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 taken 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. Students who have received credit for MATH 220 may not subsequently receive credit for MATH 120. Not open for credit to students who have taken PSYC 370 or GC 202 courses.

MATH 130 - Essentials of Calculus (4 Credit Hours)

A one-semester introduction to single-variable calculus for students who have not taken a calculus course. This class focuses on functions, graphs, limits, exponential, logarithmic and trigonometric functions, differentiation, integration, techniques and applications of integration, and applications from the natural and social sciences. Students who have received credit for MATH 135 or MATH145 may not subsequently receive credit for MATH130.

MATH 135 - Single Variable Calculus (4 Credit Hours)

A one-semester single-variable calculus course starting from a knowledge of basic derivatives and integrals. This class focuses on advanced topic in derivatives, advanced topics in integrals, optimization, applications of calculus to the natural and social sciences, sequences and series, probability, and differential equations. Students who have received credit for MATH 145 may not subsequently receive credits for MATH 135.

Prerequisite(s): Placement or MATH 130.

MATH 145 - Multivariable Calculus (4 Credit Hours)

A one-semester multivariable calculus course with an introduction to linear algebra. This class focuses on vectors, partial derivatives, gradients, optimization of functions of several variables, multiple integrals and their applications, and elementary linear algebra.

Prerequisite(s): Placement or MATH 135.

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 using calculus-based probability. Topics include a basic study of probability models, sampling distributions, confidence intervals, hypothesis testing, categorical data analysis, ANOVA, multivariate regression analysis, logistic regression, and other statistical methods. Scopes of conclusion, model building and validation principles, and common methodological errors are stressed throughout.

Prerequisite(s): MATH 135 and DA 101 or MATH 145.

Crosslisting: DA 220.

MATH 221 - Applied Probability (4 Credit Hours)

This course introduces students to the key conceptual ideas of probability. Students explore these concepts through the use of empirical investigation with data. The course offers substantial programming labs where students increase their algorithmic design and coding proficiency, and also gain familiarity with various Python modules and toolboxes.

Prerequisite(s): MATH 145 and CS 145 or one course from CS 109-CS 114 range.

Crosslisting: ML 221.

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 247 - Calculus III for Pre-engineering students (4 Credit Hours)

A one semester overview of mathematics applied to physical systems, with extensive use of examples from introductory and intermediate physics. Topics covered include operators, functions, vectors, complex numbers, integration & differentiation in multiple dimensions, geometry (gradients, divergence, & curl), differential equations, and linear algebra. The unity of linear systems will be emphasized, though non-linearity will also be discussed. Both hand- and computer-aided computation will be required. NOTE: MATH 247 does not fulfill any requirements for the MATH major or minor.

Prerequisite(s): PHYS 121 or PHYS 126 and MATH 145 or consent.

Crosslisting: PHYS 201.

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 will continue from where CS 234 left off in studying computers as mathematical abstractions 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 the decidability of other problems. Topics in complexity theory include the classes P and NP, NP-completeness, and other fundamental complexity classes. The Department strongly recommends that students enrolling in this course have earned a grade of C or higher in Data Structures (CS 271). This course is classified as a theory elective.

Prerequisite(s): CS 145 or one from CS 109-114, and MATH 300 or CS 234 and CS 271.

Crosslisting: CS 334.

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

A student in good standing may work intensively in areas of special interest under the Directed Study plan. A Directed Study is appropriate when, under the guidance of a faculty member, a student wants to explore a subject more fully than is possible in a regular course or to study a subject not covered in the regular curriculum. A Directed Study should not normally duplicate a course that is regularly offered. Directed Studies are normally taken for 3 or 4 credits. A one-semester Directed Study is limited to a maximum of 4 credit hours. Note: Directed Studies may not be used to fulfill General Education requirements.

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

A student in good standing may work intensively in areas of special interest under the Directed Study plan. A Directed Study is appropriate when, under the guidance of a faculty member, a student wants to explore a subject more fully than is possible in a regular course or to study a subject not covered in the regular curriculum. A Directed Study should not normally duplicate a course that is regularly offered. Directed Studies are normally taken for 3 or 4 credits. A one-semester Directed Study is limited to a maximum of 4 credit hours. Note: Directed Studies may not be used to fulfill General Education requirements.

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

Independent Study engages a student in the pursuit of clearly defined goals. In this effort, a student may employ skills and information developed in previous course experiences or may develop some mastery of new knowledge or skills. A proposal for an Independent Study project must be approved in advance by the faculty member who agrees to serve as the project advisor. Note: Independent Studies may not be used to fulfill General Education requirements.

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

Independent Study engages a student in the pursuit of clearly defined goals. In this effort, a student may employ skills and information developed in previous course experiences or may develop some mastery of new knowledge or skills. A proposal for an Independent Study project must be approved in advance by the faculty member who agrees to serve as the project advisor. Note: Independent Studies may not be used to fulfill General Education requirements.

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 several presentations during the semester on substantive, well-researched themes appropriate to their status in their major.

Prerequisite(s): MATH 300 or CS 271.

Corequisite(s): a 300-level or higher mathematics or computer science course.

Crosslisting: CS 395.

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 213 and 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.

Prerequisite(s): MATH 213 and MATH 300.

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.

Prerequisite(s): MATH 213 and MATH 300.

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): CS 145 or one from CS 109-114, and MATH 213, and MATH 220 or DA 220 or MATH 221 or ML 221.

Crosslisting: CS 337.

MATH 420 - Statistical Modeling (4 Credit Hours)

This course is a survey of advanced statistical modeling techniques, building off the models studied in Applied Statistics, and introducing models based on more sophisticated mathematical structures such as matrices. The course emphasizes how to create, fit, test, and use statistical models in a variety of situations, and how to verify the conditions required for each model introduced. Course topics will be demonstrated via real-world applications, using the statistical computing language R to carry out the necessary computations. The selection of statistical models will depend on the instructor, but the course will contain elementary time series analysis regardless of who is teaching it. Other possible models include hierarchical linear models, longitudinal models, mixed models, Bayesian models, non-parametric models, statistical learning and data mining models, and principal component analysis.

Prerequisite(s): CS 145 or one from CS 109-114 and MATH 213, and MATH 220 or DA 220 or MATH 221 or ML 221.

MATH 421 - Bayesian Statistics (4 Credit Hours)

Statistics is the science of turning data into informed conclusions. The Bayesian approach to statistics uses probability theory to model both the uncertainty inherent in sampling from a population, and also our own degree of uncertainty in the answers we expect to get. That is, we begin with a prior distribution representing what we know or believe about a population, then we integrate new data that arrives into this distribution, resulting in a posterior distribution representing our updated beliefs. With this framework, an analysis can be informed by all past analyses on the topic (even if they analyzed other datasets), and we can construct an iterative approach that converges to the truth. This course introduces the Bayesian perspective, Bayesian approaches to statistical models you have seen in other classes, and new models that are only possible in the Bayesian approach. Course topics are demonstrated via real-world applications.

Prerequisite(s): CS 145 or one from CS 109-114 and MATH 145 and MATH 220 or DA 220 or MATH 221 or ML 221.

MATH 422 - Time Series Analysis (4 Credit Hours)

Time series data is data that is sampled at several points in time. Such data is best analyzed with the tools of time series analysis. This course introduces numerous examples of, and exploratory data analysis techniques for, time series data. The course then covers the essential time series models, including ARIMA, SRIMA, Spectral Analysis, GARCH models, and the GLMM. We learn to fit these models via maximum likelihood estimation and numerical analysis techniques, how to verify the conditions required by each of the models, how to interpret the output of each model, and how to use the models for forecasting. After a detailed study of time series analysis, we will sample additional statistical modeling techniques in the latter half of the course. Course topics are demonstrated via real-world applications, using the statistical computing language R to carry out the necessary computations.

Prerequisite(s): CS 145 or one from CS 109-114 and MATH 213, and MATH 220 or DA 220 or MATH 221 or ML 221.

MATH 423 - Statistical Theory (4 Credit Hours)

Statistical theory will provide an introduction to the mathematical ideas underlying multivariate statistical models, point estimation, hypothesis testing, and decision theory. In addition to providing a basis for understanding the theoretical underpinnings of statistics, the course will also highlight applications. Special topics may include statistical machine learning, Monte Carlo methods, and large sample theory.

Prerequisite(s): CS 145 or one from CS 109-114 and MATH 213 and MATH 220 or DA 220 or MATH 221 or ML 221.

MATH 425 - Probabilistic Modeling (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): CS 145 or one from CS 109-114, and MATH 213, and MATH 220, or DA 220 or MATH 221 or ML 221.

MATH 426 - Statistical Modeling in Sports (4 Credit Hours)

In this course, students will use statistical modeling and machine learning techniques to evaluate and predict player performance, team performance, economic performance, and health performance in sports. Students will use these findings to determine desired strategies and outcomes for team performance and player evaluation. In performing this analysis, students will carefully consider the ways that confounding bias, overfitting, variable specification, and interpretability arise in sports problems. Through the application of statistical modeling methods, students will practice communicating results to stakeholders so that they can make the best decisions for the players, teams, and organizations.

Prerequisite(s): MATH 145 and MATH 220 or DA 220 or MATH 221 or ML 221 and CS 145 or one from CS 109-114.

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. This course also covers the basics of graph theory, and several randomized algorithms on graphs. Graphs are often used to mathematically model phenomena of interest to computer scientists, including the internet, social networks 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. The department strongly recommends that students enrolling in this course have earned a grade of C or higher in Data Structures (CS 271). This course is classified as a theory elective.

Prerequisite(s): MATH 300 and CS 271 or MATH 220 or DA 220 and CS 105 or CS 145 or one from CS 109-114.

Crosslisting: CS 335.

MATH 428 - Intermediate Machine Learning (4 Credit Hours)

This course offers a mathematical treatment of modern machine learning, building on prior exposure to linear models and optimization, emphasizing the analysis and implementation of core algorithms from both a theoretical and practical perspective. Emphasis is placed on connecting mathematical theory with algorithmic implementation and on developing a rigorous understanding of how and why modern machine learning techniques work. Topics include convex optimization, regularization techniques, kernel methods, Bayesian learning, support vector machines, dimensionality reduction, neural networks, and reinforcement learning.

Prerequisite(s): MATH 213 and ML 221 or MATH 221.

Crosslisting: ML 310.

MATH 430 - Fourier Analysis (4 Credit Hours)

A study of a widely used and applied subfield of advanced Linear Algebra and Calculus harnessing the power of orthogonal bases. 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 use the power of changing bases, differentiation, and integration 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): CS 145 or one from CS 109-114 and MATH 213, and MATH 220 or DA 220 or MATH 221 or ML 221.

MATH 433 - Mathematical Methods for Engineering (4 Credit Hours)

This course will provide a survey of mathematical methods pervasive in engineering. We will explore integral transforms of various kinds, including the Laplace and Fourier transforms, and their applications to solving ordinary and partial differential equations. We will also explore foundational concepts like vector calculus, Sturm-Liouville problems and orthogonal functions, and separation of variables for boundary value problems in partial differential equations. All of these will be presented in the context of real-world problems and applications.

Prerequisite(s): CS 145 or one from CS 109-114 and MATH 213 and DA 220 or MATH 220 or MATH 221 or ML 221.

MATH 434 - Advanced Differential Equations (4 Credit Hours)

Differential equations arise in many areas of study. Indeed, any parameter that changes with respect to time may be described using a differential equation. This course covers both analytic and qualitative approaches to differential equations, as well as numerical methods. Topics may also include applications in biology, circuit theory, and mechanics.

Prerequisite(s): CS 145 or one from CS 109-114, and MATH 213 and MATH 220 or DA 220 or MATH 221 or ML 221.

MATH 435 - Mathematical Modeling (4 Credit Hours)

A course in mathematical modeling, including linear and nonlinear optimization models, linear and nonlinear 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): MATH 213 and MATH 220 or DA 220 or MATH 221 or ML 221 and CS 145 or one from CS 109-114.

MATH 440 - Real Analysis (4 Credit Hours)

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

Prerequisite(s): MATH 213 and MATH 300.

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 213 and MATH 300.

MATH 447 - Complex Analysis (4 Credit Hours)

A study of the complex plane and functions of complex variables. Topics include analytic functions, derivatives, singularities, integrals, Taylor series, Laurent Series, conformal mappings, residue theory, analytic continuation, Cauchy-Riemann equations, Cauchy’s Theorem, the Cauchy Integral Formula, Picard’s Theorems, the Riemann Mapping Theorem, and Rouche's Theorem.

Prerequisite(s): MATH 213 and MATH 300.

MATH 451 - Senior Research (4 Credit Hours)

Students may enroll in Senior Research in their final year at Denison. Normally, Senior Research requires a major thesis, report, or project in the student's field of concentration and carries eight semester-hours of credit for the year. Typically, a final grade for a year-long Senior Research will not be assigned until the completion of the year-long Senior Research at the end of the second semester. Each semester of Senior Research is limited to a maximum of 4 credit hours. Note: Senior Research may not be used to fulfill General Education requirements.

MATH 452 - Senior Research (4 Credit Hours)

Students may enroll in Senior Research in their final year at Denison. Normally, Senior Research requires a major thesis, report, or project in the student's field of concentration and carries eight semester-hours of credit for the year. Typically, a final grade for a year-long Senior Research will not be assigned until the completion of the year-long Senior Research at the end of the second semester. Each semester of Senior Research is limited to a maximum of 4 credit hours. Note: Senior Research may not be used to fulfill General Education requirements.

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

Special topics offered at an advanced level not covered in regular courses.

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 395 or 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.