Course Catalog 2011-2012
Associate Professor Matthew Neal, 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 Mathematics curriculum is designed so that students will have a sound theoretical understanding of mathematics and an understanding of a variety of applications of mathematics. The study of mathematics is a challenging activity that sharpens logical reasoning and improves problem solving skills.
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.
For research opportunities in mathematics see the Oak Ridge Science Semester described at www.orss.denison.edu. Summer research opportunities may also be available on campus.
Students interested in taking only one or two courses in Mathematics should choose Math 102, 121, or 123.
Bachelor of Arts Degree. The core courses consist of Mathematics 123, 124, 210, 231 and 232. The minimum requirements for a B.A. in mathematics are the core plus Computer Science 110 or 111, either Mathematics 321 or 332, one course in discrete mathematics (331 or 337), one course in continuous mathematics (329 or 357), and two additional courses chosen from either 242 or 275 and all 300-level courses, excluding 361-362 and 363-364.
Bachelor of Science Degree. The B.S. requirements consist of the core courses, CS 110 or 111, the real analysis sequence (321-322), the abstract algebra sequence (332-333), one course in discrete mathematics (331 or 337), one course in continuous mathematics (329 or 357), and two additional courses chosen from either 242 or 275 and all 300-level courses, excluding 361-362 and 363-364. A yearlong senior research project may count as one elective.
A minor in Mathematics consists of 123, 124, 210, 231, 232, and two courses chosen from either CS 110 or 111, Math 242 or 275 and all 300-level courses, excluding 361-362 and 363-364. It is possible that a class in another department may substitute for a mathematics elective, but it must have a very strong and persistent mathematical component and must be approved on a case-by-case basis by the Math and Computer Science department.
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: five core courses (MATH 123, 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 simple linear regression. Not open for credit to students who have taken Psychology 370. (Offered each year) 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, relations and functions, cardinality and mathematical induction. 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 and differential equations with applications to the physical and social sciences. Includes a further study of infinite series and its applications. Prerequisite: Math 124. (Offered each spring) 4
Mathematical Modeling (MATH-232). A course in mathematical modeling including linear and nonlinear optimization models, 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 analyses, 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 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 I (MATH-321). Thorough analysis of limits, continuity, differentiation, integration, uniform convergence of infinite series and topology. Prerequisites: Math 210, 231. (Offered every other fall) 4
Advanced Analysis II (MATH-322). A continuation of Math 321. Topics may include topology, measure theory and differential geometry. Vector calculus and differential geometry. Prerequisites: Math 321. (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 I (MATH-332). The study of abstract vector spaces and introduction to the structure and properties of groups, rings and fields. Prerequisites: Math 210, 231. (Offered every other fall) 4
Abstract Algebra II (MATH-333). A continuation of Math 332. A continued study of the structures of groups, rings and fields, and other topics in abstract algebra. Prerequisite: Math 332. (Offered every other spring) 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
Probability (MATH-341). The probability is developed by studying combinatorics, probability models, moment generating functions, limit theorems and conditional probability. Applications include but are not limited to statistical decision theory and inference. Prerequisite: Math 231. 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 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). 4