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Mathematics

Mathematics (MATH)

Professors: George R. Exner, Michael R. Frey, Pamela B. Gorkin, Paul J. McGuire, Allen R. Schweinsberg, Howard Smith

Associate Professors: Carmen O. Acuña, Gregory T. Adams, M. Lynn Breyfogle, Thomas Cassidy, Ulrich Daepp (Chair), James E. Hutton, Karl A. Voss

Assistant Professors: Karen Boomer, Peter A. Brooksbank, Emily Dryden, Sharon A. Garthwaite, Peter McNamara, Matthew S. Miller (visiting), Angela Pile (visiting), Nathan C. Ryan, Linda B. Smolka

Instructor: Amy M. Donner (visiting)

Mathematics has for centuries been the basic language of the natural sciences, and it has been studied for its own sake since ancient times. More recently, mathematics has found itself used more and more in the social sciences, and an understanding of the basics of calculus and statistics is fast becoming a requirement for proficiency in many of these disciplines. Quite apart from its importance to so many fields, the study of mathematics has its own rewards as accomplishment in the subject, even at a relatively elementary level, requires and promotes clarity of thought and clarity of expression.

A major in mathematics may be seen as the first step toward obtaining a graduate degree in one of the mathematical sciences, or it may constitute preparation for a professional degree program in a field such as education, medicine, law, or business. It also opens the door to a whole range of employment opportunities, as the analytical skills that a student develops in pursuing a major in mathematics are greatly valued by potential employers. There are, for example, excellent career prospects in actuarial work and in the rapidly growing areas of biomathematics and biostatistics (interpreting results of clinical trials), modeling (in industry, government, and finance) and cryptology (in banking, television, the Internet, and elsewhere).

Students may choose to major in mathematics in the bachelor of arts program or in the bachelor of science program. Students in either program complete an introductory year of calculus either by taking MATH 201 or 202 during their first year, or by achieving a high score on the Advanced Placement Test of the College Entrance Examination Board.

The choice of degree program depends largely upon the student’s mathematical objective and interest in fields other than mathematics. Students with a strong interest in a career in mathematics or science – and in particular, students planning to continue on to Ph.D. programs in the mathematical sciences – are strongly advised to take courses beyond the minimum requirements for the major. Since a maximum of 12 courses in any one department may be counted toward the bachelor of arts degree, such students may be best advised to enroll in the bachelor of science program. On the other hand, students with strong interests outside of science and mathematics probably will prefer the bachelor of arts program.

The bachelor of arts major in mathematics consists of eight mathematics courses beyond the introductory year of calculus, plus one additional course in a related field. Five of the mathematics courses are specified: MATH 211, 213, 280, 308, and 320. The three remaining mathematics courses must be mathematics electives at the 300 level. The "related" course may be a fourth mathematics course at the 300 level, or MATH 212, or MATH 216, or any course in which mathematics plays a significant role at a reasonable level of sophistication.

Subject to the approval of the mathematics department chair, this course may be:

• almost any full-credit course in computer science at or above the 200 level, for example CSCI 203
  
  
• a third science course (beyond the two required of all liberal arts students) in which college-level mathematics plays a major role. Included among the courses in this category are nearly all courses in physics at or above the 200 level
  
  
• secondary school student teaching in mathematics
  
  
• an appropriate course from the humanities, social sciences, or engineering
  
  
• a Capstone course in which college-level mathematics or statistics plays a major role.

The bachelor of science major in mathematics requires 10 mathematics courses beyond the introductory year of calculus. Six of the 10 mathematics courses are specified: MATH 211, 212, 213, 280, 308, and 320. The remaining four courses are mathematics electives at the 300 level. BS students also must take PHYS 211 and 212 (or PHYS 211E and 212E) and two additional laboratory science courses.

The additional laboratory science courses may be chosen from any discipline in the division of natural sciences or from computer science. Any course in physics beyond PHYS 212 may be chosen, excluding those used to meet the basic two-course physics requirement; any laboratory course in computer science at the level of CSCI 203 or beyond may be chosen.

The recommended sequence for the bachelor of science major is as follows:

Prospective secondary school teachers (grades 7 – 12) must complete either the bachelor of arts or the bachelor of science degree with a major in mathematics. Students seeking teacher certification should confer as early as possible with the mathematics and education departments to devise a program of study, which normally will include all requirements for certification in the Commonwealth of Pennsylvania. For this certification, students must include MATH 303 and MATH 335 among their mathematics electives within the mathematics major; additional mathematics requirements include MATH 207, MATH 240, and either MATH 216 or MATH 307. Required courses in education include EDUC 101, 201, 240, 359 (student teaching), 459, and either 334 or 335. Also required is a course in English literature, which must be in addition to the basic W1 course.

Students majoring in mathematics with a special interest in pure mathematics or statistics can earn formal concentration in these areas by selecting their 300-level electives appropriately and taking one additional course. In particular, those intending to pursue graduate study in mathematics or statistics should plan to complete the relevant concentration.

The pure mathematics concentration consists of MATH 309, 345, 346, and two of the following: MATH 311, 333, 362. The statistics concentration consists of MATH 303, 304, 305, 307, and either MATH 309 or MATH 345. Students majoring in mathematics with a special interest in computer science are encouraged to consider minoring in computer science.

Students who, by the end of their junior year, have completed MATH 308 and 320 and a total of at least three mathematics courses at the 300 level, and who have achieved a grade point average of at least 3.50 both in their mathematics courses and overall are encouraged to apply for departmental honors. If an appropriate mathematics department faculty adviser is available and the student is eligible under the above criteria, then the student can work for departmental honors. To achieve departmental honors, he or she completes at least two half-credit semesters of independent study in mathematics (MATH 391), writes an honors thesis under the adviser’s direction, and satisfies all other requirements as put forth by the University Honors Council.

A minor in mathematics consists of either four credits from mathematics courses numbered 211 or above, at least one of them at the 300 level; or of three credits from courses in mathematics numbered 211 or above, at least two of them at the 300 level. All credits must come from courses taken at Bucknell University.

The minor can be specified as mathematics (statistics), if at least two of the required credits are from among the courses MATH 217, 303, 304, 305, and 307.

The minor can be specified as mathematics (applied/modeling mathematics) if at least two of the required credits are from among the courses MATH 212, 222, 226, 343, 350, and 358.

111. Mathematics from a Humanist Perspective (II; 3, 0)
Provides the nonspecialist with an appreciation for what mathematics is and what mathematicians do.

117. Introduction to Mathematical Thought (II; 3, 1.5)
An investigation of number, numeration, and operations from the perspective of elementary school teachers and pupils. Open only to BS elementary education or early childhood students. Required fieldwork.

118. Elementary Geometry and Statistics (II; 3, 0)
Investigation of geometric, probablistic, and statistical concepts related to elementary school mathematics and how children learn and make sense of these concepts. Prerequisite: MATH 117 or permission of the instructor.

192. Topics in Calculus (II; 3, 0)
Elementary calculus and applications taken primarily from economics. Topics include algebraic, exponential, and logarithmic functions of one and several variables, graphs, limits, derivatives, and integration. Not open to students who have taken MATH 201 or 205.

201. Calculus I (I and II; 4, 0)
An introduction to the calculus of algebraic, trigonometric, and transcendental functions. Interpretation, significance, and calculations of a derivative. Applications to geometry, biology, physics, economics, and other subjects. Introduction to the definite integral, including the Fundamental Theorem of Calculus. Not open to students who have taken MATH 192 or MATH 205.

202. Calculus II (I and II; 4, 0)
Methods of integration including substitution, integration by parts, numerical approximations, and improper integrals. Series, including Taylor series. Complex numbers, polar coordinates, differential equations and applications. Prerequisite: MATH 201 or 205. Not open to students who have taken MATH 206.

205. Accelerated Calculus I (I; 4, 0)
For students intending to complete Calculus I and II in one semester, this course covers the material of MATH 201 during the first half of the semester. (Students normally complete MATH 206 during the second half of the semester.) Prerequisite: Placement or permission of the instructor. Not open to students who have taken MATH 201 or MATH 192.

206. Accelerated Calculus II (I; 4, 0)
Covers the material of MATH 202 in the second half of the semester. Prerequisite: completion of MATH 205 during the first half of the same semester.

207. The Teaching of Mathematics in Secondary Schools (II; 3, 0)
Investigation into the components of effective secondary school mathematics instruction, including lesson design/implementation (curriculum, tasks, discourse, and assessment). Required fieldwork. Prerequisite: EDUC 201 or permission of the instructor.

209. Mathematical Problem Solving (I; 1-5, 0) Half course.
Mathematical problem solving, with an emphasis on problems and topics that appear in contests such as the Putnam Competition. Prerequisite: permision of the instructor.

211. Calculus III (I and II; 3, 0)
Calculus of vector-valued functions and functions of several variables. Applications, including local, absolute, and constrained extrema. Multiple integrals and applications. Line integrals and surface integrals. Prerequisite: MATH 202.

212. Differential Equations (I and II; 3, 0)
Basic methods of solving ordinary differential equations. Systems of linear differential equations, Laplace transform, applications and selected topics. Prerequisite: MATH 211. Not open to students who have taken MATH 222.

213. Elementary Linear Algebra (I and II; 3, 0)
Linear equations, matrices, vector spaces, linear transformations, determinants, eigenvalues. Prerequisite: MATH 202 or 206.

216. Statistics I (I and II; 3, 1)
Exploratory data analysis, sampling distributions, regression, sampling designs, confidence intervals, hypothesis testing, ANOVA. Statistical software is used and applications, including projects, are undertaken. Not open to students who have taken MATH 226, MGMT 242, or PSYC 215.

217. Statistics II (I; 3, 1)
Exploratory data analysis, design of experiments and inference emphasizing applications in biology and environmental science. Includes multiple linear regression, analysis of variance, categorical data analysis, nonparametric statistics. Prerequisite: MATH 216 or equivalent.

222. Differential Equations for Engineers (II; 3, 0) Half course.
First order differential equations, second order linear equations, higher order linear equations, numerical approximations. Prerequisite: MATH 211. Open only to civil engineering and computer science engineering students. Not open to students who have taken MATH 212.

226. Probability and Statistics for Engineers (I; 3, 0) Half course.
Descriptive modeling and statistics, sampling and experimental design, discrete and continuous random variables, central limit theorem, and elementary inference. Prerequisite: MATH 202 or 206. Open only to engineering students and students in computer science. Not open to students who have taken MATH 216.

240. Combinatorics and Graph Theory for Secondary Mathematics (II; 3, 0) Half course.
Combinatorics (permutations, combinations) and graph theory (Eulerian paths, trees, directed graphs). Does not count toward the mathematics major. Students will join a section of MATH 241 mid-semester. Prerequisite: MATH 280. Open only to students seeking certification in secondary mathematics who have not taken MATH 241.

241. Discrete Structures (II; 3, 0)
Sets, logic, and relations, mathematical induction, functions, combinatorics, graph theory. Does not count toward the mathematics major. Corequisite: MATH 211 or MATH 213.

280. Logic, Sets, and Proofs (I and II; 3, 0)
Logic, sets; proof techniques; relations, functions, sequences and convergence; cardinality. Skills and tools for independent reading, problem solving, and exploration. Prerequisite: Corequisite: MATH 211 or MATH 213.

291. Undergraduate Readings (I or II; R; 2-8, 0) Half to two courses.
Readings and research in special topics at an intermediate level. Prerequisite: permission of the instructor, adviser, and department chair.

303. Probability (I and II; 3, 0)
Elementary probability, random variables, moments, central limit theorem, conditional expectation, statistical distributions derived from the normal distribution. Probability simulations and applications from various fields. Prerequisite: MATH 211.

304. Mathematical Statistics (II; 3, 0)
Point and interval estimation, hypothesis testing, theory of least squares and its relation to the design and analysis of experiments. Prerequisites: MATH 216 or equivalent and MATH 303 or permission of the instructor.

305. Statistical Modeling (I; 3, 0)
Regression and analysis of (co)variance. Model diagnosis and remediation. Generalized linear models and nonlinear regression. Multicollinearity and ridge regression. Use of advanced statistical software. Prerequisites: MATH 216 or equivalent, and either MATH 213 or MATH 303 or permission of the instructor.

307. Statistical Design of Scientific Studies (II; 3, 0)
Sampling, design of experiments and observational studies. Includes completely randomized, block, factorial, and nested designs. Simple random stratified, systematic, and cluster sampling. Estimation procedures and sample size calculations. Prerequisite: MATH 217 or MATH 303.

308. Introduction to Real Analysis I (I and II; 3, 0)
Real numbers and elementary topology of Cartesian spaces, convergence, continuity, differentiation, and history of the development of analysis. Prerequisites: MATH 211, MATH 213 and MATH 280.

309. Introduction to Real Analysis II (AI or II; 3, 0)
Continuation of MATH 308. Integration theory and advanced topics in analysis. Prerequisite: MATH 308.

311. Theory of Numbers (AI or II; 3, 0)
Classical number theory in an algebraic setting. Topics include unique factorization, diophantine equations, and linear and quadratic congruences. Advanced topics from algebraic or analytic number theory. Prerequisites: MATH 213 and MATH 280, or permission of the instructor.

319. Topics in Advanced Mathematics (I or II; R; 3, 0)
Special topics, to be selected from algebra, analysis, geometry, statistics, etc.

320. Introduction to Algebra (I and II; 3, 0)
Groups and rings; homomorphisms and isomorphism theorems; history of the development of algebra. Additional selected topics. Prerequisites: MATH 213 and MATH 280.

333. Topology (AI or II; 3, 0)
Topological spaces, connectedness, compactness, continuity, separation, and countability axioms. Metric, product, function, and uniform spaces. Prerequisites: MATH 211 and MATH 280, or permission of the instructor.

335. Geometry (I; 3, 0)
Historical and axiomatic foundations of geometry. Euclidean and non-Euclidean geometries. Prerequisite: MATH 280 or permission of the instructor.

343. Numerical Analysis (I; 3, 2)
Floating point arithmetic, development of computational algorithms and error estimates for root approximation, interpolation and approximation by polynomials, numerical differentiation and integration, cubic splines, least-squares, linear systems; lab component. Prerequisites: MATH 211 and CSCI 203, or permission of the instructor.

345. Linear Algebra (AI or II; 3, 0)
Systems of linear equations, determinants, vector spaces, canonical forms for linear transformations and matrices, bilinear forms, inner product spaces, applications to such other areas as geometry, differential equations, linear programming. Prerequisites: MATH 213 and either MATH 280 or permission of the instructor.

346. Modern Algebra (AI or II; 3, 0)
Advanced topics in algebra including group theory, field theory, Galois theory. Prerequisite: MATH 320.

350. Methods in Applied Mathematics (AI or II; 3, 0)
Techniques drawn from partial differential equations, transform methods, Fourier and complex analysis, and variational calculus. Prerequisite: MATH 212 or MATH 222 or permission of the instructor.

358. Topics in Operations Research (AI or II; 3, 0)
Mathematical and statistical techniques in operations research. Queueing theory. Additional topics may include simulation, forecasting, non-linear programming, inventory models. Methods and applications drawn from various fields. Prerequisite: MATH 303 or permission of the instructor.

362. Introduction to Complex Analysis (AI or II; 3, 0)
Limits, analytic functions, integrals including contour integrals. Cauchy’s Integral Theorem, entire functions and singularities. Prerequisites: MATH 211 and MATH 280, or permission of the instructor.

378. Seminar (I or II; R; 2, 0) Half course.
Informal seminars in various topics as the need arises. Topics may deal with algebra, analysis, topology, differential equations, statistics. Prerequisite: permission of the instructor.

391 and 392. Reading and Research (I or II; R; 2-8, 0) Half to two courses.
Reading and research in various topics for qualified undergraduates or graduate students. Prerequisite: permission of the instructor, adviser, and department chair.