Course Catalog 2012-2013

Associate Professor Thomas C. Bressoud, Chair

Professors Daniel D. Bonar, Todd H. Feil, Michael D. Westmoreland; Associate Professors Lewis D. Ludwig, Matthew Neal; Assistant Professors William (Andy) Lorenz, Sarah Rundell; Visiting Instructor Timothy DeGenero; Academic Administrative Assistant Dee Ghiloni

The study of mathematics is a challenging and exciting activity that sharpens logical reasoning and improves problem solving ability. The curriculum is designed so that students can apply these skills to analyze both real-world quantitative questions and fascinating theoretical puzzles.

Students interested in mathematics should take 123-124 followed by 231 and 232 by the end of the sophomore year. Prospective mathematics majors or minors should also take 210 the first semester of the sophomore year. It is recommended that Mathematics students take CS 110 or 111 by the end of their sophomore year.

300-level courses fall into two categories. Foundations courses focus on teaching abstract reasoning and the reading, creation, and writing of rigorous proofs in the study of the foundational structures of mathematics. Currently these courses are MATH 321, 322, 331, and 332. Applied courses, while not devoid of proofs, include a significant study of how mathematical techniques can be used to model and analyze real world problems. Currently, we regularly offer MATH 329, 334, 337, 341, and 357 as applied courses. Other courses, such as MATH 361-362, MATH 363-364, or MATH 400, have variable topics.

Denison offers several research opportunities, including funding for summer projects. Interested students should consult a faculty member as early as possible in the fall semester.

Students interested in taking only one or two courses in Mathematics should choose MATH 102, 120, 121, or 123 or a First Year studies course.

Bachelor of Arts Degree: The "Core" courses consist of MATH 124, 210, 231, 232 and CS 110/111. The "Foundations" courses are MATH 321, 322, 331, and 332.

Students who did not receive a 4 or 5 on the Calculus AB advanced placement exam will usually need to take MATH 123 before MATH 124 but may count MATH 123 toward the Math Major (see category D below).

The minimum requirements for a B.A. are the Core and five distinct courses chosen from the following categories:

Any student who takes MATH 124 and subsequently takes a 100-level math course may not count that course toward elective D) in the Math Major. However, any student who takes a 100-level math course before taking MATH 124 may count that course toward elective D) in the Math Major. Any Computer Science course aside from CS 110/111 may count toward elective D) in the Math Major regardless of when the student takes MATH 124.

Bachelor of Science Degree: The "Core" courses consist of MATH 124, 210, 231, 232 and CS 110/111. The "Foundations" courses are MATH 321, 322, 331, and 332.

Students who did not receive a 4 or 5 on the Calculus AB advanced placement exam will usually need to take MATH 123 before MATH 124 but may count MATH 123 toward the Math Major (see category D below).

The minimum requirements for a B.S. are the "Core" courses, all four "Foundations" courses, and four elective courses chosen from the following categories:

The minimum requirements for a Mathematics Minor are MATH 124, 210, 231, 232 and an elective that must be CS 110, CS 111, or any 200 or 300-level math course excluding 361-362 and 363-364.

Computational Science is the field of study concerned with constructing mathematical models and numerical solution techniques, and using computer algorithms and simulation to analyze and solve scientific, social scientific, and engineering problems. The Computational Science concentration consists of: four core courses (MATH 124, 231, CS 110 or 111, and CS 173), and an additional course at the 200-level or above. This additional course, which may be in another department, must have a strong and persistent mathematical modeling or computing component and must be pre-approved by the Mathematics and Computer Science department. In addition, the student must take a two (2) semester sequence of courses in another department besides Mathematics and Computer Science. A written plan for completing the concentration must be approved by the Mathematics and Computer Science department prior to enrollment in the elective course. In particular, the elective course and cognate requirements above must be chosen consistently with a valid educational plan for the study of Computational Science (as defined above). Any Mathematics major who wishes to complete this concentration must choose a computer science course as their elective course. Any Computer Science major who wishes to complete this concentration must choose a mathematics course for their elective course. A double Mathematics and Computer Science major is not eligible for this concentration.

It is recommended that a B.A. candidate in Mathematics consider a second major or a strong minor. Economics would be a reasonable second major or minor for students planning to go into business or into an MBA program following graduation. Computer Science would also be a strong second major or minor.

Students who intend to pursue graduate study in mathematics should pursue a Bachelor of Science degree.

Elements of Statistics (MATH-102).  An introduction to statistical reasoning and methodology. Topics include exploratory data analysis, elementary probability, a standard normal-theory approach to estimation and hypothesis testing and linear and multi-variable regression. Not open for credit to students who have taken Psychology 370. (Offered each year) 4

Special Topics in Mathematics (MATH-120).  Previous topics included in thie area: Mathematical Methods for the Natural and Social Sciences. 4

Essentials of Calculus (MATH-121).  A one-semester introduction to single-variable differential and integral calculus and selected topics in multi-variable calculus. Emphasis is given to applications from the natural and social sciences. (Offered each semester) 4

Calculus I (Single Variable) (MATH-123).  An accelerated introduction to the calculus of single variable functions. Topics include limits, derivatives, integrals, and applications of calculus to the natural and social sciences. Prerequisite: Placement or Math 121. (Offered each semester) 4

Calculus II (Multivariable) (MATH-124).  A continuation of the study of single variable calculus, together with an introduction to the calculus of multivariable functions. Topics include: an introduction to infinite sequences and series, vectors, partial and directional derivatives, gradient, optimization of functions of several variable, integration techniques, double integrals, elementary linear algebra, and an introduction to differential equations with applications to the physical and social sciences. Prerequisite: AP Calculus AB or BC score of 4 or 5 or Math 123. (offered each semester) 4

Introductory Topics in Mathematics (MATH-199).  A general category used only in the evaluation of transfer credit. 1-4

Topics in Mathematics (MATH-200).  (Also listed under Computer Science offerings.) 4

Introduction to Proof Techniques (MATH-210).  An introduction to proof writing techniques. Topics will include logic and proofs, set theory, mathematical induction, relations, modular arithmetic, functions, cardinality, number theory, and calculus. Prerequisite: Math 124. (Offered each year) 4

Technical Communication I (MATH-215).  This course aims to enhance mathematics and computer science students' proficiency and comfort in orally communicating content in their disciplines. Students will present three talks during the semester on substantive, well-researched themes appropriate to their status in their major. Prerequisite: Math 210 or CS 271. (Offered each year) 1

Linear Algebra and Differential Equations (MATH-231).  A continued study of Linear Algebra with applications to linear differential equations and mathematical models in the physical and social sciences. Topics include abstract vector spaces over the real and complex numbers, bases and dimension, change of basis, the Rank-Nullity Theorem, orthogonal bases, linear transformations, the matrix of a linear transformation, eigenvectors and eigenvalues, diagonalization, the matrix exponential, linear differential equations of order n, linear systems of first order differential equations, and a continued study of infinite series, power series, and series solutions of linear differential equations. Prerequisite: Math 124. (Offered each spring) 4

Mathematical Modeling (MATH-232).  A course in mathematical modeling including linear and nonlinear optimization models, linear and non-linear dynamic models, and probability models. This course focuses on applying mathematics to open ended, real world problems, and effectively communicating conclusions. Sensitivity analysis and model robustness are emphasized throughout. This course also strongly features approximation and simulation methods along side analytic methods. Prerequisite: MATH 231. (Offered each spring) 4

Applied Statistics (MATH-242).  Statistics is the science of reasoning from data. This course will introduce you to the fundamental concepts and methods of statistics, including calculus-based probability. Topics include experimental design, data collection, and the scopes of conclusion, sampling, the application of probability models to statistical analysis, hypothesis testing, and regression analysis. Prerequisite: Math 124. (Offered each fall) 4

Elementary Graph Theory (MATH-275).  Graphs are mathematical structures that are used to model a great variety of phenomena ranging from the internet to social networks to phylogenetic clusters, In this class, we will study the mathematical properties of graphs and develop algorithms to solve many common graph problems. Prerequisite: CS 110 or 111 and 174 or MATH 210. (Offered each year) 4

Intermediate Topics in Mathematics (MATH-299).  A general category used only in the evaluation of transfer credit. 1-4

Technical Communication II (MATH-315).  This course is a capstone experience in oral and written communication for mathematics and computer science majors. Students will research a substantive topic, write a rigorous expository article, and make a presentation to the department. Prerequisite: Math/CS 215. Corequisite: a 300-400 level mathematics or computer science course. (Offered each year) 1

Advanced Analysis (MATH-321).  A rigorous analysis of limits, continuity, differentiation, integration, uniform convergence, infinite series and basic topology. Prerequisites: Math 210, 231. (Offered every other fall) 4

Topology (MATH-322).  A study of general topological spaces, including interiors, closures, boundaries, subspace, product, and quotient topologies, continuous functions, homeomorphisms, metric spaces, connectedness, and compactness together with applications of these concepts. Additional topics may include algebraic topology, including homotopy and homology groups, and/or a parallel study of general measure spaces, including inner and outer measure. Prerequisite: Math 321 or permission of instructor. (Offered every other spring) 4

Complex Analysis (MATH-329).  An introduction to complex numbers, analytic functions, derivatives, singularities, integrals, Taylor series, Laurent Series, conformal mappings, residue theory, analytic continuation. Cauchy-Riemann Equations, Cauchy's Theorem, Cauchy Integral Formula, Big and Little Picard Theorems, Riemann Mapping Theorem, Rouche's Theorem. Prerequisite: Math 210, 231. (Offered every other year) 4

Combinatorics (MATH-331).  The basic ideas of sets and functions are used to explore the three basic problems in combinatorics: the counting problem, the existence problem, and the optimization problem. Topics may include: combinatorial proof, the principle of inclusion-exclusion, induction, generating functions, recurrence relations, the Pigeonhole principle, Ramsey theory, basic graph theory, shortest path problems, minimum spanning tree problems, transversal theory, and graph coloring. Prerequisite: Math 210. (Offered every other year) 4

Abstract Algebra (MATH-332).  A rigorous analysis of the structure and properties of abstract groups, rings, fields, and vector spaces. Prerequisites: Math 210, 231. (Offered every other fall) 4

Theory of Computation (MATH-334).  This course is a study of formal languages and their related automata. Turing machines, unsolvable problems and NP-complete problems. Prerequisites: CS 110 or CS 111 and Math 210 or CS 174. 4

Operations Research (MATH-337).  This course involves mathematical modeling of real-world problems and the development of approaches to find optimal (or nearly optimal) solutions to these problems. Topics include: Modeling, Linear Programming and the Simplex Method, the Karush-Kuhn Tucker conditions for optimality, Duality, Network Optimization, and Nonlinear Programming. Prerequisite: Math 231. (Offered every other fall) 4

Applied Probability (MATH-341).  A study of single variable, multi-variable, and stochastic probability models with application to problems in the physical and social sciences. Includes problems in Biology, Finance, and Computer Science. Prereqs: Math 231. Co-requisite: CS 110 or 111. 4

Vector Calculus and Fourier Analysis (MATH-357).  A study of vector calculus, Fourier series, and Fourier transforms together with applications to ordinary and partial differential equations and mathematical models in the sciences. Prerequisite: MATH 231. (Offered every other fall) 4

Directed Study (MATH-361). 1-4

Directed Study (MATH-362). 1-4

Independent Study (MATH-363). 1-4

Independent Study (MATH-364). 1-4

Advanced Topics in Mathematics (MATH-399).  A general category used only in the evaluation of transfer credit. 1-4

Advanced Mathematical Topics (MATH-400).  Advanced topics in Abstract Algebra, Analysis, Geometry or Applied Math. 4

Advanced Mathematical Topics (MATH-401).  Advanced topics in Abstract Algebra, Analysis, Geometry or Applied Math. 4

Senior Research (MATH-451). 4

Senior Research (MATH-452). 4