University of Dayton
Academic Catalog 2013-14

Mathematics

Joe Mashburn, Department Chairperson 
Paul Eloe, MAS and MFM Program Director
Rebecca J. Krakowski, MME 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).

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.

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

Course List

MTH 531Advanced Differential Equations3
MTH 535Partial Differential Equations3
Six additional hours of mathematics courses approved by the committee6
Total Hours12

Engineering Systems

Course List

EGM 503Introduction to Continuum Mechanics3
EGM 533Theory of Elasticity3
Six additional hours of engineering courses (of a mathematicsl nature) approved by the committee6
Total Hours12

Computational Systems

Course List

MTH 555Numerical Analysis I3
MTH 556Numerical Analysis II3
Six additional hours of computer science courses approved by the committee6
Total Hours12

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.

Mathematics Education (MME)

The MME program has been developed to meet the professional needs of high school mathematics teachers. Ohio Department of Education licensure guidelines now require that all K-12 grade teachers complete a master's degree program in their content area or general education in order to maintain a provisional teaching license for a second renewal. The Department of Mathematics has created the MME to meet this requirement, and has been designed to address issues that are especially important to high school mathematics educators.

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; students' continued growth as leaders in education; an emphasis on the latest technological advances - both computer-based and using hand-held graphing utilities; students will consistently experience "best practices" as 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 classses, three graduate semester credit 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.

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

Course List

MTH 556Numerical Analysis3
MTH 563Computational Finance3
MTH 558Financial Math I3
MTH 559Financial Math II3
MBA 621Fin Deriva&Risk Mgt3
Total Hours15

Certificate in Statistical Finance

Course List

MTH 543Linear Models3
or ENM 501 Appl Engr Statistics
MTH 544Time Series3
or ENM 530 Engineering Economy
MTH 563Computational Finance3
MTH 558Financial Math I3
MTH 559Financial Math II3
Total Hours15

Certificate in Financial Risk Management

Course List

MBA 621Fin Deriva&Risk Mgt3
MBA 628Fixd Income Analysis3
MTH 558Financial Math I3
MTH 559Financial Math II3
MTH 563Computational Finance3
Total Hours15

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.

Master of Financial Mathematics (fim)

Course List

MBA 620APrin of Corp Fin Mgt1.5
MBA 620BPrn Crp Invt&Ast Mgt1.5
MTH 535Partial Diff Equatns **3
MTH 541Mathematics Clinic3
MTH 544Time Series3
MTH 556Numerical Analysis3
MTH 558Financial Math I3
MTH 559Financial Math II3
MBA 621Fin Deriva&Risk Mgt3
or MTH 557 Fincl Drvtvs&Rsk Mgt
MTH 563Computational Finance3
Select four courses: ***12
Database Mgt Sys I
Database Mgt Sys II
Investmnt&Fin Markts
Fixd Income Analysis
Adv Diffrntl Equatns
Difference Equations
Linear Models
Stat for Exprimntrs
Methods Mathmtl Phys
Meth of Applied Math
Linear Algebra
Fourier Analysis
Total Hours39

 

Footnotes

*

 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. 

***

 ECO 441 and FIN 480 can be taken for graduate credit.

Master of Mathematics Education (mme)


 

Course List

EDT 500 Models of Teaching 3
EDT 502 Philosphcl Study-Edu 3
EDT 650 Prf Dev-Teach Ldrs 3
MTH 512 Geom for Sec Tchrs 3
MTH 513 Alg for SecTeachers 3
MTH 514 Adv Math Sec Tchrs 3
MTH 515 Gphs&com - Soc Tchrs 3
MTH 516 Ap Lin Abs Alg Sc Tr 3
MTH 517 Res Meth in Mth Ed 3
MTH 541 Mathematics Clinic 3
Total Hours 30

Master of Science in Applied Mathematics (mas)

MTH 404Complex Variables3
or MTH 525 Complex Variables I
MTH 430Real Analysis3
MTH 531Adv Diffrntl Equatns3
or MTH 535 Partial Diff Equatns
MTH 541Mathematics Clinic3
MTH 555Numerical Analysis3
or MTH 556 Numerical Analysis
MTH 565Linear Algebra3
Select five of the following: *15
Statistical Infer
Statistical Infer
Real Variables
Real Variables
Complex Variables I
Complex Variables II
Difference Equations
Mathematicl Modelng
Mathematics Clinic
Linear Models
Special Functions
Stat for Exprimntrs
Methods Mathmtl Phys
Meth of Applied Math
Modern Algebra I
Modern Algebra II
Linear Algebra
Topology I
Topology II
Functional Analysis
Differential Geomtry
Vector&Tensor Anly
Fourier Analysis
Topics In Math
Total Hours33

 

Footnotes

*

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


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 302 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.