#### 112.

### Introduction to Mathematical Modeling (II; 3, 0)

Introduction for the non-specialist to mathematical modeling of real-world phenomena such as voting and networks, using graph theory, probability, and other accessible tools.

*Professors: *Gregory T. Adams, M. Lynn Breyfogle, Thomas Cassidy, Ulrich Daepp (Chair), George R. Exner, Michael R. Frey, Pamela B. Gorkin, Paul J. McGuire

*Associate Professors: *Carmen O. Acuña, Peter A. Brooksbank, Emily Dryden, Sharon A. Garthwaite, James E. Hutton, Peter McNamara, Adam Piggott, Nathan C. Ryan, Linda B. Smolka, Karl A. Voss

*Assistant Professors: *Kelly Bickel, Jodi Black, KB Boomer, Van T. Cyr, Gabrielle Flynt, Jeffrey Langford, Alex J. Rice (visiting), Bonnie B. Smith (visiting)

*Instructor:* Amy M. Donner (visiting)

Mathematics has long been the basic language of the natural sciences, and has been studied for its own sake since ancient times. An understanding of the basics of calculus, statistics, and linear algebra has become a requirement for proficiency in many of the social sciences. The study of mathematics has its own rewards because accomplishment in the subject, even at a relatively elementary level, requires and promotes clarity of both thought and expression. For many, the study of mathematics offers entrance into an exciting world of challenges where beauty and utility coexist in balanced harmony.

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).

The Mathematics Department offers three majors. Students may earn a Bachelor of Arts in Mathematics, a Bachelor of Science in Mathematics, or a Bachelor of Science in Applied Mathematical Sciences. Students in each major complete an introductory year of calculus either by taking MATH 201 or MATH 202 during their first year, or by achieving a high score on the Advanced Placement Test of the College Entrance Examination Board.

The College Core Curriculum disciplinary depth requirements for Bachelor of Science and Bachelor of Arts majors are satisfied as follows: *writing within the major *in MATH 280, MATH 308, and MATH 320 (all W2 courses); *formal presentation *in MATH 280; and *information literacy *in MATH 308 and MATH 320. The requirement for a *Culminating Experience *within the major may be satisfied in any of the following ways: (i) taking a full credit 400-level mathematics course; or (ii) completing an honors thesis, senior thesis, or other research project in the senior year that involves mathematics or statistics; or (iii) completing student teaching for secondary certification. With one exception, the Culminating Experience cannot double-count as one of the mathematics electives required in the major. The one exception is that students earning a Bachelor of Arts in Mathematics may count a 400-level mathematics course both as their mathematics-related course and as the Culminating Experience.

The choice of degree program depends largely upon the student's mathematical objectives and interests in fields other than mathematics. Students with strong interests outside mathematics have options including the Bachelor of Science in Applied Mathematical Sciences program, the Bachelor of Science in Interdisciplinary Studies in Economics and Mathematics program, and the Bachelor of Arts in Mathematics program. 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 any Bachelor of Arts degree, such students may be best advised to choose one of the Bachelor of Science majors.

The **Bachelor of Arts ****in Mathematics** major consists of eight mathematics courses beyond the introductory year of calculus, plus one additional course in a related field and a Culminating Experience.

Of the eight mathematics courses beyond the introductory year of calculus, five are specified: MATH 211, MATH 245, MATH 280, MATH 308, and MATH 320. The three remaining mathematics courses must be mathematics electives at the 300- or 400-level not including MATH 417.

The course in a related field may be:

- a fourth mathematics course at the 300- or 400-level (including Math 417)
- MATH 207, MATH 212, or MATH 216
- an additional full-credit computer science or science course (beyond those required of all liberal arts students) in which college-level mathematics or statistics plays a major role. Included among the courses in this category are nearly all courses in computer science or physics at or above the 200 level and an appropriate course from the humanities, social sciences, or engineering in which mathematics plays a significant role at a reasonable level of sophistication.

The mathematics department chair shall make the determination of whether or not a particular course outside the mathematics department may count as the course in a related field.

A single 400-level course may be used to satisfy both the Culminating Experience requirement and the related course requirement.

Students with a special interest in pure mathematics or statistics can earn formal concentration in these areas by completing an appropriate suite of 300- and 400-level courses, as described below.

The pure mathematics concentration consists of MATH 345, MATH 409, MATH 446, and two of the following: MATH 311, MATH 333, MATH 362. The statistics concentration consists of MATH 303, MATH 305, MATH 307, MATH 404, and either MATH 345 or MATH 409. Students majoring in mathematics with a special interest in computer science are encouraged to consider minoring in computer science.

The **Bachelor of Science in Mathematics** major requires 10 mathematics courses beyond the introductory year of calculus and a Culminating Experience. Six of the 10 mathematics courses are specified: MATH 211, MATH 212, MATH 245, MATH 280, MATH 308, and MATH 320. The remaining four courses are mathematics electives at the 300- or 400-level not including MATH 417. The major also requires 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 natural sciences or from computer science. Any course in physics beyond PHYS 212, and any laboratory course in computer science at the level of CSCI 203 or beyond, may be chosen.

Students with a special interest in pure mathematics or statistics can earn formal concentration in these areas by completing an appropriate suite of 300- and 400-level courses, as described below. 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 345, MATH 409, MATH 446, and two of the following: MATH 311, MATH 333, MATH 362. The statistics concentration consists of MATH 303, MATH 305, MATH 307, MATH 404, and either MATH 345 or MATH 409. Students majoring in mathematics with a special interest in computer science are encouraged to consider minoring in computer science.

The recommended sequence for the Bachelor of Science in Mathematics major is as follows:

**First Year**First Semester: MATH 201; PHYS 211

Second Semester: MATH 202; PHYS 212

**Sophomore Year**First Semester: MATH 211; MATH 245, Laboratory science

Second Semester: MATH 212; MATH 280; Laboratory science

**Junior Year**First Semester: MATH 308 or 320; Elective in mathematics

Second Semester: MATH 308 or 320; Elective in mathematics

**Senior Year**First Semester: Elective in mathematics

Second Semester: Elective in mathematics; Culminating Experience

The **Bachelor of Science in Applied Mathematical Sciences** major (with a concentration in statistics or applied mathematics) requires 10 mathematics courses beyond the introductory year of calculus, a computing course, five courses in a different program, and a Culminating Experience. More specifically, there are six required core mathematics courses consisting of MATH 211, MATH 216, MATH 245, MATH 280, MATH 303, and MATH 308 together with four concentration related courses in statistics or applied mathematics. Further, the major requires a computing course and significant coursework in a declared outside department or program as described below. While the Culminating Experience may be met in any of the ways specified above, students earning a Bachelor of Science in Applied Mathematical Sciences are strongly encouraged to consider the option of a thesis or research experience integrating the outside coursework.

The **statistics concentration** consists of MATH 217 and three elective courses chosen from MATH 305, MATH 307, MATH 345, MATH 404, or MATH 409. At least two courses must be selected from MATH 305, MATH 307, or MATH 404. Appropriate courses could be additional electives counting towards the concentration if so determined by the academic adviser in consultation with the mathematics department chair.

The **applied mathematics concentration** consists of MATH 212 and three elective courses chosen from MATH 343, MATH 345, MATH 350, MATH 358, MATH 362, or MATH 409. Appropriate courses could be additional electives counting towards the concentration if so determined by the academic adviser in consultation with the mathematics department chair.

The **computing course** can be a computer science course at or above the 200-level or a computing course appropriate to the program of study as determined through consultation with the academic adviser and the mathematics department chair.

**Outside Coursework: **For the purpose of completing a coherent sequence of courses that provide a solid introduction to the discipline all students must partner with a department or program in a discipline that applies statistics or mathematics. In this regard, a minimum of five courses chosen in consultation with the mathematics department adviser and the outside department or program is required. A partner department/program will usually be chosen from the College of Engineering, the School of Management, the Division of Social Sciences, or the Division of Natural Sciences. Entering students can declare the intended major in the summer after acceptance to Bucknell, but must consult with the Mathematics Department and formally declare the outside coursework by the end of their third semester. All other students must consult with the Mathematics Department at the point of declaring the major and specify the outside coursework. In either case the Mathematics Department will consult with the partner department or program to ensure that the coursework is appropriate and can be completed.

A sample sequence for the Bachelor of Science in Applied Mathematical Sciences major is provided below. It should be noted that each student's sequence will be unique, depending on when the program is started, how many AP or transfer credits are applied, and when the desired courses are offered.

**First Year**First Semester: MATH 201

Second Semester: MATH 202

**Sophomore Year**First Semester: MATH 211; MATH 216; Computing course

Second Semester: MATH 245; MATH 303; Outside course

**Junior Year**First Semester: MATH 280; MATH 216/217; Outside course

Second Semester: MATH 308; Concentration elective; Outside course

**Senior Year**First Semester: Concentration elective; Outside course

Second Semester: Concentration elective; Outside course; Culminating Experience

Prospective secondary school teachers (grades 7 - 12) must complete one of the three majors within the department. 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. Certain education courses are also required, as is a course in English literature that is additional to the basic W1 course.

Students who complete departmental honors normally have a grade point average of 3.5 both in their mathematics courses and overall. Students in the Bachelor of Arts in Mathematics major or Bachelor of Science in Mathematics major have usually completed MATH 308 and MATH 320 by the end of their junior year. Students in the Bachelor of Science in Applied Mathematical Sciences major have usually completed MATH 308 by the end of their junior year. To be accepted into the Honors Program, a student must submit an honors proposal for approval by the adviser, the department chairperson or program head, and the Honors Council. The student must then complete an honors thesis under the adviser’s direction, must have completed a total of at least three mathematics courses at the 300- or 400-level by the end of their junior year, and satisfy all other requirements as put forth by the University Honors Council. Such students usually complete at least two half-credit semesters of independent study in mathematics (MATH 491).

A **minor **in mathematics consists of either: four credits from mathematics courses numbered 211 or above, at least one of them at the 300- or 400- level; or of three credits from courses in mathematics numbered 211 or above, at least two of them at the 300- or 400-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, MATH 303, MATH 304, MATH 305, and MATH 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, MATH 222, MATH 226, MATH 343, MATH 350, and MATH 358.

Introduction for the non-specialist to mathematical modeling of real-world phenomena such as voting and networks, using graph theory, probability, and other accessible tools.

An investigation of number, numeration, and operations from the perspective of elementary school teachers and pupils. Open only to B.S. in Education Early Childhood Pre-K to 4 students. Required fieldwork.

Investigation of geometric, probablistic, and statistical concepts related to PreK to 4 mathematics and how children learn and make sense of these concepts. Prerequisite: MATH 117 or permission of the instructor.

Elementary calculus and applications taken primarily from economics. Topics include algebraic, exponential, and logarithmic functions, graphs, limits, derivatives, and integration. Not open to students who have taken MATH 201.

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.

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.

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.

Mathematical problem solving, with an emphasis on problems and topics that appear in contests such as the Putnam Competition. Prerequisite: permission of the instructor.

Calculus of vector-valued functions and functions of several variables. Multiple, line, and surface integrals; applications and extrema. Green's, Stokes', Divergence Theorems. Prerequisite: MATH 202.

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.

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.

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.

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.

Descriptive modeling and statistics, sampling and experimental design, discrete and continuous random variables, central limit theorem, and elementary inference. Prerequisite: MATH 202. Open only to engineering students and students in computer science. Not open to students who have taken MATH 216.

Counting techniques and traversal problems. Does not count toward the major. Students join MATH 241 mid-semester. Corequisite: MATH 280. Only for students seeking secondary certification.

Logic, sets; mathematical induction; relations, functions; combinatorics and graph theory. Does not count toward the mathematics major. Prerequisite: MATH 202.

Linear equations, matrices, vector spaces, linear transformations, determinants, eigenvalues. Prerequisite: MATH 202. Not open to students who have taken MATH 213.

Logic, sets; proof techniques; relations, functions, sequences and convergence; cardinality. Skills and tools for independent reading, problem solving, and exploration. Prerequisite: MATH 211 or MATH 245.

Readings and research in special topics at an intermediate level. Prerequisites: permission of the instructor, adviser, and department chair.

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.

Regression and analysis of (co)variance. Model diagnosis and remediation. Model selection, multicollinearity, logistic regression. R or SAS will be used. Prerequisites: MATH 216 or equivalent, and either MATH 245 or MATH 303 or permission of the instructor.

Experiments, observational studies. Completely randomized, block, mixed models, crossed, nested design. Simple random, stratified, cluster sampling. Estimation procedures, sample size calculations. Uses R or SAS. Prerequisite: MATH 217 or MATH 303.

Real numbers and elementary topology of Cartesian spaces, convergence, continuity, differentiation, and history of the development of analysis. Prerequisites: MATH 211, MATH 245 and MATH 280.

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 245 and MATH 280, or permission of the instructor.

Special topics, to be selected from algebra, analysis, geometry, statistics, applied mathematics, etc.

Groups and rings; homomorphisms and isomorphism theorems; history of the development of algebra. Additional selected topics. Prerequisites: MATH 245 and MATH 280.

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.

Historical and axiomatic foundations of geometry. Euclidean and non-Euclidean geometries. Prerequisite: MATH 280 or permission of the instructor.

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.

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 245 and either MATH 280 or permission of the instructor.

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.

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.

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.

Seminar based on topics from algebra, analysis, topology, differential equations, statistics, or applied mathematics; topics selected according to demand or interest. Prerequisite: permission of the instructor.

Reading and research in various topics for qualified undergraduate or graduate students. Prerequisites: permission of the instructor, adviser, and department chair.

Point and interval estimation, hypothesis testing, Fisher's likelihood theory, frequentist versus Bayesian approach, computational statistics. Prerequisites: MATH 216 or equivalent and MATH 303 or permission of the instructor.

Continuation of MATH 308. Integration theory and advanced topics in analysis. Prerequisite: MATH 308.

Topics in statistics and mathematics. This course is designed for seniors to satisfy the Culminating Experience requirement. Prerequisites: MATH 280 and a 300-level mathematics course.

Advanced topics in group theory including solvable groups, field theory and Galois theory. Prerequisite: MATH 320.

Reading and research in various topics for qualified undergraduates or graduate students at a level appropriate for a Culminating Experience. Prerequisites: permission of the instructor, adviser, and department chair.

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