Mathematics and Computer Science
Chair: Professor Mark Krusemeyer
Professors: David F. Appleyard, Jack Goldfeather, Mark Krusemeyer, Richard W. Nau, Samuel E. Patterson
Visiting Professor: Loren Larson
Associate Professors: Laura M. Chihara, Deanna Beth Haunsperger, Stephen F. Kennedy, Gail S. Nelson, Jeffrey R. Ondich, Eric R. Westlund
Visiting Assistant Professor: John M. Alongi
Instructor: David R. Musicant
Adjunct 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, 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 342, 315, 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, 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 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, 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. Secret Writing: A Seminar on Cryptography Attempts 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 modern computers, with a focus on the twentieth century and World War II in particular. No knowledge of mathematics beyond algebra and geometry will be assumed. 6 credits cr., S/CR/NC, MS, Spring—C. Roosenraad
MATH 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 students develop skill in mathematical reasoning. No prerequisites. 6 credits cr., MS, Spring—M. Krusemeyer
MATH 109. Calculus I with Review, Part 1 Mathematics 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—L. Larson
MATH 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: Mathematics 109 or consent of the instructor. 6 credits cr., MS, Spring—S. Kennedy
MATH 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 cr., MS, Fall,Winter,Spring—Staff
MATH 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. 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 cr., MS, Fall,Spring—Staff
MATH 121. Calculus II Integration 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 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 cr., S/CR/NC, MS, Winter—Staff
MATH 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 placement by or consent of the department. 6 credits cr., MS, Fall,Winter,Spring—Staff
MATH 215. Introduction to Probability and Statistics Frequency distributions, descriptive statistics, sampling and experimental design, normal distributions and approximations, hypothesis testing, probability, random variables, 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 and 325 instead of 215. Prerequisite: Mathematics 110 or 111 or consent of the instructor. Students may not receive credit for both Mathematics 115 and either Mathematics 215 or Psychology 124. Mathematics 215 does not count toward the Mathematics major. 6 credits cr., MS, Fall,Winter,Spring—Staff
MATH 216. Seminar: History of Mathematics This 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 223. Discrete Mathematics for Computer Science Cross-listed with CS 223. 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, and group theory. Prerequisite: Mathematics 110 or 111 and Computer Science 117 or Mathematics 110 or 111 and concurrent registration in Computer Science 117. Students may not receive credit for both CS/MATH 223 and MATH 236. 6 credits cr., MS, Spring—J. Ondich
MATH 232. Linear Algebra Vector 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—J. Alongi, S. Kennedy, G. Nelson
MATH 234. Philosophy of Mathematics Cross-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, Not offered in 2000-2001.
MATH 236. 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 CS/MATH 223 and MATH 236. 6 credits cr., MS, Fall,Winter,Spring—D. Appleyard, D. Haunsperger, M. Krusemeyer
MATH 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 cr., MS, Winter—D. Appleyard
MATH 244. 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 secondary school teachers. Prerequisite: Mathematics 236 6 credits cr., MS, Offered in alternate years. Not offered in 2000-2001.
MATH 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 cr., MS, Winter—D. Haunsperger
MATH 285. Topics in Statistics: Applied Linear Regression This course is an introduction to regression analysis methods. We will cover simple linear regression, multiple regression and analysis of variance, as well as regression diagnostics and model-building strategies. Applications will be emphasized. Prerequisites: Mathematics 215 or 325, and 232 or permission of the instructor 6 cr., MS, Spring—L. Chihara
MATH 285. Topics in Statistics: Applied Linear Regression This course is an introduction to regression analysis methods. We will cover simple linear regression, multiple regression and analysis of variance, as well as regression diagnostics and model-building strategies. Applications will be emphasized. Prerequisites: Mathematics 215 or 325, and 232 or permission of the instructor. 6 cr., MS, Spring—L. Chihara
MATH 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 cr., MS, Fall—R. Nau
MATH 312. Elementary Theory of Numbers Properties 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 315. Probability Theory 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 may include survey sampling. Prerequisite: Mathematics 236 or consent of the instructor. 6 credits cr., MS, Fall—L. Chihara
MATH 321. Real Analysis I A 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—S. Kennedy
MATH 325. Mathematical Statistics Confidence intervals, parameter estimation, hypothesis testing, regression, analysis of variance, and decision theory. A computer package will be used to analyze real data sets. Prerequisite: Mathematics 315. 6 credits cr., MS, Winter—L. Chihara
MATH 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 cr., MS, Winter—G. Nelson
MATH 332. Advanced Linear Algebra Advanced topics in linear algebra. Prerequisite: Mathematics 236 or consent of the instructor. 6 credits cr., MS, Offered in alternate years. Not offered in 2000-2001.
MATH 333. Combinatorial Theory Deciding 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—M. Krusemeyer
MATH 341. Partial Differential Equations Fourier 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—J. Alongi
MATH 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 cr., MS, Spring—L. Larson
MATH 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 cr., MS, Spring—G. Nelson
MATH 349. Methods of Teaching Mathematics Cross-listed with EDUC 350. 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: Senior standing and permission of the instructor. 6 credits cr., ND, Winter—C. Roosenraad
MATH 351. Functions of a Complex Variable Algebra 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 2000-2001.
MATH 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 cr., MS, Offered in alternate years. Not offered in 2000-2001.
MATH 354. Topology An introduction to point-set, geometric, and algebraic topology. Topics include continuous transformations, compactness, connectedness, the fundamental group, simplicial complexes and surfaces. Prerequisite: Mathematics 236 or consent of the instructor. 6 credits cr., MS, Offered in alternate years. Not offered in 2000-2001.
MATH 395. Seminar in Statistics 6 credits cr., MS, Spring—L. Chihara
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—Staff
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 majors 3 credits cr., S/NC, ND, Spring—Staff
Computer Science Courses (CS)
CS 117. Introduction to Computer Science Problem solving and implementation of algorithms in C++. 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 cr., MS, Fall,Winter,Spring—Staff
CS 127. Data Structures An introduction to abstract data types, classes in C++, recursion, searching, sorting, stacks, queues, linked lists, trees, graphs and hash tables. Prerequisite: Computer Science 117 or consent of the instructor. 6 credits cr., MS, Fall,Winter,Spring—Staff
CS 207. Computer Organization and Architecture 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 cr., MS, Fall—J. Ondich
CS 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 cr., MS, Spring—D. Musicant
CS 218. Seminar on the History of Computer Science This seminar will survey computer science history from the mid-nineteenth century to the present. Topics will include the development of programming languages, operating systems, computer networks, microprocessors, artificial intelligence, and computer graphics. Each student will give one or two presentations supplementing the weekly readings. Prerequisites: Computer Science 127 or consent of the instructor. 2 credits cr., ND, Not offered in 2000-2001.
CS 223. Discrete Mathematics for Computer Science Cross-listed with MATH 223. 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, and group theory. Prerequisite: Mathematics 110 or 111 and Computer Science 117 or Mathematics 110 or 111 and concurrent registration in Computer Science 117. Students may not receive credit for both CS/MATH 223 and MATH 236. 6 credits cr., MS, Spring—J. Ondich
CS 227. Computer Algorithms Design and analysis of algorithms. Divide and conquer, dynamic programming, greedy method, backtracking, branch-and-bound, and inductive approaches. Recurrence relations, sorting and searching, complexity theory, NP-completeness. Prerequisites: Computer Science 223 and 127 and Mathematics 121. 6 credits cr., MS, Winter—R. Nau
CS 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 cr., MS, Fall—D. Appleyard
CS 247. Digital Electronics Cross-listed with PHYS 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. 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 cr., MS, Spring—R. Nau
CS 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 cr., MS, Offered in alternate years. Not offered in 2000-2001.
CS 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, 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 2000-2001.
CS 327. Artificial Intelligence Heuristic search; knowledge representation using logic, frames and rules; expert systems; uncertain knowledge and reasoning; machine learning; LISP. Prerequisite: Computer Science 127 or consent of the instructor. 6 credits cr., MS, Winter—D. Musicant
CS 337. Computer Networks Introduction 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 organizational structure of the Internet, client-server programming, protocol implementation and distributed algorithms. 6 credits cr., MS, Offered in alternate years. Not offered in 2000-2001.
CS 395. Natural Language Processing Computers 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, Spring—J. Ondich
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, 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