# Mathematics

- Master of Science in Applied Mathematics
- Master of Financial Mathematics
- Master of Mathematics Education
- Certificate, Computational Finance
- Certificate, Financial Risk Management
- Certificate, Statistical Finance

Joe Mashburn, Department Chairperson

Paul Eloe, Master of Science in Applied Mathematics and Master of Financial Mathematics Program Director

Rebecca J. Krakowski, Mathematics Education Program Director

The Department of Mathematics offers three masters degrees, the Master of Science in Applied Mathematics (MAS), the Master of Financial Mathematics (MFM) and the Master of Mathematics Education (MME).

Master of Science in Applied Mathematics (MAS)

The MAS program is interdisciplinary in nature. A plan of study may include up to a four-course concentration in computer science, engineering, or business for students with appropriate backgrounds. The primary objective of the program in applied mathematics is to train students to do professional work in the applications of mathematics. The program provides a background in mathematical, numerical, and statistical analyses and students will gain valuable experience in modeling and computation. Students will have the opportunity to work on a semester or year-long project known as the Mathematics Clinic project.

The program strives to offer an individualized plan of study that meets the needs and career goals of the student. This is achieved by offering a core of courses blending analysis, linear algebra, modeling, and numerical analysis in the Department of Mathematics. The student, with departmental approval, will select a four-course concentration. The Mathematics Clinic project, the capstone requirement, is a research project in which the student applies mathematical, numerical, or statistical modeling methods to a problem related to the student's four-course concentration. The Mathematics Clinic project can be a team project and can involve faculty members from several departments.

An individualized degree program consists of courses satisfying the five core areas, an area of concentration, and electives. The program is approved by the student's committee and program director, and is intended to satisfy the specific needs and interests of the individual. Any core course that is already part of the student's academic background may be replaced with an elective consistent with the other requirements of the program.

MTH 404 | Complex Variables | 3 |

or MTH 525 | Complex Variables I | |

MTH 430 | Real Analysis | 3 |

MTH 531 | Advanced Differential Equations | 3 |

or MTH 535 | Partial Differential Equations | |

MTH 541 | Mathematics Clinic | 3 |

MTH 555 | Numerical Analysis I | 3 |

or MTH 556 | Numerical Analysis II | |

MTH 565 | Linear Algebra | 3 |

Select five of the following: ^{*} | 15 | |

Statistical Inference | ||

Statistical Inference | ||

Real Variables | ||

Real Variables | ||

Complex Variables I | ||

Complex Variables II | ||

Difference Equations & Applications | ||

Mathematical Modeling | ||

Mathematics Clinic | ||

Linear Models | ||

Special Functions | ||

Statistics for Experimenters | ||

Methods of Mathematical Physics | ||

Methods of Applied Mathematics | ||

Modern Algebra I | ||

Modern Algebra II | ||

Linear Algebra | ||

Combined Designs Theory | ||

Topology | ||

Topology II | ||

Functional Analysis | ||

Differential Geometry | ||

Vector & Tensor Analysis | ||

Discrete & Continuous Fourier Analysis | ||

Topics in Mathematics | ||

Total Hours | 33 |

Footnotes

* | At most, 6 hours of approved 400-level courses may be part of the student's program. |

To satisfy the requirement of an area of concentration, a student will be required to take 12 semester hours of 500-level coursework in the selected area of concentration. Examples of areas of concentration include (but are not limited to):

**Differential Systems**

MTH 531 | Advanced Differential Equations | 3 |

MTH 535 | Partial Differential Equations | 3 |

Six additional hours of mathematics courses approved by the committee | 6 | |

Total Hours | 12 |

**Engineering Systems**

EGM 503 | Introduction to Continuum Mechanics | 3 |

EGM 533 | Theory of Elasticity | 3 |

Six additional hours of engineering courses (of a mathematicsl nature) approved by the committee | 6 | |

Total Hours | 12 |

**Computational Systems**

MTH 555 | Numerical Analysis I | 3 |

MTH 556 | Numerical Analysis II | 3 |

Six additional hours of computer science courses approved by the committee | 6 | |

Total Hours | 12 |

## Master of Financial Mathematics (FIM)

The MFM is a professional program in quantitative methods in finance to support a growing local and regional market in financial services. It is offered in cooperation with the Department of Economics and Finance. The program integrates statistics, computation and modeling with training in the professional domain and graduates will find employment opportunities in the banking, insurance and financial trading industries. A plan of study includes six core courses that include coursework in the MBA program, and four electives courses, selected, in consultation with a faculty advisor, from a broad set of electives from Mathematics, MBA, Management Science and Computer Science. There are two introductory courses, one in partial differential equations and one in principles of finance; one or both of these courses can be waived for students with appropriate background in mathematics or finance.

As with the MAS program, the MFM program requires a capstone experience of a Mathematics Clinic project. Teams of students will report to a faculty member and work on a project that is posed by the financial industry.

MBA 620A | Principles of Corporate Financial Management | 1.5 |

MBA 620B | Principles of Corporate Investment & Assets Management | 1.5 |

MTH 535 | Partial Differential Equations ^{**} | 3 |

MTH 541 | Mathematics Clinic | 3 |

MTH 544 | Time Series | 3 |

MTH 556 | Numerical Analysis II | 3 |

MTH 558 | Financial Mathematics I-Discrete Model | 3 |

MTH 559 | Financial Mathematics II-Continuous Model | 3 |

MBA 627 | Financial Derivatives & Risk Management | 3 |

or MTH 557 | Financial Derivatives & Risk Management | |

MTH 563 | Computational Finance | 3 |

Select four courses: ^{***} | 12 | |

Database Management Systems | ||

Database Management Systems II | ||

Investments & Finance Marketing | ||

Fixed Income Analysis | ||

Advanced Differential Equations | ||

Difference Equations & Applications | ||

Linear Models | ||

Statistics for Experimenters | ||

Methods of Mathematical Physics | ||

Methods of Applied Mathematics | ||

Linear Algebra | ||

Combined Designs Theory | ||

Discrete & Continuous Fourier Analysis | ||

Total Hours | 39 |

* | This is a required introductory finance-related course. It can be replaced for students with sufficient background. |

** | This is a required introductory mathematics course. It can be replaced with an appropriate elective for students with sufficient background. |

*** |

## Master of Mathematics Education (MME)

The MME program was developed primarily to meet the professional needs of high school mathematics teachers. Although Ohio Department of Education licensure guidelines no longer require all K-12 grade teachers complete a master's degree program in their content area or general education, the MME completed in its entirety or in part will certainly satisfy ongoing professional development requirements for license renewal, and has been designed by the Department of Mathematics to address issues that are especially important to high school mathematics educators. The MME may also be of interest to upper grades middle school mathematics teachers, provided they have the AYA 7-12 license in mathematics. (Note: The MME is NOT an initial licensure program.)

Key features of the MME include: curriculum that focuses on pedagogical content knowledge - the special knowledge that distinguishes the mathematics knowledge of teachers from that of mathematicians; student development of a stronger mathematics content knowledge, as well as the ability and opportunity to apply this knowledge to the 9-12 grade curriculum; introduction to major research issues and both quantitative and qualitative methods in mathematics education; continued growth as leaders in education; an emphasis on the latest technological advances - both computer-based and using hand-held graphing utilities; consistent experience of "best practices" modeled by program faculty whose area of expertise is mathematics education.

This is primarily a summer program that offers a solid base in the teaching of secondary school mathematics. The curriculum includes both mathematics and education coursework consisting of 10 classes, three graduate semester hours each, that may be completed over the course of three summers, with minimal requirements during the regular school year.

As is the case with other graduate programs within the Department of Mathematics, the MME program requires a capstone experience of a Mathematics Clinic project. Each student will work with a faculty member to design and implement an action research project in mathematics education. A "journal ready" report will be required, as well as a presentation of their findings in one of our departmental colloquia.

EDT 500 | Models of Teaching | 3 |

EDT 502 | Philosophical Studies in Education | 3 |

EDT 650 | Professional Development of Teacher Leaders | 3 |

MTH 512 | Geometry for Secondary Teachers | 3 |

MTH 513 | Algebra for Secondary Teachers | 3 |

MTH 514 | Advanced Mathematics for Secondary Teachers | 3 |

MTH 515 | Applications of Graph Theory & Combinatorics in Modern Mathematics | 3 |

MTH 516 | Applications of Linear & Abstract Algebra in Modern Education | 3 |

MTH 517 | Research Methods & Issues in Mathematics Education | 3 |

MTH 541 | Mathematics Clinic | 3 |

Total Hours | 30 |

Certificate Programs

Certificate programs appeal to students who do not want to commit to the full MFM program. Upon successful completion of five courses focused on a specific set of concepts, a student will earn a post-baccalaureate certificate in that area. The certificate programs and the associated five courses are:

Certificate in Computational Finance (CFN)

Certificate Requirements:

MTH 556 | Numerical Analysis II | 3 |

MTH 563 | Computational Finance | 3 |

MTH 558 | Financial Mathematics I-Discrete Model | 3 |

MTH 559 | Financial Mathematics II-Continuous Model | 3 |

MBA 627 | Financial Derivatives & Risk Management | 3 |

Total Hours | 15 |

Certificate in Statistical Finance (STF)

Certificate Requirements:

MTH 543 | Linear Models | 3 |

or ENM 501 | Applied Engineering Statistics | |

MTH 544 | Time Series | 3 |

or ENM 530 | Engineering Economy | |

MTH 563 | Computational Finance | 3 |

MTH 558 | Financial Mathematics I-Discrete Model | 3 |

MTH 559 | Financial Mathematics II-Continuous Model | 3 |

Total Hours | 15 |

## Certificate in Financial Risk Management (FRM)

Certificate Requirements:

MBA 627 | Financial Derivatives & Risk Management | 3 |

MBA 628 | Fixed Income Analysis | 3 |

MTH 558 | Financial Mathematics I-Discrete Model | 3 |

MTH 559 | Financial Mathematics II-Continuous Model | 3 |

MTH 563 | Computational Finance | 3 |

Total Hours | 15 |

The certificate programs are designed as mini-programs in focus areas. Thus, each includes the capstone applied research experience of Mathematics Clinic.

**Entrance, performance, and exit standards**

Students seeking admission to the Certificate Programs will satisfy the entrance requirements to the MFM program. These are:

- Completion of a graduate application for admission to a certificate program at the University of Dayton
- Bachelor's degree in a science or technical area such as mathematics, physics, computer science, engineering, economics or finance, and at least a 3.0 GPA on a 4.0 scale
- Prerequisite mathematics coursework in calculus, differential equations, linear algebra, elementary probability and statistics
- Programming skills

Students applying for a Certificate must be enrolled in the Certificate program and must have completed the requirement of five courses with a minimum G.P.A. of 3.0.

Students cannot simultaneously be admitted to the Master of Financial Mathematics and one of the certificate programs. Students can be simultaneously enrolled in any other post-baccalaureate program at the University of Dayton and a certificate program. Students must meet the entrance standards of the Master of Financial Mathematics to gain admission to a certificate program. To learn more about the application process for admission to a certificate program, please contact the Department of Mathematics.

**Assistantships**

Financial assistance is available to qualified students through graduate teaching assistantships. A graduate assistant receives a stipend, tuition remission and health benefits. Most graduate assistants require two years to complete the requirements for a master's degree. Internships in the MFM program are recommended and the Department facilitates finding internship opportunities.

**Facilities**

Departmental PCs and the MATHSCI Computer Learning Environment are available for student use in conjunction with projects or coursework.

### Courses

**MTH 512. Geometry for Secondary Teachers. 3 Hours**

Investigation of traditional secondary school topics in Euclidean geometry, introduction to similar ideas in non-Euclidean spaces, examination of the impact of mathematics education research on the teaching and learning of geometry, and exploration of real-world applications. Extensive use of the dynamic software package The Geometer's Sketchpad® will also be incorporated into every aspect of the course. Topics to be explored may include transformations, symmetry, tessellations, centers of triangles (incenter, centroid, orthocenter, and circumcenter), similarity, coordinate geometry, and spherical or hyperbolic geometry.
Prerequisite(s): MTH 370 or permission of instructor.

**MTH 513. Algebra for Secondary Teachers. 3 Hours**

Investigation of traditional secondary school topics from introductory and advanced algebra courses, examination of appropriate use of manipulatives (e.g., algebra tiles) to explore algebraic concepts, integration of hand-held graphing technology and data collection devices in the study of algebra, and implications of research in mathematics education on the teaching and learning of algebra. Topics discussed in the course may include basic properties and mechanics of equations and functions, functions that model real-world phenomena, models for factoring polynomial expressions, and integration of physical science and mathematics.
Prerequisite(s): Permission of instructor.

**MTH 514. Advanced Mathematics for Secondary Teachers. 3 Hours**

Investigation of concepts related to trigonometry, analytic geometry, precalculus, and calculus; integration of appropriate uses of graphing technology and data collection devices to enhance students' understanding in their investigation of real-world examples; and implications of research in mathematics education on the teaching and learning of the concepts discussed in this course. A variety of topics that may be explored include: trigonometric functions and applications; rate of change in business, physics, and society; limits, continuity, and differentiability; and applications of area and volume.
Prerequisite(s): MTH 218 or permission of instructor.

**MTH 515. Applications of Graph Theory & Combinatorics in Modern Mathematics. 3 Hours**

An opportunity to study selected topics in graph theory and combinatorics in depth. Appropriate uses of computing technology will be included. Topics may include an introduction to circuits and graph coloring theorems, traveling salesperson problems, and sorting algorithms, problems, and methods in counting, networks, and finding winning strategies for Nim-type games.
Prerequisite(s): (MTH 367 or MTH 411) or permission of instructor.

**MTH 516. Applications of Linear & Abstract Algebra in Modern Education. 3 Hours**

Study of topics connected to real-world applications in both linear and abstract algebra, and an introduction to matrix operations with EXCEL and TI graphing technology. Topics discussed may include: introductory coding theory and cryptography; symmetry groups in mathematics, science, engineering, architecture, and art; permutation groups; linear programming problems and the simplex method; and Markov chains.
Prerequisite(s): (MTH 302, MTH 361) or permission of instructor.

**MTH 517. Research Methods & Issues in Mathematics Education. 3 Hours**

Review of related literature and research in education and mathematics education, and a study of key concepts necessary to analyze, evaluate, and conduct educational research. Application of both qualitative and quantitative research methods specifically related to the development of a research proposal. The focus on quantitative research methods provides ample opportunities to review fundamental concepts and properties of both descriptive and inferential statistics. Introduction to SAS or SPSS, both statistical programming packages appropriate for use in educational research, will be included in the course.
Prerequisite(s): (MTH 367 or MTH 412) or permission of instructor.

**MTH 519. Statistical Inference. 3 Hours**

Sample spaces, Borel fields, random variables, distribution theory, characteristic functions, exponential families, minimax and Bayes' procedures, sufficiency, efficiency, Rao-Blackwell theorem, Neyman-Pearson lemma, uniformly most powerful tests, multi-variate normal distributions.

**MTH 520. Statistical Inference. 3 Hours**

Sample spaces, Borel fields, random variables, distribution theory, characteristic functions, exponential families, minimax and Bayes' procedures, sufficiency, efficiency, Rao-Blackwell theorem, Neyman-Pearson lemma, uniformly most powerful tests, multi-variate normal distributions.

**MTH 521. Real Variables. 3 Hours**

The topology of the real line, continuity and differentiability, Riemann and Stieltjes integrals, Lebesgue measure and Lebesgue integral. Measure and integration over abstract spaces, Lp-spaces, signed measures, Jordan-Hahn decomposition, Radon-Nikodym theorem, Riesz representation theorem, and Fourier series.

**MTH 522. Real Variables. 3 Hours**

The topology of the real line, continuity and differentiability, Riemann and Stieltjes integrals, Lebesgue measure and Lebesgue integral. Measure and integration over abstract spaces, Lp-spaces, signed measures, Jordan-Hahn decomposition, Radon-Nikodym theorem, Riesz representation theorem, and Fourier series.

**MTH 525. Complex Variables I. 3 Hours**

Analytic functions, integration on paths, the general Cauchy theorem. Singularities, residues, inverse functions and other applications of the Cauchy theory.

**MTH 526. Complex Variables II. 3 Hours**

Infinite products, entire functions, the Riemann mapping theorem and other topics as time permits.
Prerequisite(s): MTH 525 or equivalent.

**MTH 527. Biostatistics. 3 Hours**

Introduction to statistical concepts and skills including probability theory and estimation, hypothesis tests of means and proportions for one or two samples using normal or t-distributions, regression and correlation, one- and two-way ANOVA, selected nonparametric tests.
Prerequisite(s): MTH 149 or MTH 169 or permission of instructor.

**MTH 531. Advanced Differential Equations. 3 Hours**

Existence and uniqueness theorems, linear equations and systems, self-adjoin systems, boundary value problems and basic nonlinear techniques.
Prerequisite(s): MTH 403 or equivalent.

**MTH 532. Difference Equations & Applications. 3 Hours**

The calculus of finite differences, first order equations, linear equations and systems, z-transform, stability, boundary value problems for nonlinear equations, Green's function, control theory and applications.

**MTH 535. Partial Differential Equations. 3 Hours**

Classification of partial differential equations; methods of solution for the wave equation, Laplace's equation, and the heat equation; applications.
Prerequisite(s): MTH 403 or equivalent.

**MTH 540. Mathematical Modeling. 3 Hours**

An introduction to the use of mathematical techniques and results in constructing and modifying models designed to describe and/or predict behavior of real-world situations.
Prerequisite(s): Permission of instructor.

**MTH 541. Mathematics Clinic. 3 Hours**

Student teams will be responsible for developing or modifying and testing a mathematical model designed for a particular purpose. Faculty guidance will be provided. May be repeated once for a maximum of 6 credit hours.
Prerequisite(s): Permission of department chairperson or program director.

**MTH 543. Linear Models. 3 Hours**

Least square techniques, lack of fit and pure error, correlation, matrix methods, F test, weighted least squares, examination of residuals, multiple regression, transformations and dummy variables, model building, ridge regression, stepwise regression, multiple regression applied to analysis of variance problems.
Prerequisite(s): MTH 368 or equivalent.

**MTH 544. Time Series. 3 Hours**

Estimation and elimination of trend and seasonal components; stationary time series, autocovariance, autocorrelation and partial autocorrelation functions; spectral analysis; modeling and forecasting with ARMA processes; nonstationary and seasonal time series.
Prerequisite(s): Courses in single and multivariate calculus; courses in statistics and probability; courses in linear algebra.

**MTH 545. Special Functions. 3 Hours**

The special functions arising from solutions of boundary value problems which are encountered in engineering and the physical sciences. Hypergeometric functions, Bessel functions, Legendre polynomials.
Prerequisite(s): MTH 403 or equivalent.

**MTH 547. Statistics for Experimenters. 3 Hours**

Covers those areas of design of experiments and analysis of quantitive data that are useful to anyone engaged in experimental work. Designed experiments using replication and blocking. Use of transformations. Applications of full and fractional factorial designs. Experimental design for developing quality into products using Taguchi methods.
Prerequisite(s): MTH 367 or equivalent.

**MTH 551. Methods of Mathematical Physics. 3 Hours**

Linear transformations and matrix theory, linear integral equations, calculus of variations, eigenvalue problems.
Prerequisite(s): MTH 403 or equivalent.

**MTH 552. Methods of Applied Mathematics. 3 Hours**

Dimensional analysis and scaling, regular and singular perturbation methods with boundary layer analysis, the stability and bifurcation of equilibrium solutions, other asymptotic methods.
Prerequisite(s): MTH 403 or equivalent.

**MTH 555. Numerical Analysis I. 3 Hours**

Solutions of nonlinear equations, Newton's methods, fixed point methods, solutions of linear equations, LU decomposition, iterative improvement, QR decomposition, SV decomposition.
Prerequisite(s): (CPS 132 or CPS 150) or equivalent; MTH 310 or equivalent.

**MTH 556. Numerical Analysis II. 3 Hours**

Interpolating functions, numerical differentiation, numerical integration including Gaussian quadrature, numerical solutions of differential equations.
Prerequisite(s): (CPS 132 or CPS 150) or equivalent; MTH 219 or equivalent.

**MTH 557. Financial Derivatives & Risk Management. 3 Hours**

This course provides a theoretical foundation for the pricing of contingent claims and for designing risk-management strategies. It covers option pricing models, hedging techniques, and trading strategies. It also includes portfolio insurance, value-at-risk measure, multistep binomial trees to value American options, interest rate options, and other exotic options.
Prerequisite(s): MBA 620.

**MTH 558. Financial Mathematics I-Discrete Model. 3 Hours**

Discrete methods in financial mathematics. Topics include introduction to financial derivatives, discrete probability theory, discrete stochastic processes (Markov chain, random walk, and Martingale), binomial tree models for derivative pricing and computational methods (European and American options), forward and futures, and interest rate derivatives.
Prerequisite(s): MTH 411 or equivalent.

**MTH 559. Financial Mathematics II-Continuous Model. 3 Hours**

Continuous methods in financial mathematics. Topics include review of continuous probability theory, Ito's Lemma, the Black-Scholes partial differential equation, option pricing via partial differential equations, analysis of exotic options, local and stochastic volatility models, American options, fixed income and stopping time. Computational methods are introduced.
Prerequisite(s): MTH 558.

**MTH 561. Modern Algebra I. 3 Hours**

Groups, rings, integral domains and fields; extensions of rings and fields; polynomial rings and factorization theory in integral domains; modules and ideals.

**MTH 562. Modern Algebra II. 3 Hours**

Finite and infinite field extensions, algebraic closure, constructible numbers and solvability by use of radicals, Galois theory, and selected advanced topics.
Prerequisite(s): MTH 561.

**MTH 563. Computational Finance. 3 Hours**

The purpose of this course is to introduce students to numerical methods and various financial problems that include portfolio optimization and derivatives valuation that can be tackled by numerical methods. Students will learn the basics of numerical analysis, optimization methods, monte carlo simulations and finite difference methods for solving PDEs.
Prerequisite(s): MBA 620 or permission of instructor.

**MTH 565. Linear Algebra. 3 Hours**

Vector spaces, linear transformations and matrices; determinants, inner product spaces, invariant direct-sum decomposition and the Jordan canonical form.

**MTH 567. Combined Designs Theory. 3 Hours**

Latin squares, mutally orthogonal Latin squares, orthogonal and perpendicular arrays, Steiner triple systems, block designs, difference sets and finite geometries.
Prerequisite(s): MTH 308 or instructor's permission.

**MTH 571. Topology. 3 Hours**

An axiomatic treatment of the concept of a topological space; bases and subbases; connectedness, compactness; continuity, homeomorphisms, separation axioms and countability axioms; convergence in topological spaces.

**MTH 572. Topology II. 3 Hours**

Compactification theory, para-compactness and metrizability theorems, uniform spaces, function spaces, and other advanced topics of current interest.
Prerequisite(s): MTH 571 or equivalent.

**MTH 573. Functional Analysis. 3 Hours**

The study of linear metric spaces with emphasis on Banach and Hilbert spaces. The Hahn-Banach theorem, the Banach fixed point theorem, and their consequences. Approximations and other selected advanced topics.

**MTH 575. Differential Geometry. 3 Hours**

Vector and tensor algebra; covariant differentiation. An introduction to the classical theory of curves and surfaces treated by means of vector and tensor analysis.

**MTH 582. Vector & Tensor Analysis. 3 Hours**

The differential and integral calculus of scalar and vector fields with emphasis on properties invariant under transformations to curvilinear coordinate systems. An introduction to tensor analysis via Cartesian tensors and then more general tensors. Derivation of the divergence, gradient, and curl in generalized coordinates.
Prerequisite(s): (MTH 218, MTH 302) or equivalent.

**MTH 583. Discrete & Continuous Fourier Analysis. 3 Hours**

Fourier representations of complex-valued functions, rules for finding Fourier transforms, mathematical operators associated with Fourier analysis, fast algorithms, wavelet analysis, selected applications.
Prerequisite(s): (MTH 219 or MTH 319) or equivalent; MTH 302 or equivalent.

**MTH 590. Topics in Mathematics. 1-6 Hours**

This course, given upon appropriate occasions, deals with specialized material not covered in the regular courses. May be taken more than once as topics change. Prerequisite(s): Permission of advisor.