The College Catalogue

Introductory, Intermediate, and Advanced Courses

50 {1050} - MCSR. Quantitative Reasoning. Every semester. Eric Gaze.

Explores the ways and means by which we communicate with numbers; the everyday math we encounter on a regular basis. The fundamental quantitative skill set is covered in depth providing a firm foundation for further coursework in mathematics and the sciences. Topics include ratios, rates, percentages, units, descriptive statistics, linear and exponential modeling, correlation, logic, probability. A project-based course using Microsoft Excel, emphasizing conceptual understanding and application. Reading of current newspaper articles and exercises involving personal finance are incorporated to place the mathematics in real-world context.

155 {1200} a - MCSR. Introduction to Statistics and Data Analysis. Every fall. Rosemary Roberts.

A general introduction to statistics in which students learn to draw conclusions from data using statistical techniques. Examples are drawn from many different areas of application. The computer is used extensively. Topics include exploratory data analysis, planning and design of experiments, probability, one and two sample t-procedures, and simple linear regression. Not open to students who have credit for Mathematics 165, Psychology 252, or Economics 257.

161 {1600} a - MCSR. Differential Calculus. Every semester. The Department.

Functions, including the trigonometric, exponential, and logarithmic functions; the derivative and the rules for differentiation; the anti-derivative; applications of the derivative and the anti-derivative. Four to five hours of class meetings and computer laboratory sessions per week, on average. Open to students who have taken at least three years of mathematics in secondary school.

165 {1300} a - MCSR. Biostatistics. Every semester. Fall 2012. John O’Brien. Spring 2013. The Department.

An introduction to the statistical methods used in the life sciences. Emphasizes conceptual understanding and includes topics from exploratory data analysis, the planning and design of experiments, probability, and statistical inference. One and two sample t-procedures and their non-parametric analogs, one-way ANOVA, simple linear regression, goodness of fit tests, and the chi-square test for independence are discussed. An average of four to five hours of class meetings and computer laboratory sessions per week. Not open to students who have credit for or are concurrently enrolled in Mathematics 155, Psychology 252, or Economics 257.

171 {1700} a - MCSR. Integral Calculus. Every semester. The Department.

The definite integral; the Fundamental theorems; improper integrals; applications of the definite integral; differential equations; and approximations including Taylor polynomials and Fourier series. An average of four to five hours of class meetings and computer laboratory sessions per week.

Prerequisite: Mathematics 161.

172 {1750} a - MCSR. Integral Calculus, Advanced Section. Every fall. The Department.

A review of the exponential and logarithmic functions, techniques of integration, and numerical integration. Improper integrals. Approximations using Taylor polynomials and infinite series. Emphasis on differential equation models and their solutions. An average of four to five hours of class meetings and computer laboratory sessions per week. Open to students whose backgrounds include the equivalent of Mathematics 161 and the first half of Mathematics 171. Designed for first-year students who have completed an AB Advanced Placement calculus course in their secondary schools.

181 {1800} a - MCSR. Multivariate Calculus. Every semester. The Department.

Multivariate calculus in two and three dimensions. Vectors and curves in two and three dimensions; partial and directional derivatives; the gradient; the chain rule in higher dimensions; double and triple integration; polar, cylindrical, and spherical coordinates; line integration; conservative vector fields; and Green’s theorem. An average of four to five hours of class meetings and computer laboratory sessions per week.

Prerequisite: Mathematics 171 or 172.

200 {2020} a - MCSR. Introduction to Mathematical Reasoning. Every semester. Fall 2012. Manuel Reyes. Spring 2013. The Department.

An introduction to logical deductive reasoning and mathematical proof through diverse topics in higher mathematics. Specific topics include set and function theory, modular arithmetic, proof by induction, and the cardinality of infinite sets. May also consider additional topics such as graph theory, number theory, and finite state automata.

Prerequisite: Mathematics 181 or permission of the instructor.

201 {2000} a - MCSR. Linear Algebra. Every semester. Fall 2012. Leon Harkleroad and Sarah Iams. Spring 2013. The Department.

Topics include vectors, matrices, vector spaces, inner product spaces, linear transformations, eigenvalues and eigenvectors, and quadratic forms. Applications to linear equations, discrete dynamical systems, Markov chains, least-squares approximation, and Fourier series.

Prerequisite: Mathematics 181 or permission of the instructor.

204 {2108} a - MCSR. Biomathematics. Every fall. Fall 2012. Mary Lou Zeeman.

A study of mathematical modeling in biology, with a focus on translating back and forth between biological questions and their mathematical representation. Biological questions are drawn from a broad range of topics, including disease, ecology, genetics, population dynamics, and neurobiology. Mathematical methods include discrete and continuous (ODE) models and simulation, box models, linearization, stability analysis, attractors, oscillations, limiting behavior, feedback, and multiple time-scales. Three hours of class meetings and 1.5 hours of computer laboratory sessions per week. Within the biology major, this course may count as the mathematics credit or as biology credit, but not both. Students are expected to have taken a year of high school or college biology prior to this course. (Same as Biology 174 {1174}.)

Prerequisite: Mathematics 161 or higher, or permission of the instructor.

224 {2208} a - MCSR. Ordinary Differential Equations. Fall 2012. Mary Lou Zeeman.

A study of some of the ordinary differential equations that model a variety of systems in the physical, natural and social sciences. Classical methods for solving differential equations with an emphasis on modern, qualitative techniques for studying the behavior of solutions to differential equations. Applications to the analysis of a broad set of topics, including population dynamics, oscillators and economic markets. Computer software is used as an important tool, but no prior programming background is assumed.

Prerequisite: Mathematics 181 or permission of the instructor.

225 {2206} a - MCSR. Probability. Every semester. Fall 2012. John O’Brien. Spring 2013. The Department.

A study of the mathematical models used to formalize nondeterministic or “chance” phenomena. General topics include combinatorial models, probability spaces, conditional probability, discrete and continuous random variables, independence and expected values. Specific probability densities, such as the binomial, Poisson, exponential, and normal, are discussed in depth.

Prerequisite: Mathematics 181 or permission of the instructor.

229 {2109} a - MCSR. Optimization. Every other year. Spring 2013. Adam Levy.

A study of optimization problems arising in a variety of situations in the social and natural sciences. Analytic and numerical methods are used to study problems in mathematical programming, including linear models, but with an emphasis on modern nonlinear models. Issues of duality and sensitivity to data perturbations are covered, and there are extensive applications to real-world problems.

Prerequisite: Mathematics 181 or permission of the instructor.

232 {2302} a - MCSR. Number Theory. Every other fall. Fall 2012. Michael King.

A standard course in elementary number theory, which traces the historical development and includes the major contributions of Euclid, Fermat, Euler, Gauss, and Dirichlet. Prime numbers, factorization, and number-theoretic functions. Perfect numbers and Mersenne primes. Fermat’s theorem and its consequences. Congruences and the law of quadratic reciprocity. The problem of unique factorization in various number systems. Integer solutions to algebraic equations. Primes in arithmetic progressions. An effort is made to collect along the way a list of unsolved problems.

Prerequisite: Mathematics 200 or permission of the instructor.

233 {2303} a - MCSR. Functions of a Complex Variable. Every other fall. Fall 2013. The Department.

The differential and integral calculus of functions of a complex variable. Cauchy’s theorem and Cauchy’s integral formula, power series, singularities, Taylor’s theorem, Laurent’s theorem, the residue calculus, harmonic functions, and conformal mapping.

Prerequisite: Mathematics 181 or permission of the instructor.

244 {2209} a - MCSR. Numerical Methods. Every other fall. Fall 2013. Adam Levy.

An introduction to the theory and application of numerical analysis. Topics include approximation theory, numerical integration and differentiation, iterative methods for solving equations, and numerical analysis of differential equations.

Prerequisite: Mathematics 201 or permission of the instructor.

247 {2404} a - MCSR. Geometry. Every other spring. Spring 2013. William Barker.

A survey of modern approaches to Euclidean geometry in two and three dimensions. Axiomatic foundations of metric geometry. Transformational geometry: isometries and similarities. Klein’s Erlangen Program. Symmetric figures. Scaling, measurement, and dimension.

Prerequisite: Mathematics 200 or permission of the instructor.

252 {2502} a - MCSR. Mathematical Cryptography. Every other spring. Spring 2014. Jennifer Taback.

Classical and modern methods of cryptography and cryptanalysis. Topics include public key cryptography and the RSA encryption algorithm, factoring techniques, and recently proposed cryptosystems based on group theory and graph theory.

Prerequisite: Mathematics 200 or permission of the instructor.

253 {2501} a - MCSR. Vector Calculus. Every other year. Fall 2012. William Barker.

A study of vector calculus based on linear algebra. The central unifying theme is the theory and application of differential forms. Topics include the derivative as a linear transformation between Euclidean spaces; the Inverse Function Theorem and the Implicit Function Theorem; multiple integration and the Change of Variables Theorem; vector fields, tenors, and differential forms; line and surface integration; integration of differential forms; the exterior derivative; closed and exact forms; the generalized Stokes’ Theorem; gradient, curl, divergence and the integral theorems of Green, Gauss, and Stokes; manifolds in Euclidean space; applications in the physical sciences.

Prerequisite: Mathematics 201.

258 {2601} a - MCSR. Combinatorics and Graph Theory. Every other fall. Fall 2012. Aba Mbirika.

An introduction to combinatorics and graph theory. Topics to be covered may include enumeration, matching theory, generating functions, partially ordered sets, Latin squares, designs, and graph algorithms.

Prerequisite: Mathematics 200 or permission of the instructor.

262 {2602} a - MCSR. Introduction to Algebraic Structures. Every year. Spring 2013. The Department.

An introduction to the theory of finite and infinite groups, with examples ranging from symmetry groups to groups of polynomials and matrices. Properties of mappings that preserve algebraic structures are studied. Topics include cyclic groups, homomorphisms and isomorphisms, normal subgroups, factor groups, the structure of finite abelian groups, and Sylow theorems.

Prerequisite: Mathematics 200 and 201 or permission of the instructor.

263 {2603} a - MCSR. Introduction to Analysis. Every year. Fall 2012. William Barker.

Building on the theoretical underpinnings of calculus, develops the rudiments of mathematical analysis. Concepts such as limits and convergence from calculus are made rigorous and extended to other contexts, such spaces of functions. Specific topics include metric spaces, point-set topology, sequences and series, continuity, differentiability, the theory of Riemann integration, and functional approximation and convergence.

Prerequisite: Mathematics 200 or a 200-level mathematics course approved by the instructor.

264 {3209} a - MCSR. Partial Differential Equations. Every other year. Fall 2014. Adam Levy.

A study of some of the partial differential equations that model a variety of systems in the natural and social sciences. Classical methods for solving partial differential equations, with an emphasis where appropriate on modern, qualitative techniques for studying the behavior of solutions. Applications to the analysis of a broad set of topics, including air quality, traffic flow, and imaging. Computer software is used as an important tool, but no prior programming background is assumed.

Prerequisite: Mathematics 201 and 224, or permission of the instructor.

265 {2606} a - MCSR. Statistics. Every spring. Spring 2013. Rosemary Roberts.

An introduction to the fundamentals of mathematical statistics. General topics include likelihood methods, point and interval estimation, and tests of significance. Applications include inference about binomial, Poisson, and exponential models, frequency data, and analysis of normal measurements.

Prerequisite: Mathematics 225 or permission of the instructor.

267 {2604} a - MCSR. Topology. Every other spring. Spring 2014. The Department.

Topology studies those properties of objects that are preserved under continuous changes. Examines abstract definition of a topology and examples of topological spaces, connectedness and compactness, countability and separation axioms, classification of surfaces, algebraic topology—including homotopy, the fundamental group, covering spaces, and introductory category theory.

Prerequisite: Mathematics 200 or permission of the instructor.

291–294 {2970–2973} a. Intermediate Independent Study in Mathematics. The Department.

299 {2999} a. Intermediate Collaborative Study in Mathematics. The Department.

302 {3602} a. Advanced Topics in Algebra. Every other spring. Spring 2014. The Department.

Introduction to rings and fields. Vector spaces over arbitrary fields. Additional topics may include Galois theory, algebraic number theory, finite fields, and symmetric functions.

Prerequisite: Mathematics 262 or permission of the instructor.

303 {3603} a. Advanced Topics in Analysis. Every other spring. Spring 2013. Thomas Pietraho.

One or more selected topics from advanced analysis. Possible topics include Lebesque measure and integration theory, Fourier analysis, Hilbert and Banach space theory, and stochastic calculus with applications to mathematical finance.

Prerequisite: Mathematics 201 and 263, or permission of the instructor.

304 {3108} a. Advanced Topics in Applied Mathematics. Fall 2012. Adam Levy. Spring 2013. Mary Lou Zeeman.

One or more selected topics in applied mathematics. Material selected from the following: Fourier series, partial differential equations, integral equations, optimal control, bifurcation theory, asymptotic analysis, applied functional analysis, and topics in mathematical physics.

Prerequisite: Mathematics 200, 201, and 224, or permission of the instructor.

305 {3606} a. Advanced Topics in Probability and Statistics. Every other fall. Fall 2013. John O’Brien.

One or more specialized topics in probability and statistics. Possible topics include regression analysis, nonparametric statistics, logistic regression, and other linear and nonlinear approaches to modeling data. Emphasis is on the mathematical derivation of the statistical procedures and on the application of the statistical theory to real-life problems.

Prerequisite: Mathematics 201 and 265, or permission of the instructor.

307 {3404} a. Advanced Topics in Geometry. Every other fall. Fall 2013. William Barker.

A survey of affine, projective, and non-Euclidean geometries in two-dimensions, unified by the transformational viewpoint of Klein’s Erlanger Programm. Special focus will be placed on conic sections and projective embeddings. Additional topics as time permits: complex numbers in plane geometry, quaternions in three-dimensional geometry, and the geometry of four-dimensional space-time in special relativity. Mathematics 247 is helpful but not required.

Prerequisite: Mathematics 200 and 201, or permission of the instructor.

401–404 {4000–4003} a. Advanced Independent Study and Honors in Mathematics. The Department.

405 {4029} a. Advanced Collaborative Study in Mathematics. The Department.