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Mathematics
Associate Professor Buckmire, Chair
Professors Lengyel, McDonald, Tinberg; Associate Professor Knoerr, Naimi; Assistant Professors Gallegos, Sundberg
On Special Appointment: Adjunct Assistant Professors Hoffman, Lawrence, Lee, Tollisen
Mathematics, encompassing several of the original liberal arts, is valued for its exquisite intellectual beauty and its timeless exploration of all things spatial, quantitative and patterned through the lens of rigorous abstraction. As a vibrant modern science, it possesses an unparalleled analytical power for describing, detailing and deriving insight into numerous physical, biological, technological, economic and societal aspects of the world we all live in. The Mathematics Department is committed to engaging a diverse range of students in the active study and creative application of the principles, ideas, and methods that characterize mathematics and the mathematical sciences, and offering preparation toward a wide variety of careers and educational pursuits.
Upon graduation, some mathematics majors go on to graduate or professional school while others begin careers in teaching, business, industry, or government. The major can be structured to provide a solid foundation in the mathematical sciences—pure and applied mathematics, statistics, and operations research—and fields close to mathematics like computer science, actuarial science, and engineering. A major or minor in mathematics can also provide an excellent technical and theoretical complement to a major or minor in other fields.
Please consult the Mathematics Department’s home page, for more detailed and regularly updated information on the program.
MAJOR: The minimum requirements for the major outlined below permit students great flexibility in designing a course of study to meet their own intellectual and career goals.
Fundamental courses: Calculus 1, 2, Mathematics 210, Mathematics 212, and Mathematics 214.
Advanced courses: 24 units of Mathematics or Computer Science courses numbered 310 or above (excluding Mathematics 400). The grade point average in these courses must exceed 2.0.
Colloquium requirement: Mathematics 300 and 400.
Breadth requirement: Computer Science 211 or Mathematics 150 or Mathematics 160 coupled with a 2-unit CS course.
The Mathematics Department has prepared guidelines for majors considering future study or careers in pure and applied mathematics, education, actuarial science, and computer science. These guidelines are available on our website.
WRITING REQUIREMENT: Students majoring in Mathematics should familiarize themselves with this requirement at the time of declaring the major. The Third Year Writing Requirement (see the Writing Program) is addressed in Mathematics 300, a writing-intensive course. Upon completion of this course, students will be deemed to have satisfied the requirement without further work, but are encouraged to enroll in English Writing 401, or will be required to successfully complete English Writing 401 with a grade of C- or higher to satisfy the requirement. Students not taking Mathematics 300 (Honors students or those petitioning an exemption of this requirement) must submit an acceptable portfolio of three revised papers to the instructor of Mathematics 300 during the Spring of the Junior year. If this portfolio is not acceptable, they will be required to successfully complete English Writing 401 with a grade of C- or higher to satisfy the requirement.
COMPREHENSIVE EXAMINATION: This examination has two parts. The first part measures competence in the fundamental courses and is handled during Mathematics 300: Junior Colloquium. The second part consists of an independent project culminating in a written report and public presentation during the senior year, and is handled through Mathematics 400: Senior Colloquium. Further information is available from the department.
MINOR: Calculus 1, 2, and at least 12 units from Mathematics 150, 210, 212, and 214. In addition, at least one 300-level 4-unit course is required. Students must take at least 20 units in Mathematics at Occidental or through college transfer (not AP) credit to earn the minor in Mathematics. The grade point average for all Mathematics courses taken must be at least 2.0.
HONORS: Students who wish to be considered for honors in mathematics should complete at least the five fundamental courses in their first two years with a grade point average greater than 3.0. Honors students must complete three approved upper division courses beyond those required for the major. These courses should be chosen to prepare the student for the senior honors project. Honors students enroll in Mathematics 499 to prepare this project, which may be substituted for Mathematics 300 in satisfying the major requirements. Consult the Mathematics Department and see the Honors Program for additional details.
GRADUATE STUDY: A Master of Arts in Teaching is available in mathematics. Consult the Graduate Study section of this catalog, the Education Department, and the Graduate Office for overall requirements. The minimum mathematics requirement is 15 units of coursework approved by the Mathematics Department.
CALCULUS PLACEMENT: Placement in calculus courses (Mathematics 108, 110, 114, or 118) is determined in part by the Calculus Placement Exam, administered online prior to the beginning of Fall Semester. Students achieving a score of 3, 4, or 5 on the College Board Advanced Placement Examination in Calculus (AB or BC) are exempt from the Calculus Placement Examination.
Students will be placed into Mathematics 108, 110, 114, or 118 based on previous mathematical experience, advising, and the results of the Calculus Placement Exam. Students with qualifying scores on the Advanced Placement Examination in Calculus are most often placed in calculus courses as follows:
Students who received scores of 1 or 2 on either the AB or BC exam should take the Placement Test (and will be advised on whether to take Mathematics 108 or 110 or 114).
Students who received a score of 3 on the AB exam should take Mathematics 110 or 114. Students who received a score of 4 or 5 on the AB exam are strongly recommended to enroll in Mathematics 128.
Students who received a score of 3 on the BC exam should take Mathematics 110 or 114 or possibly 128 depending on their AB sub-score (see previous paragraph).
Students who received a score of 4 or 5 on the BC exam should take a 200-level Mathematics course.
In addition to the calculus courses, Mathematics 105, 146, 150, 160, 210, 212, 214, and Computer Science 211 may be taken by first-year students meeting the prerequisites.
Students with transfer credits should confer with the Department for advice on placement in an appropriate mathematics course.
STATISTICS PLACEMENT: Students receiving a 4 or 5 on the APStatistics Exam are exempt from Math 146; Math 150 is recommended for these students wanting to take further statistics courses.
COMPUTER SCIENCE COURSES AND PLACEMENT: The Mathematics Department offers a modest program in computer science and computer programming. See the Computer Science department, for a listing of these courses and for further information on placement based on College Board Advanced Placement Examinations in Computer Science.
MATHEMATICS COURSES: Calculus is a prerequisite for all mathematics courses with the exceptions of Mathematics 105 and 146, as well as most Computer Science courses. All students planning to take Calculus must take the online Calculus Placement Exam prior to the beginning of the Fall Semester unless they are exempt due to having received an Advanced Placement exam score. (See Calculus Placement above or contact the Mathematics Department for further details.) Prerequisites for any course may be waived with permission of the instructor.
CORE SCIENCE/MATHEMATICS REQUIREMENTS: Every four unit course in the department of Mathematics, with the exception of Mathematics 108, may be used toward the Core Science/Mathematics requirement. Mathematics 160 may be combined with one 2 unit programming course (in Computer Science) to meet one of these course requirements. None of the courses in Mathematics (or Computer Science) meets the laboratory science part of the requirement.
104. WOMEN IN MATHEMATICS.
This course is designed to introduce a variety of mathematical topics stemming from the research of women mathematicians both past and present, from Hypatia to current professors. In discussing the work of these women, we will also discuss the gender issues that are associated with being a female mathematician. Course material will be covered in lecture, research, in-class visitors and activities. Course work will include research papers, a course project and problem sets related to the mathematician of discussion.
Gallegos
105. MATHEMATICS AS A LIBERAL ART.
Introduction to mathematical thinking. Investigation of mathematical patterns in counting, reasoning, motion and change, shape, symmetry, and position. Not open to seniors.
Lawrence
CALCULUS: Calculus differs in some respects from the traditional courses offered at some secondary schools and most other colleges or universities. Occidental’s program is based on scientific modeling, makes regular use of computers, and requires interpretation as well as computation. A variety of courses comprise this program, accommodating different levels of preparation. The core content is described below as Calculus 1 and 2. Actual courses suited to different levels of preparation are listed under each description.
CALCULUS 1: SCIENTIFIC MODELING AND DIFFERENTIAL CALCULUS.
Many mathematical models in the natural and social sciences take the form of systems of differential equations. This introduction to the calculus is organized around the construction and analysis of these models, focusing on the mathematical questions they raise. Models are drawn from biology, economics, and physics. The important elementary functions of analysis arise as solutions of these models in special cases.
The mathematical theme of the course is local linearity. Topics include the definition of the derivative, rules for computing derivatives, Euler’s Method, Newton’s Method, Taylor polynomials, error analysis, optimization, and an introduction to the differential calculus of functions of two variables.
108. UNIFIED PRECALCULUS AND CALCULUS 1-A.
The first of a two course sequence enriching the material in Calculus 1 with additional study of elementary functions, algebra, trigonometry, graphing, and mathematical expression. Weekly lab. Prerequisites: the Calculus Readiness Examination and less than four years of high school mathematics. Does not satisfy the core math/science requirement.
Tollisen
109. UNIFIED PRECALCULUS AND CALCULUS 1-B.
Continuation of Mathematics 108. This course satisfies Calculus 1 prerequisites for subsequent courses. Weekly lab. Prerequisite: Mathematics 108.
Tinberg
110. CALCULUS 1.
This course satisfies Calculus 1 prerequisites for subsequent courses. Weekly lab. Prerequisites: the Calculus Readiness Examination and at least four years of high school mathematics.
Buckmire
114. CALCULUS 1 (EXPERIENCED).
This course satisfies Calculus 1 prerequisites for subsequent courses. Weekly lab. Prerequisites: a year of prior calculus experience and either the Calculus Readiness Examination or an appropriate Advanced Placement Calculus score.
Sundberg
128. CALCULUS 1 AND 2 (ADVANCED PLACEMENT).
A one-semester course covering applications of differential and integral calculus in biology, economics, physics and other areas through the study of models. These models take the form of differential equations or systems of differential equations. Central to the exploration of models are methods of approximation such as Euler’s Method, Taylor series, and Fourier series. The course assumes the basic skills from the successful completion of Advanced Placement (AB) calculus and uses this background to develop techniques for analyzing mathematical models algebraically, graphically and numerically. This course satisfies Calculus 1 and Calculus 2 prerequisites for subsequent courses. Weekly lab. Prerequisite: permission of instructor or AP Calculus AB score of 4 or 5.
Lawrence
CALCULUS 2: SCIENTIFIC MODELING AND INTEGRAL CALCULUS.
This course continues the study of the calculus through scientific modeling. While Calculus 1 is concerned with local changes in a function, Calculus 2 focuses on accumulated changes. Models solved by accumulation functions lead to the definition of the integral and the Fundamental Theorem of Calculus. Additional topics include numerical and analytic techniques of integration, trigonometric functions and dynamical systems modeling periodic or quasiperiodic phenomena, local approximation of functions by Taylor polynomials and Taylor series, and approximation of periodic functions on an interval by trigonometric polynomials and Fourier series.
120. CALCULUS 2.
This course satisfies Calculus 2 prerequisites for subsequent courses. Weekly lab. Prerequisites: Mathematics 109 or 110 or 114.
Knoerr, Lee, Naimi
124. CALCULUS 2 (EXPERIENCED).
This course satisfies Calculus 2 prerequisites for subsequent courses. Weekly lab. Prerequisites: Mathematics 114.
Not given in 2007-2008
146. STATISTICS.
Comprehensive study of measures of central tendency, variation, probability, the normal distribution, sampling, estimation, confidence intervals and hypothesis testing. Introduction to use of technology in statistics. Real-life problems are used to illustrate methods. Not open to students who have completed or are currently enrolled in Psychology 201, Biology 368, Mathematics 150 or any Mathematics course above 200.
Knoerr, Lee
150. STATISTICAL DATA ANALYSIS.
An introductory course in statistics emphasizing modern techniques of data analysis. Exploratory data analysis and graphical methods; random variables, statistical distributions, and linear models; classical, robust, and nonparametric methods for estimation and hypothesis testing; introduction to modern multivariate methods. Students will make significant use of a computer application specifically designed for data exploration. The course is strongly recommended for students who are going to use graphical techniques and statistics for research in their fields. Weekly lab. Prerequisite: a Calculus 1 course or permission of instructor.
Lee, Lengyel
160. CREATIVE PROBLEM-SOLVING.
Formal and informal techniques for problem-solving, developed by working on an intriguing collection of puzzles and problems which go beyond those encountered in the usual curriculum. These include problems which can be posed in elementary mathematical or logical terms but which require strategy and ingenuity to solve. This course may be taken up to two times for credit. Prerequisite: a genuine desire to solve problems!
2 units
Lengyel
195. DIRECTED RESEARCH.
Intensive study in an area of mathematics or computer science of the student’s choosing under the direct supervision of a member of the faculty. Prerequisite: permission of the supervising instructor. May be repeated once for credit.
1 unit
Staff
201. MATHEMATICS, EDUCATION, AND ACCESS TO POWER.
This seminar course is a writing-intensive CBL based course designed to expose students to the complicated ways that mathematics affects the community. The CBL component of this course involves tutoring and mathematics assistance at Franklin High School in nearby Highland Park. The seminar component involves meeting weekly with processing discussions and discussion of readings. Topics will include the teaching and learning of mathematics as well as the role of mathematics in individuals’ lives and their community. Prerequisites: Mathematics 110 or 114, and permission of instructor. May be repeated twice for credit.
Buckmire, Gallegos, Knoerr, Sundberg
210. DISCRETE MATHEMATICS.
The language of sets and logic, including propositional and predicate calculus. Formal and informal proofs using truth tables, formal rules of inference and mathematical induction. Congruences and modular arithmetic. Elementary counting techniques. Discrete probability. Abstract relations including equivalence relations and orders. Prerequisite: a Calculus 1 course.
Sundberg, Tollisen
212. MULTIVARIABLE CALCULUS.
Calculus of functions of several variables, parametric curves and surfaces, and vector fields in 2- and 3-space, with applications. Vectors, graphs, contour plots. Differentiation, with application to optimization. Lagrange multipliers. Multiple and iterated integrals, change of variable and Jacobians. Line and surface integrals. Vector analysis, Green’s, Gauss’, and Stokes’ Theorems. Applications to physics, economics, chemistry, and mathematics. Prerequisite: a Calculus 2 course.
Knoerr, Lengyel
214. LINEAR SYSTEMS.
Introduction to linear algebra through a study of linear algebraic systems and systems of first-order linear differential equations. Vector and matrix algebra, Gaussian elimination and the LU decomposition. Determinants. Real vector spaces, subspaces, and the Fundamental Theorem of Linear Algebra. Orthogonality, the QR decomposition, and least squares. First-order linear systems, eigenvalues, and the matrix exponential function. Computing with MATLAB is integrated into the course and projects treat applications to a variety of fields. Prerequisite: a Calculus 2 course.
Lawrence, Naimi
300. JUNIOR COLLOQUIUM.
Preparation for the comprehensive examination and senior project. Completion of Third Year Writing Requirement. Emphases on problem-solving, clear written expression and verbal presentation. Open to junior mathematics majors.
2 units
Buckmire
310. REAL ANALYSIS.
A beginning course in advanced calculus and real analysis. Properties of the real number system, sequences and series of real numbers, the Heine-Borel and Bolzano-Weierstrass Theorems, continuity and uniform continuity, sequences and series of functions. Prerequisite: Mathematics 210.
Not given in 2007-2008
312. COMPLEX ANALYSIS.
The differential and integral calculus of complex-valued functions of a complex variable, emphasizing geometry and applications. The complex number system, analytic functions and the Cauchy-Riemann equations, elementary functions and conformal mappings, contour integration, Taylor and Laurent series, function theory. Applications to physics, engineering and real analysis. Prerequisite: Mathematics 212.
Lawrence
320. ALGEBRA.
A first course in group theory: basic axioms and theorems, subgroups, cosets, normal subgroups, homomorphisms, and extension of the theory to rings and fields. Prerequisites: Mathematics 210 and 214.
Tinberg
322. NUMBER THEORY.
Classical theory of numbers, from ancient to modern. Prime numbers and factorization. Divisors, numerical functions, linear and quadratic congruences. Diophantine problems, including the Fermat conjecture. Factoring methods. Prerequisite: Mathematics 210.
Tollisen
330. PROBABILITY.
Standard methods of calculus are used to study probability: sample spaces, random variables, distribution theory, estimating unknown parameters of distributions. Various applications to real life problems will be discussed. Moment-generating functions and other techniques to calculate moments and characterize distributions. Probabilistic inequalities and the central limit theorem. Point estimators and unbiasedness. Prerequisites: Mathematics 212 and 214.
Lengyel
332. MATHEMATICAL STATISTICS.
Theory and applications of statistical inference. Both Bayesian and classical parametric methods are considered. Point and interval estimation, hypothesis testing. Limit theorems and their use in approximation, maximum likelihood estimation and the generalized likelihood ratio test. Introduction to linear models, nonparametric methods, and decision theory. Prerequisite: Mathematics 330.
Lengyel
341. ORDINARY DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS.
The first half of the course will focus on theoretical, qualitative, and quantitative analyses of ordinary differential equations. First-order linear and nonlinear equations and first order linear systems will be examined from analytical, graphical, and numerical points of view. The second half of the course will be devoted to the study of linear and nonlinear discrete and continuous dynamical systems with special emphasis on qualitative analysis. Prerequisite: Mathematics 214.
Gallegos
342. PARTIAL DIFFERENTIAL EQUATIONS.
An introduction to the study of partial differential equations. This course will include the study of Fourier series, the separation of variables methods, and specifically the wave, heat and Laplace’s equations as well as other elementary topics is PDEs. Numerical approximation techniques and applications to specific topics such as traffic flow, dispersive waves or other areas may be included. Given in alternate years. Prerequisite: Mathematics 212.
Gallegos
350. MATHEMATICAL LOGIC.
A metamathematical investigation of the main formal language used to symbolize ordinary mathematics: first order logic. The focus is on the two fundamental theorems of logic: completeness and compactness. Gödel’s completeness theorem says that every intuitively valid consequence is formally provable from the hypotheses, while compactness says that every intuitively valid consequence of an infinite premise set really depends on only finitely many premises. Prerequisite: Mathematics 210 or permission of instructor. Given in alternate years.
Not given in 2007-2008
352. COMPUTABILITY AND COMPLEXITY.
The logical foundation of the notion of a computable function underlying the workings of modern computers. Representation of the informal mathematical idea of calculability by canonical proxies: “general recursive functions,” “Turing computable functions.” Discussion of Church’s Thesis, which asserts the adequacy of these representations. Survey of decidable and undecidable problems. Prerequisites: Mathematics 210 or permission of instructor.
Not given in 2007-2008
360. AXIOMATIC GEOMETRY.
Axiomatic development of Euclidian and non-Euclidian geometries, including neutral and hyperbolic geometries, and, possibly, brief introductions to elliptic and projective geometries. The course will emphasize a rigorous and axiomatic approach to geometry and consequences of Euclid’s Parallel Postulate and its negations. Prerequisite: two college mathematics courses or permission of instructor.
Not given in 2007-2008
362. TOPOLOGY.
General topology studies those properties (such as connectedness and compactness) which are preserved by continuous mappings. A disk and a solid square are topologically equivalent; so are a doughnut and a coffee cup; but a disk is different from a doughnut. This course enables you to construct your own proofs and counterexamples while getting to know the basic concepts behind modern mathematics. Prerequisites: Mathematics 210 or permission of instructor.
Naimi
370. NUMERICAL ANALYSIS.
Analysis of methods for approximating solutions to algebraic and differential equations by computer. Error estimation and stability are themes throughout. Topics include iterative methods for linear and nonlinear systems, condition numbers and Gaussian elimination, function interpolation and approximation, explicit and implicit methods for initial value problems. Prerequisite: Mathematics 212 or 214 or permission of instructor.
Buckmire
372. OPERATIONS RESEARCH.
Optimal decision-making and modeling of deterministic and stochastic systems. Different systems of constraints lead to different methods. Linear, integer, dynamic programming, and combinatorial algorithms. Practical problems from economics and game theory. Inventory strategies and stochastic models are analyzed by queuing theory. Prerequisites: Mathematics 210 and 214.
Not given in 2007-2008
380. COMBINATORICS.
Investigation of the existence and classification of arrangements. Topics to include principles of enumeration, inclusion-exclusion, the pigeon-hole principle, Ramsey theory, generating functions, special counting sequences, and introductory graph theory. Prerequisite: Mathematics 210.
Not given in 2007-2008
382. GRAPH THEORY.
Graph Theory is a beautiful area of mathematics with many applications. It is used in computer science, biology, urban planning, and many other contexts. Like other areas of discrete mathematics, Graph Theory has the property that the problems are often quite approachable and understandable. Sometimes the solutions to Graph Theory problems can be complex and often require clever arguments, thus the subject is quite pleasing to study. This class will build a solid foundation in Graph Theory for the students. Possible topics are graph isomorphisms, coverings, and colorings; independence number, clique number, connectivity, network flows, and matching theory. Prerequisite: Mathematics 210. Suggested corequisite: Mathematics 380.
Sundberg
392. MATHEMATICAL MODELS IN BIOLOGY.
This course is intended to introduce students to common models used in biology. A variety of models in terms of both biology and mathematics will be covered. Biological topics include action potential generation, genetic spread, cell motion and pattern formation, and circulation. These topics span a range of mathematical models as well, including finite difference equations and differential equations, both linear and non-linear. The focus will be on model analysis and the translation between the mathematical language and the biological meaning. Such analysis will be done both quantitatively and qualitatively. Towards this end, topics seen in previous mathematical courses, such as eigenvalues, phase portraits, and stability, will be revisited. Relevant biology will be presented with each model. The course will be project based. Prerequisite: Mathematics 212 or 214, or permission of instructor.
Gallegos
396. MATHEMATICAL MODELING.
A project-oriented introduction to mathematical modeling. Techniques from calculus, linear algebra and other areas of mathematics will be used to solve problems from the life, physical and social sciences. Familiarity with a programming language is desirable but not required. This course may be taken up to two times for credit. Prerequisites: Mathematics 212 and 214.
2 units
Not given in 2007-2008
397. INDEPENDENT STUDY.
Directed individual study of advanced topics. Prerequisite: permission of instructor.
2 or 4 units
Staff
400. SENIOR COLLOQUIUM.
Senior comprehensive projects. Required of senior mathematics majors.
2 units
Buckmire
499. HONORS.
Prerequisite: permission of department.
Staff
501. ELEMENTARY MATHEMATICS EDUCATION.
Intended for fifth-year students pursuing a credential or MAT in elementary education. We will examine the National mathematics standards, pedagogy specific to mathematics learning, and research issues in elementary mathematics learning. Content and methods will be discussed and utilized in the classroom when appropriate. Prerequisite: enrollment in the fifth year credential or MAT programs or permission of the instructor.
Not given in 2007-2008
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