Ohio University
Graduate Catalog
Mathematics
The Department of Mathematics offers the Master of Science degree and the Doctor of Philosophy degree. At the master's level, programs are available in applied mathematics, computer science , mathematics for secondary school teachers , and pure mathematics. At the doctoral level, you may specialize in algebra, analysis, topology, or applied mathematics. The principal feature of graduate study in mathematics is the possibility of designing a study plan to meet your individual needs and interests.
To be admitted to graduate study, you should have an undergraduate average of at least a B (3.0 on a 4.0 scale). If you plan to pursue the study of pure or applied mathematics, your undergraduate program should have included advanced calculus and junior or senior-level courses in abstract and linear algebra. Prospective computer science majors should have completed at least a calculus sequence, differential equations, PASCAL, and assembly language. Secondary education majors should have completed the calculus sequence and courses in geometry and algebra. If you are admitted with deficiencies, you will be expected to make up the deficiencies during the first year.
The Master of Science degree may be taken with or without a thesis--no examination is required. Under the nonthesis option for pure and applied mathematics majors, the minimum amount of coursework is 45 quarter hours, half of which should be earned in course sequences numbered 600 or higher. If most or all of your work is on the 500 level, the graduate committee may require more than 45 hours, but not over 60. (The usual requirement is 55 hours.) The coursework should include at least two mathematics sequences, e.g., algebra, analysis, topology, etc.
To pursue the computer science option, you must complete a minimum of 56 graduate hours distributed in a manner to assure a sound program of study. Full-time students normally complete the program in four quarters, while graduate associates may take two years.
Specific minimum requirements for the computer science option are as follows: 12 hours of mathematics sequence (see below), four hours of mathematics in addition to a sequence, 30 hours of computer science including one concentration described below, and 10 hours of electives in computer science or mathematics or a research project.
The acceptable mathematics sequences for this degree program are MATH 511, 513A,B Linear and Abstract Algebra; MATH 544, 545, 546 Numerical Methods; MATH 550A,B,C Mathematical Statistics; MATH 560A, B, C Advanced Calculus; MATH 613A, B, C Abstract Algebra; MATH 660A, B, C Real Analysis.
The areas of concentration in computer science are CS 504, 506, 510 Theoretical Computer Science; CS 511, 512, 613 Concurrent and Parallel Processing; CS 542, 544, 558 Operating System and Communications; CS 562, 564, 568 Information Retrieval and Databases; CS 580, 582, 583 Artificial Intelligence; CS 657A,B,C, 612 Software Engineering and Real Time Systems.
With the assistance of your faculty advisor, you must submit a plan of study approved by the graduate chair by the end of your first quarter. Any changes to this study plan must be approved by your faculty advisor and the graduate chair at least one quarter before you apply for graduation.
The Department of Mathematics, together with the College of Education, offers a joint program for secondary school teachers. The master's degree may be taken either in the College of Education or in the Department of Mathematics. Expect at least half of your credits to be earned in mathematics. Topics studied are geometry, algebra, number theory, and analysis. A minimum of 50 hours is required.
There are no specific courses required for the Ph.D. degree, but each student must pass a comprehensive examination and write an acceptable dissertation.
The dissertation is expected to be a scholarly work demonstrating your ability to understand, organize, improve, and present mathematical ideas of outstanding importance, depth, or interest. It should be worthy of publication.
The Department of Mathematics encourages its students to develop the ability to read mathematics in those languages which predominate the literature of the discipline. Students in post-master's courses will be expected to understand mathematics written in one or more of the following languages: French, German, or Russian.
All graduate-level computer science courses, except for 521, 522, and 599, may be used to satisfy requirements for a graduate degree in mathematics.
You may apply for admission for any quarter. To apply for financial aid for the following academic year, you must submit application materials by March 1, although late applications will be considered if vacancies exist.
Faculty
Mathematics (MATH) Courses
500 History of Mathematics (4) Main lines of mathematical development in terms of contributions made by great mathematicians: Euclid, Archimedes, Descartes, Newton, Gauss, etc.
506 Foundations of Mathematics II (4) Introductory topics in set theory and axiomatic development of real number system.
507 Number Theory (4) Prereq: 307. Topics in number theory.
510 Matrix Theory (4) Primarily intended for science and engineering majors. Topics include matrix algebra and matrix calculus, matrix solutions of systems of linear equations, eigenvector and eigenvalue problems, quadratic forms, and inner product spaces.
511 Linear Algebra (4) Vector spaces and linear transformations; matrices and determinants; characteristic roots and similarity; dual spaces; classification of quadratic and Hermitian forms.
512 Introduction to Algebraic Coding Theory (4) Prereq: 211, 410. Encoding and decoding. Vector spaces over finite fields. Linear Codes, parity-check matrices, syndrome decoding, Hamming Codes, and Cyclic Codes.
513A Introduction to Modern Algebra (4) Prereq: 511 or equivalent mathematical experience. Groups, permutation groups, subgroups, normal subgroups, quotient groups. Conjugate classes and class equation formula and its application to p-groups. Fundamental theorem on homomorphisms.
513B Introduction to Modern Algebra (4) Prereq: 513A. Fundamental theorem on finite abelian groups and its consequences. Cauchy theorem and first Sylow theorem. Polynomial rings. UFD and Euclidean domains. Maximal ideals. Algebraic extensions and splitting fields. Fundamental theorem of Galois theory.
529 Topics in Mathematics of Elementary and Secondary Schools (1-5) Selected topics related to teaching of mathematics in grades K-12. May be repeated for credit.
539 Topics in Geometry (1-5) When demand is sufficient, a course in some phase of geometry will be offered under this number. May be repeated for credit.
540 Vector Analysis (4) Vector algebra and its applications. Vector calculus and space curves. Scalar and vector fields, gradient, divergence, curl, and Laplacian. Line and surface integrals, divergence theorem, Stoke's theorem, and Green's theorem.
541 Fourier Analysis and Partial Differential Equations (4) Representation of functions as sums of infinite series of trigonometric functions, Bessel functions, Legendre polynomials, or other sets of orthogonal functions. Use of such representations for solution of partial differential equations dealing with vibrations, heat flow, and other physical problems.
542 Theory of Linear and Nonlinear Programming (4)
Prereq: 510 or equiv; computer programming experience desirable.
Minimization of functions subject to equality and inequality
constraints. Kuhn-Tucker theorem, algorithms for function
minimization, such as steepest descent and conjugate gradient, and
penalty function method. (Not a course in computer programming.)
543 Mathematical Modeling and Optimization (4) Prereq: 211, 340, or 410, FORTRAN. Differential equation models of wide variety of physical, social, and biological phenomena presented. Qualitative analysis introduced and used to investigate models. Optimal criteria incorporated to convert models to optimal control problems. Pontriagin's maximal principle used to find analytic solutions. Numerical solutions to optimal control problems also treated.
544 Introduction to Numerical Analysis (4) Prereq: CS 521 and undergrad course in differential equations. Iterative methods for solving nonlinear equations, polynomial interpolation and approximations, numerical differentiation and integration, numerical solution of differential equations, error analysis.
545 Advanced Numerical Methods (4)
Prereq: 541 and 544 or EE 778 and CHE 501. Initial and boundary value
problems; numerical solutions of parabolic, elliptic, and hyperbolic
equations; stability; error estimates; applications to engineering
problems. (Also offered as ET 545.)
546 Numerical Linear Algebra (4) Prereq: MATH 510 and FORTRAN. Floating point arithmetic, numerical solution of systems of linear equations using Gaussian elimination and its variants, numerical techniques for eigenvalues, error analysis, and implementation of algorithms on computer.
549 Advanced Differential Equations (4) Prereq: undergrad course in differential equations and 510 or 511. Introduction to theory of ordinary differential equations with special attention to oscillation, plane autonomous systems, Liapunov theory, and quadratic functionals.
550A Theory of Statistics (4) Probability distributions of one and several variables, sampling theory, estimation of parameters, confidence intervals, analysis of variance, correlation, and testing of statistical hypotheses.
550B Theory of Statistics (4) Prereq: 550A. Continuation of 550A. See 550A for description.
550C Theory of Statistics (4) Prereq: 550B. Continuation of 550A-B. See 550A for description.
551 Stochastic Processes (4) Prereq: 550B. Markov chains, Poisson process, birth and death process, queuing, and related topics.
560A Advanced Calculus (4) Prereq: undergrad course in introductory analysis. Critical treatment of functions of single variable. Emphasis on topics not treated in undergraduate introductory analysis course, such as compactness, nested intervals, deeper properties of continuous functions, Riemann-Stieltjes integration, and uniform convergence.
560B Advanced Calculus (4) Prereq: 560A and 511. Primarily devoted to study of differential calculus in n-space. Topics include review of inner product spaces and linear transformations, elementary topology of plane, limits and continuity of functions of several variables, directional derivative, differential, chain rule, and implicit function theorem.
560C Advanced Calculus (4) Prereq: 560B. Primarily devoted to study of integral calculus in n-space. Riemann-Darboux integral, Jordan content, iterated integrals, transformation of integrals, differential forms, and their integrals.
570 Applied Complex Variables (4) Analytic and harmonic functions, Cauchy integral and residue theorems, contour integration, Taylor and Laurent expansions, conformality and linear transformations with applications.
580A Elementary Point Set Topology (4) Topology of Euclidean spaces and general metric spaces.
580B Elementary Point Set Topology (4) Prereq: 580A. Introduction to general topological spaces.
599 Selected Topics in Mathematics (1-15) May be repeated for credit.
600A Set Theory (5) Introduction to axiomatic set theory; ordinals and cardinals; equivalents of axiom of choice.
600B Set Theory (5) Prereq: 600A. Introduction to combinatorial set theory, trees, partitions relations, closed unbounded and stationary sets, Martin's Axiom.
613A Abstract Algebra (5) Prereq: 513B. Groups, rings and fields, Jordan-Holder theorem, structure theorem for finitely generated abelian groups, integral domains, principal ideal rings, modules, linear algebras, field extensions, and Galois theory.
613B Abstract Algebra (5) Prereq: 613A. Continuation of 613A. See 613A for description.
613C Abstract Algebra (5) Prereq: 613B. Continuation of 613A-B. See 613A for description.
630A Tensor Analysis on Manifolds (5) Prereq: 511, 560C. Manifolds, tensor algebra, vector analysis on manifolds, differential forms, exterior derivatives, Stokes theorem, Riemannian and semi-Riemannian manifolds, curvature and torsion tensors.
630B Tensor Analysis on Manifolds (5) Prereq: 630A. Continuation of 630A. See 630A for description.
630C Tensor Analysis on Manifolds (5) Prereq: 630B. Continuation of 630A-B. See 630A for description.
640A Numerical Analysis (5) Prereq: 511, 570. Approximation by piecewise polynomial functions, variational principles, variational formulation of partial differential equations. The Rayleigh-Ritz-Galerkin method, convergence of approximations, time-dependent problems, isoparametric elements and nonconforming finite element methods, applications.
640B Numerical Analysis (5) Prereq: 640A. Continuation of 640A. See 640A for description.
640C Numerical Analysis (5) Prereq: 640B. Continuation of 640A-B. See 640A for description.
641A Methods of Applied Mathematics (5) Prereq: 560C, 510 and 340. Course content varies. May be repeated for credit.
641B Methods of Applied Mathematics (5) Prereq: 641A. Course content varies. May be repeated for credit.
641C Methods of Applied Mathematics (5) Prereq: 641B. Course content varies. May be repeated for credit.
642A Optimization Theory (5) Prereq: 560A,B,C; 510; 340. Classical problems of calculus of variation; Euler-Lagrange, Dubois-Reymond, Legendre, and Weierstrass necessary conditions; formulation of classical problems as nonlinear programming problems in function space.
642B Optimization Theory (5) Prereq: 642A, 660C, FORTRAN. Numerical solutions of boundary value problems; formulation and solution of optimal control problems with set, equality, and inequality constraints; applications to economics, classical mechanics, and engineering.
642C Optimization Theory (5) Prereq: 642B. Pontriagin's maximal principle is derived and applied to optimal control problems. Numerical solutions considered more fully.
645A Differential Equations (5) Prereq: 560C, 510, 541. Gronwall's inequality; existence and uniqueness; linear equations; autonomous equations; periodic solutions; stability; characteristics of first-order p.d.e.; classification of second-order equations into elliptic, parabolic, and hyperbolic types; special consideration of Laplace's equation, heat equation, and wave equation; hyperbolic systems, etc.
645B Differential Equations (5) Prereq: 645A. Continuation of 645A. See 645A for description.
645C Differential Equations (5) Prereq: 645B. Continuation of 645A-B. See 645A for description.
647A Special Functions (5) Prereq: 560C and 570, or 670A. Infinite products; Gamma, Beta, and Zeta functions; asymptotic series; cylindrical functions; spherical functions; orthogonal polynomials; Legendre, Hermite, and Laquerre polynomials.
647B Special Functions (5) Prereq: 647A. Continuation of 647A. See 647A for description.
660A Real Analysis (5) Prereq: 560C. Abstract measure and integration, Lebesgue measure on real line; Lp-spaces; Fubini and Radon-Nikodym theorems; differentiation theory.
660B Real Analysis (5) Prereq: 660A. Continuation of 660A. See 660A for description.
660C Real Analysis (5) Prereq: 660B. Continuation of 660A-B. See 660A for description.
670A Complex Analysis (5) Prereq: 560C. Analytic functions, multivalued analytic functions, power series, complex integration, Cauchy integral theorem, its extensions and consequences. Residue theorem, Taylor and Laurent expansions, max-modulus principle and its generalizations, elementary conformal mapping, conformal representations, Riemann surfaces, Weierstrass and Mittag-Leffler's factorization theorems, simple periodic functions, simple properties of elliptic functions. Dirichlet problem.
670B Complex Analysis (5) Prereq: 670A. Continuation of 670A. See 670A for description.
670C Complex Analysis (5) Prereq: 670B. Continuation of 670A-B. See 670A for description.
671A Potential Theory (5) Prereq: 560C and 570, or 670A. Newtonian and logarithmic potentials, their continuity and discontinuity properties, Dirichlet problems, subharmonic functions, harmonic functions, etc.
671B Potential Theory (5) Prereq: 671A. Continuation of 671A. See 671A for description.
680A Point Set Topology (5) Prereq: 560C. General topological spaces, product and quotient spaces, convergence, separation, countability properties, compactness and paracompactness, connectivity, metric spaces, completion, metrization, completely regular spaces, uniform spaces.
680B Point Set Topology (5) Prereq: 680A. Continuation of 680A. See 680A for description.
680C Point Set Topology (5) Prereq: 680B. Continuation of 680A-B. See 680A for description.
690 Independent Study (1-15) Independent study of topics under guidance of faculty member. May be repeated for credit.
695 Thesis (arranged) May be repeated for credit.
699 Topics in Mathematics (1-15) May be repeated for credit.
710A Group Theory (5) Prereq: 613C. Abelian groups, permutation groups, Sylow theorems, solvable groups, group extensions, free groups and free products, group representation, and characters.
710B Group Theory (5) Prereq: 710A. Continuation of 710A. See 710A for description.
711A Theory of Rings and Modules (5) Prereq: 613C. Rings with minimum condition, Wedderburn theorems, Jacobson radical, Jacobson density theorem, commutativity conditions, algebras, Goldie theorems, modules, and chain conditions.
711B Theory of Rings and Modules (5) Prereq: 711A. Continuation of 711A. See 711A for description.
730A Differential Geometry--Classical (5) Prereq: 613C, 660C, 680C. Local geometry of curves, local geometry of surfaces, tensors, Riemannian geometry, differential geometry in the large, applications.
730B Differential Geometry--Classical (5) Prereq: 730A. Continuation of 730A. See 730A for description.
731A Differential Geometry--Modern (5) Prereq: 613C, 660C, 680C. Differentiable manifolds, calculus of variations, lie groups, differential geometry in Euclidean spaces, g-structures.
731B Differential Geometry--Modern (5) Prereq: 731A. Continuation of 731A. See 731A for description.
740A Ordinary Differential Equations (5) Prereq: 645B. Advanced topics in ODE's.
740B Ordinary Differential Equations (5) Prereq: 740A. Continuation of 740A. See 740A for description.
740C Ordinary Differential Equations (5) Prereq: 740B. Continuation of 740A-B. See 740A for description.
741A Partial Differential Equations (5) Prereq: 645C. Advanced topics in PDEs.
741B Partial Differential Equations (5) Prereq: 741A. Continuation of 741A. See 741A for description.
741C Partial Differential Equations (4) Prereq: 741B. Continuation of 741A-B. See 741A for description.
760A Measure and Integration (5) Prereq: 613C, 660C, 680C. Various types of measures and integrals in modern research.
760B Measure and Integration (5) Prereq: 760A. Continuation of 760A. See 760A for description.
761A Functional Analysis (5) Prereq: 660A. Normed linear spaces, Hilbert spaces, Hahn-Banach extension theorems, Banach-Steinhaus theorem, closed graph theorem, applications to differential and integral equations.
761B Functional Analysis (5) Prereq: 761A. Topics selected from spectral theory, Banach algebras, integration in Banach spaces, linear topological vector spaces, and other topics.
761C Functional Analysis (5) Prereq: 761B. Continuation of 761B. See 761B for description.
780A General Topology (5) Prereq: 680C. Continuation of main line of development of 680A-B-C, but at deeper and more advanced level. Offered especially for those students who intend to specialize in general topology.
780B General Topology (5) Prereq: 780A. Continuation of 780A. See 780A for description.
780C General Topology (5) Prereq: 780B. Continuation of 780A-B. See 780A for description.
809 Topics in the Foundation and History of Mathematics and in Number Theory (1-15) Selected topics not offered in normal course offerings. May be repeated for credit.
819 Topics in Algebra (1-15) Detailed study of advanced topics not covered in other algebra courses. May be repeated for credit.
829 Topics in the Teaching of Mathematics (1-15) Selected topics not covered in regular course offerings. May be repeated for credit.
839 Topics in Geometry (1-15) Selected topics not covered in regular offerings. May be repeated for credit.
849 Topics in Applied Mathematics (1-15) Selected topics not covered in regular offerings. May be repeated for credit.
859 Topics in Probability, Statistics, and Stochastic Processes (1-15) Selected topics not covered in regular offerings. May be repeated for credit.
869 Topics in Real Analysis (1-15) Selected topics not covered in regular offerings. May be repeated for credit.
879 Topics in Complex Analysis (1-15) Special topics not ordinarily covered in other courses. May be repeated for credit.
889 Topics in Topology (1-15) Special topics not covered in other courses. May be repeated for credit.
890 Independent Study (1-15) Independent study under guidance of faculty member. May be repeated for credit.
891 Seminar (1-15) May be repeated for credit.
895 Dissertation (arranged) May be repeated for credit.
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) April 13, 1998.
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