite: Mathematics 211.6 credits cr., MS, Fall,Winter,Spring—L. Chihara, K. Garrett, M. Krusemeyer, D. Romano

MATH 234. Philosophy of MathematicsCross-listed with PHIL 255.Before 1800, the theorems of mathematics were generally regarded as paradigms of certainty, and philosophers (e.g., Plato and 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 nowhere-differentiable functions) forced a critical re-examination of the foundations of mathematics. We will study some of these discoveries and in light of them 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?6 credits cr., HU, Spring—S. Kennedy

MATH 236. Mathematical StructuresBasic 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 Computer Science 177 (or Computer Science/Mathematics 223) and Mathematics 236.6 credits cr., MS, Fall,Winter,Spring—J. Goldfeather, D. Haunsperger, M. Krusemeyer

MATH 241. Ordinary Differential EquationsAn introduction to the theory and methods of solution of ordinary differential equations. Prerequisites: Mathematics 232 or consent of the instructor.6 credits cr., MS, Fall,Winter—L. Chihara, M. Krusemeyer

MATH 244. GeometriesEuclidean 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 secondary school teachers. Prerequisite: Mathematics 2366 credits cr., MS, Offered in alternate years. Not offered in 2002-2003.

MATH 251. Chaotic DynamicsAn 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 cr., MS, Winter—S. Kennedy

MATH 265. ProbabilityIntroduction to probability and its applications. Topics include: combinatorial analysis used in computing probabilities, random variables, independence, joint and conditional distributions, expectation, moment generating functions, law of large numbers and properties of the most common probability distributions. A computer package, such as S-PLUS or Mathematica, may be used for simulating random phenomena. Prerequisite: Mathematics 211.6 credits cr., MS, Fall—R. Dobrow

MATH 275. Mathematical StatisticsIntroduction to mathematical statistics. The mathematics underlying fundamental statistical concepts will be covered as well as applications of these ideas to real-life data. Topics include: Confidence intervals, hypothesis testing, parameter estimation, maximum likelihood, and goodness of fit tests and regressions. A statistical software package will be used to analyze data sets. Prerequisite: Mathematics 265.6 credits cr., MS, Winter—L. Chihara

MATH 285. Topics in Statistics: Applied Regression AnalysisA second course in statistics covering simple linear regression, multiple regression, model building techniques, model checking, and ANOVA. Emphasis will be placed on applications with real-life data. Additional topics, time permitting, include logistic regression, non-linear regression, etc. Prerequisite: Mathematics 215 or Mathematics 275.6 credits cr., MS, Spring—L. Chihara

MATH 311. Topics in Numerical AnalysisTopics 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 cr., MS, Fall—G. Nelson

MATH 312. Elementary Theory of NumbersProperties of the whole numbers. Topics include 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 cr., MS, Fall—M. Krusemeyer

MATH 321. Real Analysis IA systematic study of concepts basic to calculus, such as topology of the real numbers, limits, differentiation, integration, convergence of sequences, and series of functions. Prerequisite: Mathematics 236 or consent of the instructor.6 credits cr., MS, Fall—C. Roosenraad

MATH 331. Real Analysis IIFurther topics in analysis such as measure theory, Lebesgue integration or Banach and Hilbert spaces. Prerequisite: Mathematics 321 or consent of the instructor.6 credits cr., MS, Winter—G. Nelson

MATH 332. Advanced Linear AlgebraAdvanced topics in linear algebra. Prerequisite: Mathematics 236 or consent of the instructor.6 credits cr., MS, Offered in alternate years. Not offered in 2002-2003.

MATH 333. Combinatorial TheoryDeciding on the existence of, finding, and/or counting arrangements, functions, and other desired structures involving finite sets. Some graph and network theory. Counting techniques include the inclusion-exclusion principle, generating functions, and recurrence relations. Existence criteria include the pigeonhole principle, Ramsey's theorem, and Hall's ("marriage") theorem. Prerequisite: Mathematics 236 or consent of the instructor.6 credits cr., MS, Spring—K. Garrett

MATH 341. Partial Differential EquationsFourier series and their applications to boundary value problems in partial differential equations of engineering and physics. Topics include separation of variables, orthogonal sets of functions, representations of functions in series of orthogonal functions, Fourier transforms, and verification and uniqueness of solutions. Prerequisite: Mathematics 241.6 credits cr., MS, Spring—R. Nau

MATH 342. Abstract Algebra IIntroduction 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 cr., MS, Spring—J. Goldfeather

MATH 344. Differential GeometryLocal 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 cr., MS, Spring—G. Nelson

MATH 349. Methods of Teaching MathematicsCross-listed with EDUC 350.Methods of and materials and technology for teaching mathematics in grades 5-12. Issues in contemporary mathematics education. Regular visits to school classrooms and teaching a class are required. Prerequisite: Senior standing and permission of the instructor.6 credits cr., ND, Winter—C. Roosenraad

MATH 351. Functions of a Complex VariableAlgebra and geometry of complex numbers, analytic functions, complex integration, series, residues, applications. Prerequisite: Mathematics 211.6 credits cr., MS, Offered in alternate years. Not offered in 2002-2003.

MATH 352. Abstract Algebra IIAn 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 cr., MS, Offered in alternate years. Not offered in 2002-2003.

MATH 354. TopologyAn introduction to point-set topology: topological spaces, continuous functions, connectedness, and compactness. Additional topics may include the countability axioms, the separation axioms, the Tychonoff theorem, or the fundamental group. Prerequisite: Mathematics 236 or consent of the instructor.6 credits cr., MS, Offered in alternate years. Not offered in 2002-2003.

MATH 395. Seminar in Mathematics6 credits cr., MS, Spring—M. Krusemeyer

MATH 395. Operations ResearchAn introduction to linear and nonlinear optimization. Modeling of large scale problems. Linear programming. The linear algebraic and geometric bases for the simplex method. Interior point methods. Duality and sensitivity. Specialized methods. Applications to assignment, transportation, network and management science problems. Unconstrained and constrained nonlinear optimization. Search methods. Karush-Kuhn-Tucker conditions. Prerequisite: Mathematics 232.6 credits cr., MS, Spring—R. Nau

MATH 395. Graph TheoryIntroduction to graph theory with numerous applications. Topics include: graph models, trees, matchings, connectivity, graph coloring, Hamiltonian cycles, Euler's formula, and planary. Prerequisite: Mathematics 236 or consent of the instructor.6 credits cr., MS, Winter—R. Dobrow

MATH 400. 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 majors3 credits cr., S/NC, ND, Spring—Staff

MATH 400. 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 cr., S/NC, ND, Fall,Winter,Spring—StaffComputer Science Courses (CS)

PHYS 247. Digital ElectronicsCross-listed with CS 247.. 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 207.6 credits cr., MS, Winter—B. Thomas

CS 257. Software DesignSelected topics in the design and implementation of software. Topics may include program specification, software life cycles, formal verification, testing, object-oriented design, debugging techniques, maintenance, and documentation. Prerequisite: Computer Science 127.6 credits cr., MS, Spring—J. Ondich

CS 307. Operating SystemsIntroduction 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 cr., MS, Offered in alternate years. Not offered in 2002-2003.

CS 317. Computer GraphicsThe raster graphics representation of 2- and 3- dimensional images. Topics include frame buffers, data structures for image storage, geometric transformations, hidden surface algorithms, raytracing, splines, and lighting models. Prerequisites: Computer Science 127, Mathematics 232 or consent of instructor.6 credits cr., MS, Offered in alternate years. Not offered in 2002-2003.

CS 327. Artificial IntelligenceIntelligent agents; heuristic search; knowledge representation using logic; uncertain knowledge and reasoning; machine learning; LISP; Prolog. Prerequisite: Computer Science 127 or consent of the instructor.6 credits cr., MS, Winter—D. Musicant

CS 337. Computer NetworksIntroduction to networks and distributed computing. Topics include the OSI reference model, the TCP/IP protocol suite, local- and wide-area networks, security, routing, the technical and administrative structure of the Internet, client-server programming, protocol implementation and distributed algorithms. Prerequisite: Computer Science 207 or consent of the instructor.6 credits cr., MS, Offered in alternate years. Not offered in 2002-2003.

CS 395. Natural Language ProcessingComputers are poor conversationalists, despite decades of attempts to change that fact. This course will provide an overview of the computational techniques developed in the attempt to enable computers to interpret and respond appropriately to ideas expressed using natural languages such as English and French. Topics will include parsing, semantic analysis and logical form, machine translation, and statistical methods in speech recognition. Prerequisite: Computer Science 237 or permission of the instructor.6 credits cr., MS, Fall—J. Ondich

CS 395. Data MiningSurvey of current techniques in data mining and knowledge discovery such as association rules, classification and prediction, clustering, and text analysis. The class will focus both on the theory behind the techniques, and the issues that arise when implementing them. Prerequisite: Computer Science 127.6 credits cr., MS, Spring—D. Musicant

CS 400. Integrative Exercise (Senior Examination)A three-hour written test on material from Computer Science 117, 127, 207, 217, 227, and 237. Required of all senior majors.3 credits cr., S/NC, ND, Spring—Staff

CS 400. 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 121, Computer Science 117, 127, 177 or 223 (or Mathematics 236); three courses from among Computer Science 207, 217, 227, 237; one course from among Computer Science courses numbered 247 or above or Mathematics 311.3 credits cr., S/NC, ND, Fall,Winter,Spring—Staff