Professors Brooks (on leave, winter semester and Short Term) and Haines; Associate Professors Ross (on leave, winter semester and Short Term), Rhodes, Chair, Wong, and Shulman; Assistant Professor Johann; Ms. Mosbo and Ms. Harder

Winter 2000 Mathematics Addendum Notes

Short Term 2000 Mathematics Addendum Notes

Mathematics today is a dynamic and ever-changing subject, and an important part of a liberal-arts education. Mathematical skills such as data analysis, problem solving, and abstract reasoning are increasingly vital to science, technology, and society itself. Entry-level courses introduce students to basic concepts and tools and hint at some of the power and beauty behind these fundamental results. Upper-level courses and the senior thesis option provide majors with the opportunity to explore mathematical topics in greater depth and sophistication, and delight in the fascination of this "queen of the sciences."

During new-student orientation the department assists students planning to study mathematics in choosing an appropriate starting course. Based on a student's academic background and skills, the department recommends Mathematics 101, 105, 106, 205, 206, or a more advanced course.

The mathematics department offers a major in mathematics, a secondary concentration in mathematics, and a secondary concentration in computing science.

Mathematics Major. The mathematics major requirements accommodate a wide variety of interests and career goals. The courses provide broad training in undergraduate mathematics and computer science, preparing majors for graduate study, and for positions in government, industry, and the teaching profession.

The major in mathematics consists of: 1) Mathematics 205 and 206; 2) Mathematics s21, which should be taken during Short Term of the first year; 3) Mathematics 301 and 309; 4) five elective mathematics or computer science courses numbered 200 or higher; 5) a one-hour oral presentation; and 6) completion of either a two-semester thesis (Mathematics 457-458) or the senior seminar (Mathematics 395). Thesis proposals require departmental approval. Entering students may be exempted from either of the courses in 1) on the basis of work before entering college. Any mathematics or computer-science Short Term unit numbered 30 or above may be used as one of the electives in 4). One elective may also be replaced by a departmentally approved course from another department.

While students must consult with their major advisors in designing appropriate courses of study, the following suggestions may be helpful: For majors considering a career in secondary education the department suggests Mathematics 312, 314, 315, 341, and Computer Science 101 and 102. Students interested in operations research, business, or actuarial science should consider Mathematics 218, 239, 314, 315, 341, s32, and the courses in computer science. Students interested in applied mathematics in the physical and engineering sciences should consider Mathematics 218, 219, 308, 314, 315, 341, and the courses in computer science. Majors planning on graduate study in pure mathematics should particularly consider Mathematics 308, 313, and 457-458. Mathematics majors may pursue individual research through 360 (Independent Study), s50 (Individual Research), or 457-458 (Senior Thesis).

Pass/Fail Grading Option: Pass/fail grading may not be elected for courses applied towards the major. Added 11/5/99. Effective beginning with Winter 2000 semester.

Mathematics Secondary Concentration. Designed either to complement another major, or to be pursued for its own sake, the secondary concentration in mathematics provides a structure for obtaining a significant depth in mathematical study. It consists of seven courses, four of which must be Mathematics 105, 106, 205, and 206. (Successful completion of Mathematics 206 is sufficient to fulfill the requirements for Mathematics 105 and 106, even if no course credit for these has been granted by Bates.)

In addition, the concentration must include at least two courses forming a coherent set. Approved sets include: 1) Analysis: s21 and 301; 2) Algebra: s21 and 309; 3) Geometry: 312 and 313; 4) Mathematical Biology: 155 and either 219 or 341; 5) Actuarial Science: 314 and either 218, 239, 315, or s32; 6) Statistics: 314 and 315; 7) Decision-making/Optimization: 239 and s32; 8) Applied/Engineering Mathematics: 219 and either 218, 308, or 341.

The final course in the concentration can be any Mathematics or Computer Science course at the 200 level or above (or a unit at the 20 level or above), or Computer Science 102.

Pass/Fail Grading Option: Pass/fail grading may not be elected for courses applied towards the secondary concentration. Added 11/5/99. Effective beginning with Winter 2000 semester.

Computer Science and Secondary Concentration in Computing Science. Students normally begin study of computer science with Computer Science 101. New students who have had the equivalent of 101 should consult with the department.

The secondary concentration in computing science consists of seven courses. These include: 1) Computer Science 101, 102; 2) either Computer Science 205 or Mathematics s21; 3) at least two of Computer Science 301, 302, 303, and 304; and 4) two additional courses or units from the following list: all computer science courses at the 200 level or above (or units at the 20 level or above), Mathematics 218, 239, Physics s30, Music 237, and Biology s45.

Students interested in a career in computer science should consider not only computer science courses, but also Mathematics 205, 218, 239, 314, and 315.

Pass/Fail Grading Option: Pass/fail grading may not be elected for courses applied towards the secondary concentration. Added 11/5/99. Effective beginning with Winter 2000 semester.

General Education. The quantitative requirement is satisfied by any of the mathematics or computer science courses or units.

Courses

101. Working with Data. Techniques for analyzing data are described in ordinary English without emphasis on mathematical formulas. Graphical and descriptive techniques for summarizing data, design of experiments, sampling, analyzing relationships, statistical models, and hypothesis testing. Applications from everyday life: drug testing, legal discrimination cases, public-opinion polling, industrial quality control, and reliability analysis. Students are instructed in the use of the computer, which is used extensively throughout the course. Enrollment limited to 30. M. Harder.

105. Calculus I. While the word "calculus" originally meant any method of calculating, it has come to refer more specifically to the fundamental ideas of differentiation and integration that were first developed in the seventeenth century. The subject's early development was intimately connected with understanding rates of change within the context of the physical sciences. Nonetheless, it has proved to be of wide applicability throughout the natural sciences and some social sciences, as well as crucial to the development of most modern technology. This course develops the key notions of derivatives and integrals and their interrelationship, as well as applications. An emphasis is placed on conceptual understanding and interpretation, as well as on calculational skills. Graphing calculators are used in the course for graphical and numerical explorations. Enrollment limited to 25 per section. Staff.

106. Calculus II. A continuation of Calculus I. Further techniques of integration, both symbolic and numerical, are studied. The course then treats applications of integration to problems drawn from fields such as physics, biology, chemistry, economics, and probability. Differential equations and their applications are also introduced, as well as approximation techniques such as Taylor series. Graphing calculators are used in the course for graphical and numerical explorations. Prerequisite(s): Mathematics 105. Enrollment limited to 25 per section. Staff.

155. Mathematical Models in Biology. Mathematical models are increasingly important throughout the life sciences. This course provides an introduction to deterministic and statistical models in biology. Examples are chosen from a variety of biological and medical fields such as ecology, molecular evolution, and infectious disease. Computers are used extensively for modeling and for analyzing data. Recommended background: a course in biology. This course is the same as Biology 155. Enrollment limited to 30. Not open to students who have received credit for Biology 255. J. Rhodes.

205. Linear Algebra. Vectors and matrices are introduced as devices for the solution of systems of linear equations with many variables. Although these objects can be viewed simply as algebraic tools, they are better understood by applying geometric insight from two and three dimensions. This leads to an understanding of higher dimensional spaces and to the abstract concept of a vector space. Other topics include orthogonality, linear transformations, determinants, and eigenvectors. This course should be particularly useful to students majoring in any of the natural sciences or economics. Prerequisite(s): one 100-level mathematics course. Open to first-year students. J. Rhodes.

206. Multivariable Calculus. This course extends the ideas of Calculus I and II to deal with functions of more than one variable. Because of the multidimensional setting, essential use is made of the language of linear algebra. While calculations make straight forward use of the techniques of single-variable calculus, more effort must be spent in developing a conceptual framework for understanding curves and surfaces in higher-dimensional spaces. Topics include partial derivatives, derivatives of vector-valued functions, vector fields, integration over regions in the plane and three-space, and integration on curves and surfaces. This course should be particularly useful to students majoring in any of the natural sciences or economics. Prerequisite(s): Mathematics 106 and 205. Open to first-year students. J. Rhodes.

218. Numerical Analysis. This course studies the best ways to perform calculations that have already been developed in other mathematics courses. For instance, if a computer is to be used to approximate the value of an integral, one must understand both how quickly an algorithm can produce a result and how trustworthy that result is. While students will implement algorithms on computers, the focus of the course is the mathematics behind the algorithms. Topics may include interpolation techniques, approximation of functions, solving equations, differentiation and integration, solution of differential equations, iterative solutions of linear systems, and eigenvalues and eigenvectors. Prerequisite(s): Mathematics 106 and 205 and Computer Science 101. J. Rhodes.

219. Differential Equations. A differential equation is a relationship between a function and its derivatives. Many real-world situations can be modeled using these relationships. This course is a blend of the mathematical theory behind differential equations and their applications. The emphasis is on first- and second-order linear equations. Topics include existence and uniqueness of solutions, power series solutions, numerical methods, and applications such as populations modeling and mechanical vibrations. Prerequisite(s): Mathematics 206. B. Shulman.

239. Linear Programming and Game Theory. Linear programming grew out of the recognition that a wide variety of practical problems reduces to maximizing or minimizing a linear function whose variables are restricted by a system of linear constraints. A closely related area is game theory, which deals with decision problems in a competitive environment, where conflict, risk, and uncertainty are often involved. The course focuses on the underlying theory, but applications to social, economic, and political problems abound. Topics include the simplex method for solving linear-programming problems and two-person zero-sum games, the duality theorem of linear programming, and the min-max theorem of game theory. Additional topics will be drawn from such areas as n-person game theory, network and transportation problems, and relations between price theory and linear programming. Computers are used regularly. Prerequisite(s): Computer Science 101 and Mathematics 205. This course is the same as Economics 239. R. Brooks.

301. Real Analysis. An introduction to the foundations of mathematical analysis, this course presents a rigorous treatment of elementary concepts such as limits, continuity, differentiation, and integration. Elements of the topology of the real numbers will also be covered. Prerequisite(s): Mathematics 206 and s21. P. Wong.

308. Complex Analysis. This course extends the concepts of calculus to deal with functions whose variables and values are complex numbers. Instead of producing new complications, this leads to a theory that is not only more aesthetically pleasing, but is also more powerful. The course should be valuable to those interested in pure mathematics, as well as those who need additional computational tools for physics or engineering. Topics include the geometry of complex numbers, differentiation and integration, representation of functions by integrals and power series, and the calculus of residues. Prerequisite(s): Mathematics 206. J. Rhodes.

309. Abstract Algebra I. An introduction to basic algebraic structures common throughout mathematics. These include the integers and their arithmetic, modular arithmetic, rings, polynomial rings, ideals, quotient rings, fields, and groups. Prerequisite(s): Mathematics 205 and s21. D. Haines.

312. Foundations of Geometry. The study of the evolution of geometric concepts starting from the ancient Greeks (800 B.C.E.) and continuing to current topics. These topics are studied chronologically as a natural flow of ideas: conic sections from the Greek awareness of astronomy, continuing to Kepler and Newton; perspective in art and geometry; projective geometry including the Gnomic, Mercator, and Stereographic terrestrial maps; Euclidean and non-Euclidean geometries with their respective axiomatic structure; isometries; the inversion map in the plane and in three-space; curvature of curves and surfaces; graph theory including tilings (tessellations); fixed point theorems; and space-time geometry. Geometers encountered are Euclid, Apollonius, Pappus, Descartes, Dürer, Kepler, Newton, Gauss, Riemann, A.W. Tucker, and others. Staff.

313. Topology. The notion of "closeness" underlies many important mathematical concepts, such as limits and continuity. Topology is the careful study of what this notion means in abstract spaces, leading to a thorough understanding of continuous mappings and the properties of spaces that they preserve. Topics include metric spaces, topological spaces, continuity, compactness, and connectedness. Additional topics, such as fundamental groups or Tychonoff's theorem, may also be covered. Prerequisite(s): Mathematics 206 and s21. J. Rhodes.

314. Probability. Probability theory is the foundation on which statistical data analysis depends. This course together with its sequel, Mathematics 315, covers topics in mathematical statistics. Both courses are recommended for math majors with an interest in applied mathematics and for students in other disciplines, such as psychology and economics, who wish to learn about some of the mathematical theory underlying the methodology used in their fields. Prerequisite(s): Mathematics 106. P. Wong.

315. Statistics. The sequel to Mathematics 314. This course covers estimation theory and hypothesis testing. Prerequisite(s): Mathematics 314. M. Harder.

341. Mathematical Modeling. Often analyzing complex situations (like the weather, a traffic flow pattern, or an ecological system) is necessary to predict the effect of some action. The purpose of this course is to provide experience in the process of using mathematics to model real-life situations. The first half examines and critiques specific examples of the modeling process from various fields. During the second half each student creates, evaluates, refines, and presents a mathematical model from a field of his or her own choosing. Prerequisite(s): Mathematics 206. B. Shulman.

360. Independent Study. Independent study by an individual student with a single faculty member. Permission of the department is required. Students are limited to one independent study per semester. Staff.

365. Special Topics. Content varies from semester to semester. Possible topics include chaotic dynamical systems, number theory, mathematical logic, representation theory of finite groups, measure theory, algebraic topology, combinatorics, and graph theory. Prerequisites vary with the topic covered but are usually Mathematics 301 and/or 309. Staff.

395. Senior Seminar. While the subject matter varies, the seminar addresses an advanced topic in mathematics. The development of the topic draws on students' previous course work and helps consolidate their earlier learning. Students are active participants, presenting material to one another in both oral and written form, and conducting individual research on related questions.

395B. Einstein's Theory of Relativity. The main focus of this course is the mathematics behind Einstein's special theory of relativity. Students discuss the Lorentz group, study the geometry of Minkowski's space, and compare special relativity to Galilean relativity. Possible additional topics include hyperbolic geometry, pseudo-Riemannian geometry, and curved space-time. Prerequisite(s): Mathematics 301 and 309. Written permission of the instructor is required. P. Wong.

457-458. Senior Thesis. Prior to entrance into Mathematics 457, students must submit a proposal for the work they intend to undertake toward completion of a two-semester thesis. Open to all majors upon approval of the proposal. Required of candidates for honors. Students register for Mathematics 457 in the fall semester and Mathematics 458 in the winter semester. Staff.

Short Term Units

s21. Introduction to Abstraction. An intensive development of the important concepts and methods of abstract mathematics. Students work individually, in groups, and with the instructors to prove theorems and solve problems. Students meet for up to five hours daily to explore such topics as proof techniques, logic, set theory, equivalence relations, functions, and algebraic structures. The unit provides exposure to what it means to be a mathematician. Prerequisite(s): one semester of college mathematics. Required of all majors. Enrollment limited to 30. Staff.

s32. Topics in Operations Research. An introduction to a selection of techniques that have proved useful in management decision-making: queuing theory, inventory theory, network theory (including PERT and CPM), statistical decision theory, computer modeling, and dynamic programming. Prerequisite(s): Mathematics 105 and a course in probability or statistics. Enrollment limited to 20. Written permission of the instructor is required. R. Brooks.

s45. Seminar in Mathematics. The content varies. Recent topics have included number theory and an introduction to error correcting codes.

s45B. Inverse Problems in the Mathematical Sciences. Many important problems with applications in all the sciences are posed in a way that inverts a direct problem. Traditional undergraduate mathematics is dominated by these direct problems, but modern science and technology confront students with inverse problems (i.e., in remote sensing, medical imaging, non-destructive testing, environmental monitoring, seismic surveys). This unit treats the mathematical theory and practical applications of inverse problems with examples drawn from geology, biology, chemistry, and physics. B. Shulman.

s45E. Combinatorics. Combinatorics is concerned with the counting , analysis, existence, and optimization of finite structures. Its problems can be as elementary as constructing a magic square or as difficult as the four-color conjecture, which was solved in 1976 after over a hundred years of research. Topics in this unit include: the pigeonhole principle and Ramsey's theorem; permutations, combinations, and their generation; binomial coefficients; the inclusion-exclusion principle; recurrence relations and generations functions; and, time permitting, graph theory. Prerequisite(s): any two mathematics or computer science courses. Open to first-year students. D. Haines.

s50. Individual Research. The department permits registration for this unit only after the student submits a written proposal for a full-time research project to be completed during the Short Term and obtains the sponsorship of a member of the department to direct the study and evaluate its results. Students are limited to one individual research unit. Staff.

Computer Science
[see requirements and general information for Computer Science]

Short Term 2000 Computer Science Addendum Notes

101. Computer Science I. An introduction to computer science, with the major emphasis on the design, development, and testing of computer software. It introduces the student to a disciplined approach to algorithmic problem-solving in a modern programming environment using an object-based event-driven programming language. Students develop programs in Visual BASIC to run under the Windows operating system. The course has an associated laboratory that provides hands-on experience. Students complete a substantial individual or group project. P. Johann.

102. Computer Science II. A continuation of Computer Science I. The major emphasis of the course is on advanced program design concepts and techniques, and their application to the development of high-quality software. Specific topics covered include the software development cycle, abstract datatypes, files, recursion, and object-oriented programming. Computer Science 101 and 102 provide a foundation for further study in computer science. Language of instruction for 1999-2000: Visual BASIC. Prerequisite(s): Computer Science 101. Enrollment limited to 25. P. Johann.

205. Discrete Structures. This course provides an introduction to logic, mathematical reasoning, and the discrete structures that are fundamental to computer science. Learning to reason effectively about discrete structures and, thereby, about the behavior of computer programs is the primary goal of the course. Learning to read and write clear and correct mathematical proofs is an important secondary aim. Specific topics include propositional and predicate logic, logic circuits, basic set theory, relations, functions, induction, recursion, and graph theory. Prerequisite(s) or Corequisite(s): Computer Science 101. Not open to students who have received credit for Mathematics s21. Open to first-year students. D. Haines.

301. Algorithms. The course covers specific algorithms (e.g., searching, sorting, merging, numeric, and network algorithms), related data structures, an introduction to complexity theory (O-notation, the classes P and NP, NP-complete problems, and intractable problems), and laboratory investigation of algorithm complexity and efficiency. Prerequisite(s) or Corequisite(s): Computer Science 102, and either Computer Science 205 or Mathematics s21. Open to first-year students. P. Johann.

302. Theory of Computation. A course in the theoretical foundations of computer science. Topics include finite automata and regular languages, pushdown automata and context-free languages, Turing machines, computability and recursive functions, and complexity. Prerequisite(s): Computer Science 102, and either Computer Science 205 or Mathematics s21. P. Johann.

303. Principles of Computer Organization. Computer and processor architecture and organization including topics such as operating systems, buses, memory organization, addressing modes, instruction sets, input/output, control, synchronization, interrupts, multiprocessing, and multitasking. The course may include training in digital logic, machine language programming, and assembly language programming. Prerequisite(s): Computer Science 101. Open to first-year students. Not open to students who have received credit for Computer Science 201. S. Ross.

304. Principles of Programming Languages. An introduction to the major concepts and paradigms of contemporary programming languages. Concepts covered include procedural abstraction, data abstraction, tail-recursion, binding and scope, assignment, and generic operators. Paradigms covered include imperative (e.g., Pascal and C), functional (e.g., LISP), object-oriented (e.g., Smalltalk), and logic (e.g., Prolog). Students write programs in SCHEME to illustrate the paradigms. Prerequisite(s): Computer Science 102, and either Computer Science 205 or Mathematics s21. Not open to students who have received credit for Computer Science 202. D. Haines.

360. Independent Study. Independent study by an individual student with a faculty member. Permission of the department is required. Students are limited to one independent study per semester. Staff.

365. Special Topics. A seminar usually involving a major project. Recent topics have been: the mathematics and algorithms of computer graphics, in which students designed and built a computer-graphics system, and contemporary programming languages and their implementations, in which students explored new languages, in some cases using the Internet to obtain languages such as Oberon, Python, Haskell, and Dylan. Written permission of the instructor is required. Staff.

Short Term Units

s45. Seminar in Computer Science. The content varies. Recent topics include cryptography and data security, and functional programming. Prerequisites vary with the topic covered. Staff.

s45B. Computers and Contemporary Society. Recent advances in computing Ñ and the changing technologies to which those advances have given rise Ñ have brought about tremendous changes in our everyday lives. The past decade has seen, for example, the birth of the World Wide Web, and explosion in the growth of the Internet, and an increasing reliance on expert systems. It has also seen an increase in software piracy, hacking, and other computer crime, as well as the rise of the new culture of simulation. This unit examines a number of issues of concern as modern culture becomes increasingly infused with technology and the technological. Topics covered include legal, ethical, and social issues in computing, such as privacy, security, free speech, professional ethics for computer scientists, computers as tools for democratization, hacker culture, and the impact of computers on how we think about and relate to ourselves and one another. Open to first-year students. P. Johann.

s50. Individual Research. The department permits registration for this unit only after the student submits a written proposal for a full-time research project to be completed during the Short Term and obtains the sponsorship of a member of the department to direct the study and evaluate its results. Students are limited to one individual research unit. Staff.

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