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Mathematics and Computer Science

Professors Brooks (on leave, 2001–2002) and Haines; Associate Professors Ross, Rhodes, Wong, Chair, and Shulman (on leave, 2001–2002); Assistant Professors Tajdari and Johnson; Ms. Harder (on leave, fall semester) and Mr. Towne Mathematics today is a dynamic and everchanging 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. Entrylevel courses introduce students to basic concepts and tools and hint at some of the power and beauty behind these fundamental results. Upperlevel 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 newstudent orientation the department assists students planning to study mathematics in choosing an appropriate starting course. Based on a students 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. Major Requirements. 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. 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. Courses used to complete 6) may also be applied to 4). 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 either through 360 or s50 (Independent Study), or 457–458 (Senior Thesis). Pass/Fail Grading Option. Pass/fail grading may not be elected for courses applied toward the major. Secondary Concentration in Mathematics. 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 fulfull 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 150 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 toward the secondary concentration in mathematics. 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, not including 360, (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 toward the secondary concentration in computing science. General Education. The quantitative requirement is satisfied by any of the mathematics or computer science courses or units. Credit awarded for advanced placement mathematics, computer science, or statistics may also satisfy the quantitative requirement. Courses 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 and 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. Staff. 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 100level mathematics course. Open to firstyear students. Enrollment limited to 25. 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 straightforward use of the techniques of singlevariable calculus, more effort must be spent in developing a conceptual framework for understanding curves and surfaces in higherdimensional spaces. Topics include partial derivatives, derivatives of vectorvalued functions, vector fields, integration over regions in the plane and threespace, 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 firstyear students. Staff. 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. Staff. 219. Differential Equations. A differential equation is a relationship between a function and its derivatives. Many realworld 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 population modeling and mechanical vibrations. Prerequisite(s): Mathematics 206. Staff. 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 linearprogramming problems and twoperson zerosum games, the duality theorem of linear programming, and the minmax theorem of game theory. Additional topics are drawn from such areas as nperson 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. Staff. 301. Real Analysis. An introduction to the foundations of mathematical analysis, this course presents a rigorous treatment of fundamental concepts such as limits, continuity, differentiation, and integration. Elements of the topology of the real numbers are also covered. Prerequisite(s): Mathematics 206 and s21. J. Rhodes. 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. Staff. 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. P. Wong. 312. Geometry. This course studies geometric concepts in Euclidean and nonEuclidean geometries. Topics include isometries, arc lengths, curvature of curves and surfaces, and tesselations, especially frieze and wallpaper patterns. Prerequisite(s): Mathematics 206. P. Wong. 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. Staff. 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. Staff. 315. Statistics. The sequel to Mathematics 314. This course covers estimation theory and hypothesis testing. Prerequisite(s): Mathematics 314. Staff. 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 reallife 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. D. Haines. 360. Independent Study. Students, in consultation with a faculty advisor, individually design and plan a course of study or research not offered in the curriculum. Course work includes a reflective component, evaluation, and completion of an agreedupon product. Sponsorship by a faculty member in the program/department, a course prospectus, and permission of the chair is required. Students may register for no more than 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.
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.
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 twosemester 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 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. Staff. s45. Seminar in Mathematics. The content varies. Recent topics have included number theory and an introduction to error correcting codes. Staff.
s50. Independent Study. Students, in consultation with a faculty advisor, individually design and plan a course of study or research not offered in the curriculum. Course work includes a reflective component, evaluation, and completion of an agreedupon product. Sponsorship by a faculty member in the program/department, a course prospectus, and permission of the chair is required. Students may register for no more than one independent study during a Short Term. Staff. 
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