# Mathematics

## Departmental Guidelines and Goals

Mathematics is an essential part of a liberal arts education with numerous connections to a variety of disciplines. The study of mathematics is a challenging and exciting activity that sharpens logical reasoning skills and improves problem solving ability. The curriculum is designed so students can apply these skills to analyze both real-world questions and explore sophisticated mathematical theory.

Students interested in the Mathematics major should take MATH 135 - Single Variable Calculus and MATH 145 - Multi-variable Calculus followed by MATH 213 - Linear Algebra and Differential Equations and MATH 300 - Introduction to Proofs by the end of their sophomore year. It is strongly recommended that Mathematics students take CS 109 - Discovering Computer Science, CS 110 - Discovering Computer Science: Digital Media and Games, CS 111 - Discovering Computer Science: Scientific Data and Dynamics or CS 112 - Discovering Computer Science: Markets, Polls, and Social Networks by the end of their sophomore year.

Most upper level mathematics electives fall into two categories: Foundations and Modeling.

Foundation 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:

Code | Title |
---|---|

MATH 400 | Combinatorics |

MATH 410 | Abstract Algebra |

MATH 413 | Advanced Linear Algebra |

MATH 440 | Advanced Analysis |

MATH 445 | Topology |

MATH 447 | Vector Calculus and Complex Analysis |

Modeling 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 modeling courses include:

Code | Title |
---|---|

MATH 415 | Operations Research |

MATH 425 | Applied Probability |

MATH 427 | Probability Computing and Graph Theory |

MATH 430 | Fourier Analysis |

MATH 435 | Mathematical Modeling |

Other courses, such as MATH 361 - Directed Study - MATH 362 - Directed StudyorMATH 363 - Independent Study - MATH 364 - Independent Study include various additional topics.

Non-major/minor students interested in taking a mathematics course should consider MATH 115 - Mathematical Methods for the Physical and Social Sciences or MATH 120 - Elements of Statistics.

Students who did not receive a 4 or 5 on the Calculus AB Advanced Placement exam will usually take MATH 135 before MATH 145.

## Faculty

Professor Michael D. Westmoreland, Chair

Professors Lewis D. Ludwig, Matthew P. Neal, Michael D. Westmoreland; Associate Professor Sarah Rundell; Assistant Professors May Mei, David White, Sarah Wolff; Visiting Instructors Laura Dolph Bosley, Alice Miller, William Robinson; Academic Administrative Assistant, Dee Ghiloni

## Mathematics Majors

The CORE courses consist of:

Code | Title | |
---|---|---|

MATH 145 | Multi-variable Calculus | |

MATH 213 | Linear Algebra and Differential Equations | |

MATH 220 | Applied Statistics | |

MATH 300 | Introduction to Proofs | |

MATH 395 | Technical Communication I | |

CS 109 | Discovering Computer Science | |

or CS 110 | Discovering Computer Science: Digital Media and Games | |

or CS 111 | Discovering Computer Science: Scientific Data and Dynamics | |

or CS 112 | Discovering Computer Science: Markets, Polls, and Social Networks |

The FOUNDATION courses consist of:

Code | Title |
---|---|

MATH 400 | Combinatorics |

MATH 410 | Abstract Algebra |

MATH 413 | Advanced Linear Algebra |

MATH 440 | Advanced Analysis |

MATH 445 | Topology |

MATH 447 | Vector Calculus and Complex Analysis |

MATH 334 | Theory of Computation |

The MODELING courses consist of:

Code | Title |
---|---|

MATH 415 | Operations Research |

MATH 425 | Applied Probability |

MATH 427 | Probability Computing and Graph Theory |

MATH 430 | Fourier Analysis |

MATH 435 | Mathematical Modeling |

### Bachelor of Arts Degree

The minimum requirement for the Bachelor of Arts in Mathematics are the six CORE courses plus four courses; two modeling courses and two foundations courses.

### Bachelor of Science Degree

The minimum requirement for the Bachelor of Science in Mathematics are:

Code | Title |
---|---|

The six CORE courses plus six courses; three of | |

MATH 410 | Abstract Algebra |

MATH 440 | Advanced Analysis |

MATH 445 | Topology |

and | |

MATH 447 | Vector Calculus and Complex Analysis |

plus three other 400-level courses. |

Students majoring in Math or Computer Science may take up to two cross-listed math or computer science courses to count as requirements in the intended major. These cross listed courses typically satisfy electives in the major.

Code | Title |
---|---|

Current cross listed courses | |

MATH 334 | Theory of Computation |

MATH 415 | Operations Research |

MATH 427 | Probability Computing and Graph Theory |

are pre-approved for this policy. | |

MATH 220 | Applied Statistics |

is preapproved for the computer science BS major. Other math and computer science courses must be approved by the department prior to enrollment. |

## Mathematics Minor

The minimum requirements for a mathematics minor are:

Code | Title |
---|---|

MATH 145 | Multi-variable Calculus |

MATH 213 | Linear Algebra and Differential Equations |

MATH 220 | Applied Statistics |

MATH 300 | Introduction to Proofs |

and one of | |

CS 109 | Discovering Computer Science |

CS 110 | Discovering Computer Science: Digital Media and Games |

CS 111 | Discovering Computer Science: Scientific Data and Dynamics |

CS 112 | Discovering Computer Science: Markets, Polls, and Social Networks |

**Computational Science Concentration**

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:

Code | Title |
---|---|

MATH 145 | Multi-variable Calculus |

MATH 213 | Linear Algebra and Differential Equations |

CS 173 | Intermediate Computer Science |

and one of | |

CS 109 | Discovering Computer Science |

CS 110 | Discovering Computer Science: Digital Media and Games |

CS 111 | Discovering Computer Science: Scientific Data and Dynamics |

or | |

CS 112 | Discovering Computer Science: Markets, Polls, and Social Networks |

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 approved in advance by the Mathematics and Computer Science Department. In addition, the student must take a two-semester sequence of courses in a department other than 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 in consistence with a valid educational plan for the student of Computational Science (as defined above). Any mathematics major who wishes to complete this concentration must choose a computer science course as an elective course. Any computer science major who wished to complete this concentration must choose a mathematics course as an elective course. A double mathematics and computer science major is not eligible for this concentration.

## Additional Points of Interest

### Research at Denison

Denison offers a number of research opportunities, including funding for summer research projects. The Anderson Foundation and the Denison University Research Foundation (DURF) support qualified students conducting summer research. For off-campus research opportunities in Mathematics, see the Oak Ridge Science Semester or various National Science Foundation Research Experience for Undergraduates experiences. Interested students should consult a faculty member as early as possible in the fall semester.

### Off-Campus Study

The department of Mathematics and Computer Science strongly encourages students to globalize their education by completing some portion of their undergraduate education abroad. A majority of Denison students spend a semester abroad during their junior year and many more spend a second summer abroad. Denison offers a wide range of opportunities to study off-campus that are highly relevant to the Denison experience.

Going abroad allows students to enhance their knowledge while experiencing another culture and way of life. Students gain valuable international experience that will benefit future career goals and/or graduate school opportunities. Math and Computer Science majors who are fluent in another language will have special advantages in the job market.

Students may take up to two classes outside the department to transfer towards the major at Denison. Additional courses taken outside Denison may accrue credit hours towards graduation, but will not contribute to requirements in the major. Courses taken outside the university must be **pre-approved** for acceptance towards major requirements. Students should provide the department chair syllabi for the intended courses for department approval. Students may petition the department chair for exceptions to this policy. In particular, transfer students may be eligible to transfer additional courses towards major requirements.

## Courses

**MATH 115 - Mathematical Methods for the Physical and Social Sciences (4 Credit Hours)**

This course will explore three major topics of mathematics: linear algebra, probability and statistics, and Markov chains. Using these three topics, students will engage in three real world applications in biology, chemistry, and economics. This course is well suited for students who need a year of mathematics, like many pre-professional programs, and are looking for real applications of mathematics beyond the typical algebra and calculus approach. While this course would be a natural extension for pre-professional students who have take Math 130 Essentials of Calculus, this course only requires a strong background in high school Algebra II.

**MATH 120 - Elements of Statistics (4 Credit Hours)**

An introduction to statistical reasoning and methodology. Topics include experimental design, 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.

**MATH 130 - Essential of Calculus (4 Credit Hours)**

A one-semester introduction to single-variable calculus focused on functions, graphs, limits, exponential and logarithmic functions, differentiation, integration, techniques and applications of integration, and optimization. Emphasis is given to applications from the natural and social sciences.

**MATH 135 - Single Variable Calculus (4 Credit Hours)**

An accelerated introduction to the calculus of single variable functions with early transcendentals. Topics include limits, derivatives, integrals, and applications of calculus to the natural and social sciences including optimization, differential equations, curve, probability, velocity, acceleration area, volume, Net Change Theorem and Fundamental Theorem of Calculus.

**Prerequisite(s):** Placement or MATH 130.

**MATH 145 - Multi-variable Calculus (4 Credit Hours)**

A continuation of the study of single variable calculus, together with an introduction to linear algebra and 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(s):** AP Calculus AB or BC score of 4 or 5 or MATH 135.

**MATH 199 - Introductory Topics in Mathematics (1-4 Credit Hours)**

A general category used only in the evaluation of transfer credit.

**MATH 200 - Topics in Mathematics (4 Credit Hours)**

A course used to introduce new intermediate-level courses into the curriculum. (Also listed under Computer Science offerings.)

**MATH 213 - Linear Algebra and Differential Equations (4 Credit Hours)**

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, linear transformations, the matrix of a linear transformation, eigenvectors and eigenvalues, diagonalization, 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(s):** MATH 145.

**MATH 220 - Applied Statistics (4 Credit Hours)**

Statistics is the science of reasoning from data. This course will introduce the fundamental concepts and methods of statistics, including calculus-based probability. Topics include experimental design, data collection, and the scopes of conclusion, a robust study of probability models and their application to statistical inference, hypothesis testing, and regression analysis.

**MATH 225 - Analysis of Risk (4 Credit Hours)**

This course covers the essentials of asset management including the diversification of investment portfolios. The course begins with the basics of present value analysis and probability theory. Basic tools will be developed and used to study issues such as basic portfolio optimization and asset pricing.

**Prerequisite(s):** MATH 145.

**MATH 299 - Intermediate Topics in Mathematics (1-4 Credit Hours)**

A general category used only in the evaluation of transfer credit.

**MATH 300 - Introduction to Proofs (4 Credit Hours)**

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(s):** MATH 145.

**MATH 334 - Theory of Computation (4 Credit Hours)**

This course is the study of computers as mathematical abstractions in order to understand the limits of computation. In this course, students will learn about topics in computability theory and complexity theory. Topics in computability theory include Turing machines and their variations, the Universal Turing machine, decidability of the halting problem, reductions, and proving decidability of other problems. Topics in complexity theory include the classes P and NP, NP-completeness, and other fundamental complexity classes.This course is a study of formal languages and their related automata, Turing machines, unsolvable problems and NP-complete problems.

**Prerequisite(s):** CS 109, 110, CS 111, or 112, and MATH 300 or CS 234.

**Crosslisting:** CS 334.

**MATH 361 - Directed Study (1-4 Credit Hours)**

**MATH 362 - Directed Study (1-4 Credit Hours)**

**MATH 363 - Independent Study (1-4 Credit Hours)**

**MATH 364 - Independent Study (1-4 Credit Hours)**

**MATH 395 - Technical Communication I (1 Credit Hour)**

This course aims to enhance mathematics and computer science students' proficiency and comfort in orally communicating content in their disciplines. Students will develop skills in presenting technical information to a non-technical audience. In particular, students will deliver a number of presentations during the semester on substantive, well-researched themes appropriate to their status in their major.

**Corequisite(s):** a 200-level mathematics or computer science course.

**MATH 399 - Advanced Topics in Mathematics (1-4 Credit Hours)**

A general category used only in the evaluation of transfer credit.

**MATH 400 - Combinatorics (4 Credit Hours)**

This course is the study of counting techniques for discrete collections of objects. This course will include topics such as permutations and combinations, binomial coefficients, inclusion-exclusion, Fibonacci numbers, Catalan numbers, set partitions, Stirling numbers, generating functions, exponential generating functions, and Pólya counting.

**Prerequisite(s):** MATH 300.

**MATH 410 - Abstract Algebra (4 Credit Hours)**

A rigorous analysis of the structure and properties of abstract groups, rings, fields, and vector spaces.

**MATH 413 - Advanced Linear Algebra (4 Credit Hours)**

This is a second course in linear algebra, which will continue to develop a linear algebra toolkit in order to pursue a mixture of theory and applications. Topics discussed will include singular value decomposition, canonical forms, orthogonal bases and inner product spaces, harmonic analysis and the discrete Fourier transform. The course will also include applications of these concepts in mathematics, computer science, and physics.

**MATH 415 - Operations Research (4 Credit Hours)**

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 may include: modeling, linear programming and the simplex method, the Karush-Kuhn Tucker conditions for optimality, duality, network optimization, and nonlinear programming.

**Prerequisite(s):** MATH 213.

**Crosslisting:** CS 337.

**MATH 425 - Applied Probability (4 Credit Hours)**

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.

**Prerequisite(s):** MATH 213.

**MATH 427 - Probability Computing and Graph Theory (4 Credit Hours)**

This course is about the design and analysis of randomized algorithms, (i.e. algorithms that compute probabilistically). Such algorithms are often robust and fast, though there is a small probability that they return the wrong answer. Examples include Google’s PageRank algorithm, load balancing in computer networks, coping with Big Data via random sampling, navigation of unknown terrains by autonomous mobile entities, and matching medical students to residencies. The analysis of such algorithms requires tools from probability theory, which will be introduced as needed. As there have been many randomized algorithms designed to solve problems on graphs, the course introduces numerous topics from graph theory of independent mathematical interest. Graphs are often used to mathematically model phenomena of interest to computer scientists, including the internet, social network graphs, and computer networks. Lastly, this course demonstrates the powerful Probabilistic Method to non-constructively prove the existence of certain prescribed graph structures, how to turn such proofs into randomized algorithms, and how to derandomize such algorithms into deterministic algorithms.

**Prerequisite(s):** CS 271 or MATH 435 or MATH 242/220, and MATH 300 and one from CS 109, CS 110, CS 111, or CS 112.

**Crosslisting:** CS 335.

**MATH 430 - Fourier Analysis (4 Credit Hours)**

A study of a widely used and applied subfield of advanced Linear Algebra and Calculus (which also uses Calculus). For example, your ear processes a sound wave (maybe from plucking guitar strings) by changing into an orthogonal frequency basis allowing us to hear the main notes and some selected overtones. This course will essentially use the power of changing (orthogonal) bases to analyze a wide array of problems in image processing, sound processing, signal reconstruction, medical imaging, wave analysis, heat diffusion, statistical modeling, quantum mechanics, number theory, and geometry. No knowledge of these application topics is necessary.

**Prerequisite(s):** MATH 213.

**MATH 435 - Mathematical Modeling (4 Credit Hours)**

A course in mathematical modeling including linear and nonlinear optimization models, linear and non-linear dynamic models, and probability and statistical models. Both continuous and discrete models are considered. 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 in conjunction with analytic methods.

**Prerequisite(s):** CS 109, CS 110 or CS 111, CS 112, and MATH 213.

**MATH 440 - Advanced Analysis (4 Credit Hours)**

A rigorous analysis of limits, continuity, differentiation, integration, uniform convergence, infinite series and basic topology.

**MATH 445 - Topology (4 Credit Hours)**

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(s):** MATH 440 or consent.

**MATH 447 - Vector Calculus and Complex Analysis (4 Credit Hours)**

Study of Vector Calculus including vector valued functions, curves, Kepler’s laws, curvature, torsion, multiple integrals, iterated integrals, Fubini’s theorem, polar, cylindrical, spherical coordinates, center of mass, moments of inertia, determinants and n-dimensional volume, change of variables, differential forms, line integrals, Green’s Theorem, surface integrals, flux, curl, divergence, Stoke’s Theorem, Divergence Theorem, Gauss’s law, Maxwell’s equations and applications to Topology. The lens is then narrowed to study functions of a complex variable, including 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, and Rouche's Theorem.

**MATH 451 - Senior Research (4 Credit Hours)**

**MATH 452 - Senior Research (4 Credit Hours)**

**MATH 470 - Advanced Mathematical Topics (1-4 Credit Hours)**

A general category used only in the evaluation of transfer credit.

**MATH 471 - Advanced Mathematical Topics (1-4 Credit Hours)**

Advanced topics in Abstract Algebra, Analysis, Geometry or Applied Math.

**MATH 495 - Technical Communication II (1 Credit Hour)**

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(s):** MATH/CS 395 and a 300-400 level computer science course or a 400-level mathematics course.

**MATH 499 - Advanced Topics in Mathematics (1-4 Credit Hours)**

A general category used only in the evaluation of transfer credit.