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Course Criteria
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4.00 Credits
This seminar introduces problem-solving techniques used throughout the mathematics curriculum. The course focuses on solving difficult problems stated in terms of elementary combinatorics, geometry, algebra, and calculus. Each class combines a lecture describing the common tricks and techniques used in a particular field with a problem session in which students work together using those techniques to tackle some particularly challenging problems. Prerequisite: any 200-level mathematics course or permission of the instructor.
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4.00 Credits
Computer Science This lab is an introduction to mathematical computation. It starts with tutorials on the software package Mathematica and its open-source alternative, SAGE. The course then discusses algorithms for finding the zeros of nonlinear functions, solving linear systems quickly, and approximating eigenvalues. The bulk of the course is devoted to curve fitting by means of polynomial interpolation, splines, Bézier curves, and least squares. Prerequisites: Mathematics 242 and any computer science course or basic programming experience.
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4.00 Credits
This course develops the basic methods of enumeration, which include elementary counting techniques, the inclusion-exclusion principle, and generating functions. Students apply these counting methods to fundamental combinatorial structures such as trees and permutations.
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4.00 Credits
Combinatorial mathematics is the study of how to combine objects into finite arrangements. Topics covered in this course are chosen from enumeration and generating functions, graph theory, matching and optimization theory, combinatorial designs, ordered sets, and coding theory. Prerequisite: Mathematics 261 or permission of the instructor.
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4.00 Credits
Economics This course introduces students to the branch of operations research known as deterministic optimization, which tackles problems such as how to schedule classes with a limited number of classrooms, determine a diet that is both rich in nutrients and low in calories, and create an investment portfolio that meets your investment needs. Techniques covered include Integer/combinatorial optimization, linear programming, nonlinear programming, and network flows. Emphasis is on the importance of problem formulation as well as how to apply algorithms to real-world problems. Prerequisites: working knowledge of multivariable calculus and basic linear algebra.
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4.00 Credits
An introduction to the theory of abstract vector spaces, a useful concept when studying physical phenomena. Topics include linear independence and dependence, bases and dimension, linear transformations, eigenvalues, eigenvectors, diagonalization, inner product spaces, and orthogonality. Prerequisite: Mathematics 261 or permission of the instructor.
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4.00 Credits
An introduction to modern abstract algebraic systems. The structures of groups, rings, and fields are studied, together with the homomorphisms of these objects. Topics studied include equivalence relations, finite groups, group actions, integral domains, polynomial rings, and finite fields. Prerequisite: Mathematics 261 or permission of the instructor.
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4.00 Credits
An introduction to point set topology. Topics include topological spaces, metric spaces, compactness, connectedness, continuity, homomorphisms, separation criteria, and, possibly, the fundamental group. Prerequisite: Mathematics 361 or permission of the instructor.
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4.00 Credits
This course explores the mathematics of curved spaces, particularly curved surfaces embedded in three-dimensional Euclidean space. Originally developed to study the surface of the Earth, differential geometry is an active area of research that is fundamental to physics, particularly general relativity. The basic issue is to determine whether a given space is indeed curved, and if so, to quantitatively measure its curvature using multivariable calculus. This course also introduces geodesics and culminates with the Gauss-Bonnet theorem. Prerequisites: Mathematics 212 and Mathematics 261, or permission of the instructor.
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4.00 Credits
The fundamental ideas of analysis in onedimensional Euclidean space are studied. Topics covered include the completeness of real numbers, sequences, Cauchy sequences, continuity, uniform continuity, the derivative, and the Riemann integral. As time permits, other topics may be considered, such as infinite series of functions or metric spaces. Prerequisite: Mathematics 261 or permission of the instructor.
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