## Graduate Studies

### Mathematics

2018-2019

Admission | Courses | Program | Requirements | Certificate | Integrated Program | Profile

**Interim Chairperson:** Victoria Baramidze

**Committee Chairperson: **Dinesh Ekanayake

**Office:** Morgan Hall 476

**Telephone:** (309) 298-1054 or (309) 298-2467 **Fax:** (309) 298-1857

**E-mail**: mathphil@wiu.edu

**Website:** wiu.edu/mathphil

**Location of Program Offering: **Macomb

#### Graduate Faculty

Professors

- Samson A. Adeleke, Ph.D., Johns Hopkins University
- Fedor Andreev, Ph.D., St. Petersburg Steklov Mathematical Institute
- Victoria Baramidze, Ph.D., University of Georgia–Athens
- Susan Martinelli-Fernandez, Ph.D., University of Chicago
- James R. Olsen, Ph.D., University of Northern Colorado
- Gordon Pettit, Ph.D., University of Notre Dame
- Christopher A. Pynes, Ph.D., Florida State University

#### Associate Professors

- Amy Ekanayake, Ph.D., Texas Tech University
- Dinesh Ekanayake, Ph.D., Texas Tech University
- Douglas LaFountain, Ph.D., University at Buffalo, The State University of New York
- Robert Mann, Ph.D., University of Nebraska-Lincoln
- Seyfi Turkelli, Ph.D., University of Wisconsin-Madison
- Mei Yang, Ph.D., University of Canterbury

#### Associate Graduate Faculty

Professors

- J. Thomas Blackford, Ph.D., Ohio State University
- Clifton Ealy, Ph.D., University of California-Berkeley
- Kimberly Hartweg, Ph.D., University of Iowa
- David Haugen, Ph.D., University of Washington
- Brian Powell, Ph.D., University of Virginia
- Feridun Tasdan, Ph.D., Western Michigan University

#### Associate Professors

- John Chisholm, Ph.D., University of Wisconsin
- Rumen Dimitrov, Ph.D., George Washington University
- Elizabeth Hansen, Ph.D., University of Iowa
- Jana Marikova, Ph.D., University of Illinois at Urbana-Champaign
- Boris Petracovici, Ph.D., University of Illinois
- Lia Petracovici, Ph.D., University of Illinois

#### Assistant Professors

- Susan Brooks, Ph.D., University of Iowa
- Mojtaba Moniri, Ph.D., University of Minnesota-Twin Cities

#### Program Description

The graduate program in the Department of Mathematics and Philosophy prepares students for needed professions in the region and nationwide. The program provides students with a solid graduate level training in the central and fundamental methods of continuous and discrete mathematics. Both the theoretical framework and the applications of these methods will be covered in the core courses. The 500-level core courses have a significant lean toward applications but theory is present; while the 600-level core courses have a significant lean toward theory and mathematical foundation but applications are not abandoned.

**Integrated Baccalaureate and Master’s Degree Program**

Go to Integrated Programs for details and program offerings.

#### Admission Requirements

Students entering the program should normally have completed an undergraduate degree with a GPA of at least 2.75 on a 4-point scale (or a GPA of at least 3.0 for the final two years of undergraduate coursework), including coursework equivalent to a major in mathematics. Students who do not fulfill the coursework requirements may be admitted at the discretion of the Departmental Graduate Committee with admission usually conditional upon the student filling specified deficiencies. Students who do not fulfill the GPA requirements will not be considered for admission unless they have demonstrated sufficient competence in mathematics coursework (as determined by the Department Graduate Committee) and complete the general part of the Graduate Record Examination (GRE) with Quantitative Reasoning score above the 60^{th} percentile. All applicants are strongly encouraged to take the general part of the GRE, particularly if applying for an assistantship. All the applicants must also submit three letters of recommendation attesting to the applicant’s academic potential at the graduate level.

#### Degree Requirements

Degree requirements of this 36-semester hour program consist of 18 semester hours of core courses, 6 semester hours of mathematics directed electives, and 12 semester hours of focus area courses that allow for focus in a single area in mathematics, or another area of study outside the Department of Mathematics and Philosophy, as sanctioned by the Department Graduate Committee. For example, the focus area courses may be in applied mathematics, Ph.D. pursuit, statistics, teaching of mathematics, biology, business, chemistry, computer science, decision science, economics, environmental science, finance, or physics. Focus area courses (12 semester hours) will share a common thread with the first 6 semester hours taken as MATH 599 and/or MATH 596 or as directed electives from another department. The second 6 semester hours of the focus area courses may also be earned through directed electives; or in special topics (MATH 699) and/or thesis (MATH 600), and/or project (MATH 601), and/or internship (MATH 602).

**I. First-Year Core Courses: 9 s.h.**

MATH 551 Methods of Classical Analysis (3)

MATH 552 Scientific Computing (3)

STAT 553 Applied Statistical Methods (3)

**II. Second-Year Core Courses: 9 s.h.**

MATH 651 Elements of Modern Analysis (3)

MATH 652 Computational Differential Equations (3)

STAT 653 Elements of Statistical Inference (3)

**III. Focus Courses: 12 s.h.**

The focus courses must be approved by the Department Graduate Committee. Students must select 6 s.h. from A. and 6 s.h. from B.

A. MATH 599 Special Topics (1–6), and/or

MATH 596 Project in Applied Mathematics (3–6)

OR

Directed Electives from any department but in a single focus area (6)

B. MATH 699 Advanced Special Topics (3–6), and/or

MATH 600 Thesis (3), and/or

MATH 601 Advanced Project in Applied Mathematics (3–6), and/or

MATH 602 Internship in Applied Mathematics (3–6)

OR

Directed Electives from any department but in the same single focus area as selected above in A.

**IV. Directed Electives: 6 s.h.**

Must be in mathematics or statistics.

**TOTAL PROGRAM: 36 s.h.
**

#### Course Descriptions

##### Mathematics (MATH)

**402G Investigations in School Geometry. (2)** A conceptual development of geometry through the investigation of geometric relationships and informal understandings leading to formal deductions. Middle and junior high school emphasis. *Prerequisite: MATH 123 or MATH 128 or equivalent.*

**406G Problem Solving and the History of Mathematics. (3)** Various problems, their solutions, related mathematical concepts and their historical significance are analyzed through investigation of classic problems and their connection to middle school mathematics. Contributions by Archimedes, Descartes, Eratosthenes, Euler, Gauss, Pascal, Pythagoras and others are studied. *Prerequisite: MATH 123 or MATH 128 or equivalent.*

**407G Number Theory Concepts in School Mathematics. (3)** Divisibility, prime numbers, perfect numbers, modular arithmetic, linear Diophantine equations, and related topics. Open only to students majoring in an elementary education program. *Prerequisite: MATH 123 or MATH 128 or equivalent.*

**408G Mathematical Topics and Technology for Middle School. (3) **The study of programming, algorithms, and technology resources to investigate concepts and connections in the content areas of middle school mathematics. *Prerequisite: MATH 123 or MATH 128 or equivalent.*

**421G Abstract Algebra. (3)** An introduction to the basic properties of groups, rings, and fields. *Prerequisite: MATH 341.*

**424G Advanced Linear Algebra. (3)**Matrix algebra, vector spaces, linear independence, basis, linear transformations, canonical forms, inner product spaces.

*Prerequisite: MATH 311 & MATH 341, or equivalent*

**430G Multivariable Calculus. (3)** The algebra of functions, continuity, differentiation and integration of n‑place functions, and related topics. *Prerequisites: MATH 231 and 311.*

**435G Introduction to Real Variables I. (3)**Topology of the real line, sequences, limits, and series. Rigorous introduction to the study of one-variable functions, continuity and differentiability, based on the epsilon-delta method.

*Prerequisites: MATH 231 and MATH 341.*

**436G Introduction to Real Variables II. (3) **A continuation of Math 435. *Prerequisite: MATH 435.*

**441G Mathematical Logic. (3) **Introduction to some of the principal topics of mathematical logic. Topics include Propositional Calculus, Quantification Theory, the Completeness Theorem, Formal Theories, Models of Theories and Recursion Theory. *Prerequisite: MATH 341. *

**461G Introductory Topology. (3)** Basic properties of topological spaces. Open and closed sets, compactness, the intermediate value theorem, metric spaces, completeness, and uniform continuity. *Prerequisite: MATH 341.*

**481G Numerical Analysis I. (3)**A survey of current methods in numerical analysis. Error analysis, solution of nonlinear equations and systems of linear equations, polynomial interpolation and approximations, and related topics.

*Prerequisites: Math 231, 280, and 311, or permission of the instructor.*

**483G Biomathematics. (3) **Mathematical modeling of biological systems. Derivation and study of continuous time Markov chain models and corresponding ordinary differential equation models. *Prerequisites: MATH 134 and 311, and either STAT 276 or 471; or equivalent.*

**488G Models in Applied Mathematics. (3)** Theory and computer exploration of mathematical models using difference equations, differential equations, and dynamical systems. Applications from the sciences. *Prerequisites: MATH 231, MATH 311, and one of CS 211 and CS 212 or CS 225 or equivalent, or CS 240, or permission of the instructor.*

**500 Teaching of Elementary Mathematics. (3)** A study of current trends and problems in the teaching of elementary and junior high school mathematics. *Prerequisite: Permission of the instructor.*

**502 Algebraic Mathematical Modeling for Middle School Teachers. (3)** Case study analyses of mathematical models of real-world problems, using algebraic, graphical, and numerical representations. Students will use algebra and technology to model, analyze, and solve real-world problems.

**503 Methods of Teaching Secondary School Mathematics. (3)** A study of current trends and problems in the teaching of secondary school mathematics. *Prerequisite: Permission of the instructor.*

**504 Research in Secondary Mathematics Education. (3)** A survey, evaluation, and application of recent research relative to the teaching of secondary school math. *Prerequisite: Permission of the instructor.*

**505 The Teaching of Mathematics in Middle Grades and Junior High. (3)** A study of teaching strategies and current trends in mathematics as they apply to the curriculum of the middle school and the junior high school. *Prerequisites: MATH 260 (C grade or better) or equivalent.*

**508 Special Topics in Elementary Mathematics. (3, repeatable to 15)** Topics will be available on demand in the areas of probability, statistics, computer science, number theory, and history of math. *Prerequisite: Permission of the instructor.*

**509 Standards and Assessment in School Mathematics. (3)** An analysis of the current state and national standards for school mathematics and their corresponding assessments. Other assessment instruments and strategies for implementing the standards and improving student achievement for all learners will also be investigated. *Prerequisites: Teacher certification.*

**511 Modern Geometry. (3)**Topics to be chosen to reflect current trends in geometry.

*Prerequisite: Permission of the instructor.*

**521 Algebra. (3)** An introduction to higher algebra. Topics to be included are groups, homomorphisms, Sylow theorems, rings and ideals, fields, field extensions, and Galois theory. *Prerequisite: MATH 424 or permission of the instructor.*

**533 Complex Variables. (3)** Topics to be studied include the topology of the complex plane, analytic functions, complex integration, and singularities. *Prerequisite: MATH 436 or permission of the instructor.*

**536 Ordinary Differential Equations. (3)** The initial value problem, existence and uniqueness theorems, linear systems, asymptotic behavior of solutions, two‑dimensional systems. *Prerequisites: MATH 333 and 435, or permission of the instructor.*

**550 Workshop in School Mathematics. (1–6, repeatable)** (Degree candidates may receive credit toward program requirements only with the permission of the student's Graduate Committee.) Workshops focusing on specific topics may be organized as required to meet the identified needs and interests of in-service teachers or specific school districts.

**551 Methods of Classical Analysis. (3)**Introduction to complex and multivariable analysis with a significant lean toward applications. Topics include geometry of Rⁿ, differential calculus in Rⁿ, line and surface integrals; conformal mappings, complex integration, Laurent series, calculus of residues; and applications.

*Prerequisites: MATH 231 and MATH 311, or equivalents.*

**552 Scientific Computing. (3) **Design, analysis, and MATLAB or Mathematica implementation of algorithms for solving problems of continuous mathematics involving linear and nonlinear systems of equations, interpolation and approximation, numerical differentiation and integration, and ordinary differential equations with a significant lean toward applications. *Prerequisites: MATH 311 and MATH 333, or equivalents.*

**554 Methods of Symmetry in Algebra, Geometry, and Topology. (3) **A study of symmetry in algebra, geometry, and topology with a significant lean toward applications. Topics of study include group of Euclidean transformations, symmetries of planar sets, topological classification of compact surfaces, crystallographic patterns and classification of their symmetry groups. *Prerequisite: MATH 424 or permission of the instructor.*

**560 Advanced Topology. (3)** Product and quotient spaces, path-connectedness, local compactness, homotopy, fundamental group. Additional topics may include Baire category, function spaces, Brouwer Fixed Point Theorem. *Prerequisites: MATH 421 and MATH 461, or permission of the instructor.*

**581 Approximation Theory. (3)**The theory behind numerical algorithms. Remainder theory, convergence theorems, best approximation in various norms, the theory of matrices in numerical analysis including the eigenvalue problem.

*Prerequisites: MATH 435 and 481, or permission of the instructor.*

**583 Nonlinear Optimization. (3)** Unconstrained optimization; equality constrained optimization; convex optimization; optimality conditions; algorithms and applications using software such as Mathematica. *Prerequisites: MATH 481 and 424, or permission of the instructor.*

**589 Mathematical Modeling. (1–3)** A development of the group approach in applications of techniques used in applied mathematics, numerical analysis, operations research, and statistics to real problems from other disciplines. May be repeated up to six hours. *Prerequisite: Permission of the instructor. *

**590 Independent Study. (1–3, repeatable to 6)**
*Prerequisite: Approval of the Department Chair.*

**596 Project in Applied Mathematics. (3, repeatable to 6) **A project in applied mathematics or statistics, or with a professional institution, which will be presented in a final paper or portfolio, demonstrating entry into an applied mathematics field. Graded S/U. *Prerequisite: Permission of the Graduate Committee.*

**599 Special Topics. (1–3, repeatable to 6 under different titles)** Special topics in mathematics or statistics with a lean towards application. May be repeated with a change in topic. *Prerequisite: Permission of the instructor.*

**600 Thesis. (3)**The thesis may be either expository, historical, critical, or original and must be approved by the student’s advisory committee. The student must present his/her thesis to the mathematics department faculty in a colloquium.

*Prerequisite: Permission of the graduate advisor.*

**601 Advanced Project in Applied Mathematics. (3, repeatable to 6) **Project in an advanced topic of mathematics or statistics, which will be presented in a final paper or portfolio, demonstrating advanced proficiency in an applied mathematics field. Graded S/U. *Prerequisite: Permission of the Graduate Committee. *

**602 Internship in Applied Mathematics. (3, repeatable to 6) **Mathematical work or training conducted at a professional institution, university or government organization, which will be presented in a final paper or portfolio, demonstrating advanced proficiency in an applied mathematics field. Graded S/U. *Prerequisite: Permission of the Graduate Committee. *

**607 Practicum in Mathematics Education. (3)** Direct internship experience for action research in mathematics education (K‑8) under guidance of qualified faculty. *Prerequisites: MATH 500 or 505 and approval of degree plan, completion of over half of candidate's course work, including EIS 500. Modifications in the above requirements are subject to the approval of the candidate's advisor.*

**651 Elements of Modern Analysis. (3)**A study of elements of modern analysis with a significant lean toward developing theory. Topics include Riemann integrability; metric spaces; pointwise and uniform convergence; Hilbert and normed vector spaces; Banach fixed point theorem, Weierstrass approximation theorem; and applications.

*Prerequisites: MATH 435 and MATH 551, or equivalents.*

**652 Computational Differential Equations. (3) **A study of elements of computational mathematics of differential equations with a lean toward developing the theory. Topics include adaptive one-step and multi-step methods of ordinary differential equations, the method of lines for evolutionary problems, and direct and iterative methods for sparse linear systems. *Prerequisites: MATH 435 or MATH 551, and MATH 552 or MATH 481.*

**654 Applications of Logic and Computability Theory. (3) **A study of elements of modern logic and computability with a lean toward developing the theory. Topics include the mathematics of computability and incomputability, introduction to computational complexity, and additional applications of logic. *Prerequisite: Permission of the instructor. *

**655 Technology and the Secondary School Mathematics Curriculum. (3) **Strategies for using technology such as calculators, computers, and Internet resources for teaching algebra, geometry, probability, and statistics in the secondary mathematics curriculum, including research on the use of the technology for mathematics teaching and learning. *Prerequisites: Permission of the instructor.*

**656 Advanced Perspective of Secondary School Mathematics. (3)**An advanced study of the mathematics of secondary school curriculum for the purpose of developing deeper connection and representations for all students. Focus is on rigorous conceptual context knowledge, methods of inquiry, and investigative problem-solving. Topics include Algebra, Geometry, and Statistics.

*Prerequisites: Permission of the Department Chair.*

**699 Advanced Special Topics. (3, repeatable to 6 under different titles) **Advanced special topics in mathematics or statistics with a lean towards theory. May be repeated with change of topic. *
Prerequisite: Permission of the instructor.
*

#####
**Statistics (STAT)**

**409G Probability and Statistics for Middle School Teachers. (3) **Probability laws, random variables, probability distributions, estimation and inference, sampling and data analysis, emphasis on concepts and connections of probability and statistical content to the challenges of teaching statistics for middle school teachers. *Prerequisite: Math 123 or 128 or equivalent.*

**471G Introduction to Mathematical Statistics I. (3)** The mathematical foundations of probability and statistics, principals of probability, sampling, distribution, moments, and hypothesis testing. *Prerequisite: MATH 138 or MATH 231 or equivalent.*

**474G Regression and Correlation Analysis. (3)** Least squares theory, correlation theory, simple, multiple, and stepwise regression, computer-assisted model building, and applied problems. *Prerequisite: STAT 276 or equivalent.*

**478G Analysis of Variance. (3)** A study of analysis of variance and covariance. Includes experimental design with applications. *Prerequisite: STAT 276 or equivalent.*

**553 Applied Statistical Methods. (3) **Introduction to probability and statistics with a significant lean toward applications. Topics include probability, probability distributions, Central Limit Theorem, sampling distributions (t, F, Chi-Square), parameter estimation, hypothesis testing, nonparametric statistics, ANOVA, and linear regression. *Prerequisites: MATH 231 and STAT 276, or equivalents.*

**570 Probability Theory and Stochastic Processes. (3)** Nature of probability theory, sample space, combinatorial analysis, fluctuations in random events, stochastic independence, random variables, generating functions, Markov chains, and simple time‑dependent stochastic processes. *Prerequisite: STAT 471 or equivalent.*

**574 Linear Models and Experimental Designs. (3)**General linear models, Gauss‑Markov Theorem, experimental design model confounding, and types of experimental designs and their analysis.

*Prerequisite: STAT 472 or permission of the instructor.*

**653 Elements of Statistical Inference. (3) **A study of elements of statistical inference with a lean toward developing the theory. Topics include probability theory, random variables, probability distribution functions, limit theorems, estimation, testing, sufficiency, robust statistical methods, bootstrap, and linear models. *Prerequisites: STAT 471 and STAT 553.*

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