# Mathematics and Computer Science

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*Chair: *Patrick W. Keef

Barry Balof

James Cotts

Janet Davis

Stacy Edmondson

Russell A. Gordon

David Guichard *(on Sabbatical, 2015-16)*Douglas Hundley

Yaping Jing

Albert W. Schueller

**Mathematics and Computer Science Department Website »**

## Mathematics

Mathematics courses provide an opportunity to study mathematics for its own sake and as a tool for use in the physical, social, and life sciences.

All or part of the calculus sequence is required or recommended by several majors at Whitman and calculus is the most common mathematics course taken by students. However, the department offers other courses (Mathematics 128) that are intended for students who wish to take mathematics but are not interested in or not prepared for calculus. Courses in programming, or with an emphasis on computing, are Computer Science167, 270, and Mathematics 235, 350, 467.

P-D-F policy: The department places no restrictions on the use of the P-D-F option for mathematics courses for majors or nonmajors, except that students choosing the mathematics major must take Mathematics 260 for a grade. The department strongly recommends that students majoring in mathematics or completing a joint major with mathematics not use the P-D-F option in mathematics courses.

The senior assessment in mathematics consists of a comprehensive examination in two parts: a four-hour written examination and a one-hour oral examination. The written examination covers three semesters of calculus and one of linear algebra — roughly the first two years of the program. The oral examination covers general and advanced topics.

**Distribution: **Courses completed in mathematics apply to the quantitative analysis distribution area.

**Learning Goals:** Upon graduation, a student will be able to:

**Major-Specific Areas of Knowledge**- Solve mathematical problems. Learn new mathematics independently. Evaluate mathematical arguments. Have depth of study in at least one area of mathematics. Have a basic understanding of several branches of mathematics.

**Communication**- Communicate mathematical ideas effectively both orally and in writing.

**Quantitative Skills**- See other goals.

**The Mathematics major:** A total of 35 credits, to include Mathematics 225, 235, 240, 260; any two of 455, 456, 475, 476; 497 or 498; 12 additional credits in mathematics courses numbered above 200. A grade of B- (2.7) or better in Mathematics 260 is required. Grades of B (3.0) or better in Mathematics 225 and 260 are strongly recommended for any student considering mathematics as a major, and both courses ought to be completed by the end of the sophomore year. Students planning graduate study should take Mathematics 456 and 486 and should acquire a reading knowledge of either French, German, or Russian.

A student who enters Whitman College without a good working knowledge of the material in Mathematics 125 and 126 will have to complete 41 mathematics credits to fulfill the requirements for the mathematics major (including six credits for Mathematics 125, 126).

Twenty-six mathematics credits are required for the mathematics-physics major, 29 mathematics credits for the economics-mathematics major, and 29 mathematics credits for the 3-2 mathematics-computer science major.

**Honors in the major: **Students do not apply for admission to candidacy for honors. To be granted honors, a senior Mathematics Major must attain the minimum Cumulative and Major GPAs specified in the faculty code (3.300 and 3.500, respectively), pass the Senior Comprehensive Examination with distinction, register for Mathematics 497, write a thesis graded A or A- by the Mathematics and Computer Science Department faculty, and receive departmental approval. The Chair of the Mathematics and Computer Science Department will notify the Registrar of those students attaining Honors in Major Study no later than the beginning of the third week of April. Two copies of the Honors Theses must be submitted to Penrose Library no later than Reading Day.

**The Mathematics minor:** Fifteen credits or more in mathematics courses numbered 200 or above.

**The Economics-Mathematics combined major:** Computer Science 167, Mathematics 225, 235, 240, 244, 247, 349, and three additional credits chosen from mathematics courses numbered above 200. Economics 100 or 101, 102, 307, 308, 327, 428, plus one additional course in economics. Students should note that in addition to Economics 307 and 308, the prerequisites for Economics 327 include Economics 227 (or Mathematics 128 or 247). However, neither Economics 227 nor Mathematics 128 applies toward the minimum major requirements. In addition, Economics 100 or 101, 102, and Mathematics 247 are the prerequisites for Economics 327. Economics 227 does not apply toward the minimum major requirements. Economics 493, 494, and other economics courses taken P-D-F may not be used to meet the 27-credit requirement. The senior assessment consists of the written exam in mathematics, the Major Field Test (MFT) in economics, and a combined oral exam scheduled by the economics department.

**The Mathematics-Physics combined major:** Mathematics 225, 235, 240, 244, and nine additional credits in mathematics courses numbered above 200; Physics 145 or 155 or 165, 156 or 166, 245, 246, 255, 256, 325, 339, and one additional physics course numbered from 300-480, or BBMB 324. Senior assessment consists of the written exam in mathematics, the written exam in physics, and a combined oral exam scheduled by the physics department.

**Majoring in Computer Science.** Please see the *Combined Plans* section of this catalog.

**Choosing a Calculus Course **Students who wish to take calculus should note the following: Students with a strong background in high school mathematics not including calculus start with Mathematics 125. Students who have taken a high school course in calculus, but who have not taken the BC calculus Advanced Placement Test (see the statement below regarding college credit for the Advanced Placement Test) should take the Advisory Calculus Placement exam offered by the department of mathematics.

Students should note that several programs require the calculus lab, Mathematics 235, in addition to Mathematics 225. Because the lab course teaches skills that are useful in other mathematics and science courses, it is strongly recommended that students take Mathematics 235 as early as possible in their programs. Programs that require the calculus labs are the mathematics major, the economics-mathematics major, the mathematics-physics major, the physics major, the 3-2 engineering program, and the 3-2 mathematics-computer science major.

**Advanced Placement **The policy for advanced standing and credit for the College Board Advanced Placement program is as follows:

- Students with a 4 or 5 on the BC calculus test are considered to have completed the equivalent of Mathematics 125 and 126 and receive six credits in mathematics.
- Students with a 4 or 5 on the AB calculus test (or on the AB subtest of the BC test) are considered to have completed the equivalent of Mathematics 125 and receive three credits in mathematics. These students should take the placement test offered by the department of mathematics to determine whether they should enroll in Mathematics 126 or Mathematics 225.
- Students with a 4 or 5 on the computer science (A) test are considered to have completed the equivalent of Computer Science 167 and receive three credits in computer science.
- Students with a 4 or 5 on the statistics test are considered to have completed the equivalent of Mathematics 128 and receive three credits in mathematics. Students should consider taking Mathematics 247 if they have also completed the equivalent of Mathematics 125.

A student has the option of repeating a course for which AP credit has been granted, but with a commensurate reduction in advanced placement credit.

#### 119 Programming with Robots

3; not offered 2015-16

An introduction to programming techniques applicable to most languages using personal robotics kits (Lego Mindstorm NXTs provided). The programming language used is most similar to the C programming language. Frequent programming projects are required in both independent and group settings. Traditional computer science topics like logic and algorithms, simple networking, event loops, and threading also will be explored.

#### 125 Calculus I

3, x Cotts, Keef

A brief review of some precalculus topics followed by limits, continuity, a discussion of derivatives, and applications of the derivative. *Prerequisites**:* two years of high school algebra; one year of plane geometry; and knowledge of trigonometry and exponential/logarithmic functions or consent of instructor.

#### 126 Calculus II

3, 3 Fall: Balof, Hundley; Spring: Cotts, A. Gordon

A continuation of Mathematics 125, covering integration, techniques for computing antiderivatives, the fundamental theorem of calculus, applications of the definite integral, and infinite series.

#### 128 Elementary Statistics

x, 3 Edmondson

Probability and statistics including methods for exploring data and relationships in data, methods for producing data, an introduction to probability and distributions, confidence intervals, and hypothesis testing. *Prerequisite**:* two years of high school mathematics.

#### 203, 204 Special Topics in Introductory Level Mathematics

1-3

On occasion, the mathematics department will offer courses on introductory topics in mathematics that are not generally covered in other introductory courses. Possible topics include Introduction to Number Theory, Chaos and Applied Discrete Probability. Any current offerings follow.

#### 225 Calculus III

4, 4 Fall: Schueller; Spring: Keef

Topics include partial derivatives, gradients, extreme value theory for functions of more than one variable, multiple integration, line integrals, and various topics in vector analysis.

#### 235 Calculus Laboratory

x, 1 Balof

A laboratory to investigate ways in which the computer can help in understanding the calculus and in dealing with problems whose solutions involve calculus. No programming required; a variety of existing programs will be used. *Pre-* or *corequisite:* Mathematics 225.

#### 240 Linear Algebra

3, 3 Fall: Keef; Spring: Hundley

This course first considers the solution set of a system of linear equations. The ideas generated from systems of equations are then generalized and studied in a more abstract setting, which considers topics such as matrices, determinants, vector spaces, inner products, linear transformations, and eigenvalues. *Prerequisite:* Mathematics 225.

#### 244 Differential Equations

3, 3 Fall: R. Gordon; Spring: Hundley

This course includes first and second order linear differential equations and applications. Other topics may include systems of differential equations and series solutions of differential equations. *Prerequisite:* Mathematics 225.

#### 247 Statistics with Applications

3, 3 Edmondson

An introduction to statistics for students who have taken at least one course in calculus. Focuses on learning statistical concepts and inference through investigations. Topics include, but are not limited to, exploratory graphics, sampling methods, randomization, hypothesis tests, confidence intervals, and probability distributions. A statistical software package will be used. *Prerequisite: *Mathematics 125 or equivalent.

#### 248 Statistical Modeling

3; not offered 2015-16

This course follows introductory statistics by investigating more complex statistical models and their application to real data. The topics may include simple linear regression, multiple regression, non-parametric methods, and logistic regression. A statistical software package will be used. *Prerequisite: **Mathematics 128, Mathematics 247, Biology 228, Economics 227, or Environmental Studies 207.*

#### 260 An Introduction to Higher Mathematics

x, 3 A. Gordon

An introduction to some of the concepts and methodology of advanced mathematics. Emphasis is on the notions of rigor and proof. This course is intended for students interested in majoring in mathematics; students should plan to complete it not later than the spring semester of the sophomore year. *Prerequisite**:* Mathematics 225.

#### 281, 282 Independent Study

1-3, 1-3 Staff

A reading project in an area of mathematics not covered in regular courses or that is a proper subset of an existing course. The topic, selected by the student in consultation with the staff, is deemed to be introductory in nature with a level of difficulty comparable to other mathematics courses at the 200-level. May be repeated for a maximum of six credits. *Prerequisite:* consent of supervising instructor.

#### 287 Independent Study in Geometry

x, 3 A. Gordon

This independent study in geometry will include a review of high school geometry, a few topics in advanced Euclidean geometry, a reading of Books I and II of Euclid's Elements, and an introduction to hyperbolic geometry. The grading for the course will be based on a journal (20%), a two-hour written midterm exam (40%), and a one-hour oral final exam (40%). Since the student will be working independently on the material, a disciplined work ethic is required. *Prerequisite: **Mathematics 225.*

#### 299 Problem-Solving in Mathematics

1, x Balof

Students will meet weekly to discuss problem-solving techniques. Each week a different type of problem will be discussed. Topics covered will include polynomials, combinatorics, geometry, probability, proofs involving induction, parity arguments, and divisibility arguments. The main focus of the course will be to prepare students for the William Lowell Putnam Mathematics Competition, a national examination held the first Saturday in December. Students who place in the top 500 on this exam nationwide have their names listed for consideration to mathematics graduate programs. Graded credit/no credit. May be repeated for a maximum of four credits. *Prerequisite**:* consent of instructor.

#### 337 Geometry

3; not offered 2015-16

Essential for prospective high school mathematics teachers, this course includes a study of Euclidean geometry, a discussion of the flaws in Euclidean geometry as seen from the point of view of modern axiomatics, a consideration of the parallel postulate and attempts to prove it, and a discussion of the discovery of non-Euclidean geometry and its philosophical implications. *Prerequisite**:* Mathematics 126.

#### 339 Operations Research

3, x Hundley

Operations research is a scientific approach to determining how best to operate a system, usually under conditions requiring the allocation of scarce resources. This course will consider deterministic models, including those in linear programming (optimization) and related subfields of operations research. *Prerequisite:* Mathematics 240; Computer Science167 or Mathematics 235.

#### 349 Probability Theory

x, 3 Edmondson

A formal introduction to probability and randomness. The topics of the course include but are not limited to conditional probability, Bayes’ Theorem, random variables, the Central Limit Theorem, expectation and variance. Both discrete and continuous probability distribution functions and cumulative distribution functions are studied. *Prerequisite**:* Mathematics 225.

#### 350 Mathematical Modeling and Numerical Methods

3; not offered 2015-16

This course explores the process of building, analyzing and interpreting mathematical descriptions of physical processes. This may include theoretical models using statistics and differential equations, simulation modeling, and empirical modeling (meaning model building from data). The course will involve some computer programming, so previous programming experience is helpful. *Prerequisite:* Mathematics 240 and 244.

#### 358 Combinatorics and Graph Theory

3, x Balof

Topics in elementary combinatorics, including: permutations, combinations, generating functions, the inclusion-exclusion principle, and other counting techniques; graph theory; and recurrence relations. *Prerequisites**:* Mathematics 260 or consent of instructor.

#### 367 Engineering Mathematics

3; not offered 2015-16

An introduction to mathematics commonly used in engineering and physics applications. Topics may include: vector analysis and applications; matrices, eigenvalues, and eigenfunctions; boundary value problems and spectral representations; Fourier series and Fourier integrals; solution of partial differential equations of mathematical physics; differentiation and integration of complex functions, residue calculus, conformal mapping. *Prerequisite:* Mathematics 244.

#### 368 Complex Variables

x, 3 Schueller

Complex analysis is the study of functions defined on the set of complex numbers. This introductory course covers limits and continuity, analytic functions, the Cauchy-Riemann equations, Taylor and Laurent series, contour integration and integration theorems, and residue theory. *Prerequisite**:* Mathematics 225.

#### 438 Statistical Theory

4; not offered 2015-16

This course studies the mathematical theory of statistics with a focus on the theory of estimation and hypothesis tests. Topics may include properties of estimators, maximum likelihood estimation, convergence in probability, the central limit theorem, order statistics, moment generating functions, and likelihood ratio tests. A statistical software package will be used. *Prerequisites:* Mathematics 349 and one of Mathematics 128, Mathematics 247, Biology 228, Economics 227, or Environmental Studies 207.

#### 455, 456 Real Analysis

4, 4 Schueller

First semester: a rigorous study of the basic concepts of real analysis, with emphasis on real-valued functions defined on intervals of real numbers. Topics include sequences, continuity, differentiation, integration, infinite series, and series of functions. Second semester: content varies from instructor to instructor but includes topics from metric spaces, the calculus of vector-valued functions, and more advanced integration theory. *Prerequisite**:* Mathematics 260.

#### 467 Numerical Analysis

3; not offered 2015-16

An introduction to numerical approximation of algebraic and analytic processes. Topics include numerical methods of solution of equations, systems of equations and differential equations, and error analysis of approximations. *Prerequisite**:* Computer Science 167. *Pre-* or *corequisite**:* Mathematics 240.

#### 471, 472 Special Topics

1-3

On occasion, the mathematics department will offer courses on advanced topics in mathematics that are not found in other course offerings. Possible topics include topology, number theory, and problem-solving. Any current offerings follow.

#### 475, 476 Abstract Algebra

x, 4 Balof

The first semester is an introduction to groups and rings, including subgroups and quotient groups, homomorphisms and isomorphisms, subrings and ideals. Topics for the second semester may include fields, simple groups, Sylow theorems, Galois theory, and modules. *Prerequisite**:* Mathematics 260.

#### 481, 482 Independent Study

1-3, 1-3 Staff

A reading or research project in an area of mathematics not covered in regular courses. The topic is to be selected by the student in consultation with the staff. Maximum of six credits. *Prerequisite*: consent of supervising instructor.

#### 497 Senior Project

x, 4 Balof

Preparation of the senior project required of all graduating mathematics majors. Each student will be matched with a faculty member from the mathematics department who will help supervise the project. Course objectives include developing students’ abilities to independently read, develop, organize, and communicate mathematical ideas, both orally and in writing. A final written and oral report on the project is completed.

#### 498 Honors Thesis

4, 4 Staff

Preparation of an honors thesis. Required of and limited to senior honors candidates in mathematics. Students will be a part of the Mathematics 497 *Senior Project* class (described above), but their work will be held to a higher standard. P*rerequisite: *admission to honors candidacy.

## Computer Science

Students of computer science will gain insight into a technology on which we increasingly rely, while learning new ways of thinking and tools to solve problems in many domains. Central to computer science is the concept of an algorithm---a precise, repeatable procedure for solving a well-defined problem. Computer scientists discover, define and characterize computational problems; they design, implement, and evaluate algorithmic solutions.

Whitman currently offers a minor in computer science, supported by two introductory courses and a few special topics courses that span the discipline. We aim to offer a significantly expanded curriculum beginning in the 2016-17 academic year.

**Learning Goals:** Upon graduation, a student will be able to:

**Minor-Specific Areas of Knowledge**- Identify and describe computational problems; design, implement, and evaluate algorithmic solutions. Apply core algorithmic concepts and data structures. Consider multiple approaches to solving problems. Independently learn new programming languages and tools. Discuss applications of computing in society as well as relationships to other disciplines.

**Communication**- Communicate computational ideas in speech, writing, diagrams, and at least one programming language. Work with others to understand and solve computational problems.

**Quantitative Skills**- See other goals.

**Majoring in Computer Science.** Please see the *Combined Plans* section of this catalog.

**The Computer Science minor:** A minimum of 15 credits in courses numbered 200 or above.

#### 167 Introduction to Computer Programming

3, 3 Fall: Jing; Spring: J. Davis

An introduction to programming techniques applicable to most high-level programming languages. Covers core programming topics including logic, loops, functions, and objects. Uses an object-oriented programming language like C++ or Java. Frequent programming projects are required.

#### 200-204 Special Topics in Introductory Computer Science

1-4

On occasion, the department will offer courses on lower-level topics in computer science that are not generally covered in other courses. For example, topics may include: Interaction Design, Databases, Information Security, Mobile Application Development, Event-driven Programming, Computers and Society. *Prerequisite: *Computer Science 167. Distribution area: quantitative analysis. Any current offerings follow.

#### 200 ST: Human-Computer Interaction

3, x J. Davis

Introduction to fundamental principles and methods of human-centered interaction design: Human capabilities and limitations, usability and accessibility guidelines, iterative design, contextual inquiry, task analysis, ideation, prototyping, evaluation. Includes hands-on project and/or laboratory work. *Prerequisite**:* Computer Science 167. Distribution: quantitative analysis.

#### 200 ST: Elements of Computer Systems

x, 3 J. Davis

This course integrates key ideas from *digital logic, computer architecture*, *compilers, *and *operating systems* in one unified framework. This will be done constructively, by building a general-purpose computer system from ground up: from the low level details of switching circuits to the high level abstractions of modern programming languages. In the process, we will explore software engineering and algorithmic techniques used in the design of modern hardware and software systems. We will discuss fundamental trade-offs and future trends. *Prerequisite**:* Computer Science 167. Distribution: quantitative analysis.

#### 270 Data Structures

3, 3 Jing

We study fundamental methods used to store, access, and manipulate data in computers. Storage structures to be covered include files, lists, tables, graphs, and trees. We will discuss and analyze methods of searching for and sorting data in these structures. *Prerequisite**:* Computer Science 167.

#### 400-404 Special Topics in Computer Science

1-4

On occasion, the department will offer courses on mid- to upper-level topics in computer science that are not generally covered in other courses. For example, topics may include: Computer Systems, Computer Graphics, Artificial Intelligence, Software Development, Programming Languages, Analysis of Algorithms, Automata Theory. *P**rerequisite: *Computer Science 167 and 270. Distribution area: quantitative analysis. Any current offerings follow.

#### 400 ST: Algorithm Design and Analysis

x, 3 Jing

We present basic techniques for the design and analysis of efficient algorithms. We will cover sorting, searching, graph algorithms, and string processing. Design techniques such as dynamic programming and the greedy method will be included, as well as asymptotic, worst-case, average-case and amortized analyses. Further data structures including heaps, hash tables, binary search trees and red-black trees will be developed. *P**rerequisite: *Computer Science 167 and 270. Distribution area: quantitative analysis. Any current offerings follow.

#### 481, 482 Independent Study

x, 1-4 Staff

Directed study or research in selected areas of computer science. A curriculum or project is designed by the student(s) with the advice and consent of an instructor in the department. Inquiry may emerge from prior course work or explore areas not covered in the curriculum.