Graduate Studies Bulletin 2009-2010

Policies and Courses

Courses for Mathematics (MATH) +/-

800. Mathematics for Elementary School Teachers (3 cr)

Prereq: Permission

Fundamental mathematical concepts basic to the understanding of arithmetic.

*800P. Number and Operation for K-3 Mathematics Specialists (3 cr) Lec 3.

Prereq: A valid elementary or early childhood teaching certificate and permission.

MATH 800P is not open to students who are candidates for an MA or MS in mathematics or statistics.

Number and operations. Place value and its role in arithmetic operations. Development of fractions and number systems. Develop the habits of mind of a mathematical thinker and to develop a depth of understanding of number and operations sufficient to enable the teacher to be a disciplinary resource for other K-3 teachers.

*800T. Mathematics as a Second Language (3 cr)

Prereq: Admission to the MAT or MScT program in mathematics or to a graduate program in the College of Education and Human Sciences

MATH 800T is intended for mid-level mathematics teachers. Numbers and operations. Careful reasoning, problem solving, and communicating mathematics both orally and in writing. Connections with other areas of mathematics. Development of mathematical thinking habits.

801. Geometry for Elementary School Teachers (3 cr)

Prereq: Permission

Properties of congruence and similarity, lines and rays, angles, parallels and perpendiculars, bisectors, notion of area.

*801P. Geometry, Measurement, and Algebraic Thinking for K-3 Mathematics Specialists (3 cr) Lec 3.

Prereq: A valid elementary or early childhood teaching certificate and permission.

MATH *801P is not open to students who are a candidate for an MA or MS in mathematics or statistics.

Polygons, polyhedra, rigid motions, symmetry, congruence, similarity, measurement in one, two and three dimensions, functions, mathematical expressions, solving equations, sequences. Develop the habits of mind of a mathematical thinker and to develop a depth of understanding of geometry, measurement and algebraic thinking to enable the teacher to be a disciplinary resource for other K-3 teachers.

*802T. Functions, Algebra, and Geometry for Middle Level Teachers (3 cr)

Prereq: Admission to the MAT or MScT program in mathematics or to a graduate program in the College of Education and Human Sciences

MATH 802T is intended for mid-level mathematics teachers. Variables and functions. Use of functions in problem solving. Theory of measurement, especially length, area, and volume. Geometric modeling in algebra. Graphs, inverse functions, linear and quadratic functions, the fundamental theorem of arithmetic, modular arithmetic, congruence and similarity. Ways these concepts develop across the middle level curriculum.

*804T. Experimentation, Conjecture and Reasoning (3 cr)

Prereq: Admission to the MAT or MScT program in mathematics or to a graduate program in the College of Education and Human Sciences

Intended for middle-level mathematics teachers. Problem solving, reasoning and proof, and communicating mathematics. Development of problem solving skills through the extensive resources of the American Mathematics Competitions. Concepts of logical reasoning in the context of geometry, number patterns, probability and statistics

805. Discrete and Finite Mathematics (3 cr)

Prereq: MATH 814 is desirable but not required

Credit in MATH 805 will not count towards the MA or MS degree in mathematics. Not open to math majors except for dual matriculants in Teachers College. Credit is not allowed for both MATH 105 and 805, or for both CSCE 235 and MATH 805. Graphs and networks. Map coloring. Finite differences. Pascal’s triangle. The Pigeonhole Principle. Markov chains. Linear programming Game Theory.

*805T. Discrete Mathematics for Middle Level Teachers (3 cr) Lec 3.

Prereq: Admission to the MAT-MScT program in MATH or to a graduate program in the College of Education and Human Sciences

MATH 805T is intended for mid-level mathematics teachers. Concepts of discrete mathematics, as opposed to continuous mathematics, which extend in directions beyond, but related to, topics covered in middle-level curricula. Problems which build upon middle-level mathematics experiences. Logic, mathematical reasoning, induction, recursion, combinatorics, matrices, and graph theory.

*806T. Number Theory and Cryptology for Middle Level Teachers (3 cr)

Prereq: Admission to the MAT or MScT program in mathematics or to a graduate program in the College of Education and Human Sciences

MATH 806T is intended for mid-level mathematics teachers. Basic number theory results and the RSA cryptography algorithm. Primes, properties of congruences, divisibility tests, linear Diophantine equations, linear congruences, the Chinese Remainder Theorem, Wilson’s Theorem, Fermat’s Little Theorem, Euler’s Theorem, and Euler’s phi-function. Mathematical reasoning and integers’ connections to the middle school curriculum.

807. Mathematics for High School Teachers I (3 cr)

Prereq: MATH 208 and 310

Analysis of the connections between college mathematics and high school algebra and precalculus.

*807T. Using Mathematics to Understand Our World (3 cr) Lec 3.

Prereq: Admission to the MAT-MScT program in MATH or to a graduate program in the College of Education and Human Sciences

MATH 807T is intended for middle-level mathematics teachers. The mathematics underlying several socially-relevant questions from a variety of academic disciplines. Construct mathematical models of the problems and study them using concepts developed from algebra, linear and exponential functions, statistics and probability. Original documentation, such as government data, reports and research papers, in order to provide a sense of the role mathematics plays in society, both past and present.

808. Mathematics for High School Teachers II (3 cr)

Prereq: MATH 310 and 350

Analysis of the connections between college mathematics and high school algebra and geometry.

*808T. Concepts of Calculus for Middle Level Teachers (3 cr) Lec 3.

Prereq: Admission to the MAT-MScT program in MATH or to a graduate program in the College of Education and Human Sciences

MATH 808T is intended for middle -level mathematics teachers. The processes of differentiation and integration, their applications and the relationship between the two processes. Rates of change, slopes of tangent lines, limits, derivatives, extrema, derivatives of products and quotients, anti-derivatives, areas, integrals, and the Fundamental Theorem of Calculus. Connections to concepts in the middle level curriculum.

*810T. Algebra for Algebra Teachers (3 cr) Lec 3.

Prereq: A valid secondary mathematics teaching certificate and permission. An undergraduate course in modern algebra is recommended, but not required.

MATH 810T will not count toward an MA or MS degree in mathematics or statistics.

The integers. The Euclidean algorithm, the Fundamental Theorem of Arithmetics, and the integers mod n. Polynomials with coefficients in a field. The division algorithm, the Euclidean algorithm, the unique factorization theorem, and its applications. Polynomials whose coefficients are rational, real or complex. Polynomial interpolation. The habits of mind of a mathematical thinker. The conceptual underpinnings of school algebra.

814. Applied Linear Algebra (Matrix Theory) (3 cr)

Prereq: MATH 208 or 107H

A term paper and/or special project is required for graduate credit. Not open to MA or MS students in mathematics. Students in the sciences cannot count MATH 814 toward a minor in mathematics. Similarity of matrices, diagonalization of symmetric matrices, canonical forms, eigenvalues, quadratic forms, vectors, and applications to linear systems.

815. Modern Algebra with Applications (3 cr)

Prereq: MATH 310 or CSCE 235 or permission

Credit for both MATH 815 and 817 is not allowed. Boolean algebra, binary functions, groups and semigroups, homomorphisms, congruencies, quotient structures, isomorphism, theorems for groups, Jordan-Holder theorem, finite-state machines, electronic realization, Winograd’s theorem, Krohn-Rhodes algebraic decomposition theory.

*817. Introduction to Modern Algebra I (3 cr)

Prereq: MATH 310 is advisable for most students

Topics from elementary group theory and ring theory, including fundamental isomorphism theorems, ideals, quotient rings, domains. Euclidean or principal ideal rings, unique factorization, modules and vector spaces including direct sum decompositions, bases, and dual spaces.

*818. Introduction to Modern Algebra II (3 cr)

Prereq: MATH 817

Topics from field theory including Galois theory and finite fields and from linear transformations including characteristic roots, matrices, canonical forms, trace and transpose, and determinants.

821. Differential Equations (3 cr)

Prereq: A grade of “P” or “C” or better in MATH 208

Not open to MA or MS students in mathematics. Students in the sciences cannot count MATH 821 toward a minor in mathematics. First- and second-order methods for ordinary differential equations including: separable, linear, Laplace transforms, linear systems, and some applications.

822. Advanced Calculus (3 cr)

Prereq: MATH 208 or 107H

A term paper and/or special project is required for graduate credit. Credit in MATH 822 will not count towards the MA or MS degree in mathematics. Students in the sciences cannot count MATH 822 toward a minor in mathematics. Green’s theorem, Stokes’ theorem, the divergence theorem, and applications from differential and integral vector calculus, line integrals, general coordinate transformations, inverse function theorem, and uniform convergence of sequences and series of functions.

823. Introduction to Complex Variable Theory (3 cr)

Prereq: MATH 208 or 107H

Introductory course for engineering, physical sciences, and mathematics majors. Complex numbers, functions of complex variables, complex integration, calculus of residues, infinite series, conformal mapping, Schwarz-Christoffel transformation, Poisson’s integral formula, and applications of the above.

824. Introduction to Partial Differential Equations (3 cr)

Prereq: MATH 821

Credit in MATH 824 will not count towards the MA or MS degree in mathematics. Derivation of the heat, wave, and potential equations; separation of variables method of solution; solutions of boundary value problems by use of Fourier series, Fourier transforms, eigenfunction expansions with emphasis on the Bessel and Legendre functions; interpretations of solutions in various physical settings.

*825. Mathematical Analysis I (3 cr)

Prereq: MATH 208 and evidence of adequate preparation

Real number system, topology of Euclidean space and metric spaces, continuous functions, derivatives and the mean value theorem, the Riemann and Riemann-Stieltjes integral, convergence, the uniformity concept, implicit functions, line and surface integrals.

*826. Mathematical Analysis II (3 cr)

Prereq: MATH 208 and evidence of adequate preparation

827. Mathematical Methods in the Physical Sciences (3 cr)

Prereq: MATH 821

Credit in MATH 827 will not count towards the MA or MS degree in mathematics. Matrix operations, transformations, inverses, orthogonal matrices, rotations in space. Eigenvalues and eigenvectors, diagonalization, applications of diagonalization. Curvilinear coordinate systems, Jacobians, changes of variables in multiple integration. Scalar, vector and tensor fields, tensor operations, applications of tensors. Complex function theory, integration by residues, conformal mappings.

828. Principles of Operations Research (3 cr)

Prereq: MATH 814 or permission and STAT 880 or IMSE 321 or equivalent

Introduction to techniques and applications of operations research. Includes linear programming, queueing theory, decision analysis, network analysis, and simulation.

830. Ordinary Differential Equations I (3 cr)

Prereq: MATH 821

The Picard existence theorem, linear equations and linear systems, Sturm separation theorems, boundary value problems, phase plane analysis, stability theory, limit cycles and periodic solutions.

831. Ordinary Differential Equations II (3 cr)

Prereq: MATH 821 and 822

The Picard existence theorem, linear equations and linear systems, Sturm separation theorems, boundary value problems, phase plane analysis, stability theory, limit cycles and periodic solutions.

832. Linear Optimization (3 cr)

Prereq: MATH 814

Mathematical theory of linear optimization, convex sets, simplex algorithm, duality, multiple objection linear programs, formulation of mathematical models.

833. Nonlinear Optimization (3 cr)

Prereq: MATH 814

Mathematical theory of constrained and unconstrained optimization, conjugate direction and quasi-Newton methods, convex functions, Lagrange multiplier theory, constraint qualifications.

838. Mathematical Methods for Biology and Medicine (5 cr) Lec, rec.

Prereq: Grade of P, C, or better in MATH 106 or 106B

MATH 838 is not open to MA or MS degree students in MATH or STAT. Some computation and visualization will be done with Matlab. Mathematical modeling, discrete and continuous probability, parameter estimation, discrete and continuous dynamical systems, and Markov chains. Application of mathematical models in the life sciences. Regression analysis, cobweb diagrams, the phase line, nullcline analysis, eigenvalue analysis, linearization, and likelihood analysis. Applications to fisheries, stage-structured populations, pharmacokinetics, epidemiology, and medical testing.

839. Mathematical Models in Biology (3 cr)

Prereq: MATH 107

MATH 839 has a small laboratory component. Discrete and continuous models in ecology, including population models, predation and food webs, the spread of infectious disease and life histories. Probability and Random processes in nature, elementary models for molecular events and pharamacokinetics.

840. Numerical Analysis I (CSCE 840) (3 cr) Lec 3.

Prereq: Grade of “Pass” or “C” or better in CSCE 150E or 155/155H; MATH 208/208H

Credit toward the degree may be earned in only one of the following: CSCE/MATH 340/840 and ENGM 480/880. Algorithm formulation for the practical solution of problems, interpolation, roots of equations, differentiation, and integration. Effects of finite precision.

841. Approximation of Functions (CSCE 841) (3 cr) Lec 3.

Prereq: A programming language, MATH 821 and 814

Uniform approximation, orthogonal polynomials, least-first-power and least squares approximation, polynomial interpolation and spline interpolation, approximation interpolation by rational functions, and Fourier series.

*842. Methods of Applied Mathematics I (3 cr)

Prereq: MATH 821 and 814, or their equivalents

Interdependence between mathematics and the physical and applied sciences. Includes the calculus of variations, scaling and dimensional analysis, regular and singular perturbation methods.

*843. Methods of Applied Mathematics II (3 cr)

Prereq: MATH 842 or permission

Application of partial differential equation models to problems in the physical and applied sciences. Includes derivation of partial differential equations, the theory of continuous media, linear and nonlinear wave propagation, diffusion, transform methods, and potential theory.

845. Introduction to the Theory of Numbers I (3 cr)

Prereq: MATH 310

Arithmetic functions, congruencies, reciprocity theorem, primitive roots, diophantine equations, and continued fractions.

846. Introduction to the Theory of Numbers II (3 cr)

Prereq: MATH 845

Diophantine approximations, irrationality and transcendence, applications of the Euler-Maclaurin sum formula, Selberg’s proof of the prime number theorem, order of magnitude of some arithmetic functions, the lattice point problem.

847. Numerical Analysis II (CSCE 847) (3 cr) Lec 3.

Prereq: CSCE 340, MATH 814 and 821

Numerical matrix methods and numerical solutions of ordinary differential equations.

850. Combinatorics (3 cr)

Prereq: MATH 310 or 325

Theory of enumeration of arrangements of objects, recursion relations, generating functions, applications to enumeration of combinatorial structures.

852. Graph Theory (3 cr)

Prereq: MATH 850; or permission and either MATH 310 or 325

Theory of networks of points and connecting paths, structure and existence theorems for graphs and subgraphs, graph characteristics, special graphs and applications.

856. Differential Geometry I (3 cr)

Prereq: MATH 814, 821, and 822

Theory of space curves and surfaces, Gaussian curvature, differential parameters, geodesics, etc.

*858. Topics in Geometry (3 cr)

Prereq: MATH 208

Selected topics in some branch of geometry.

865. Introduction to Mathematical Logic I (3 cr)

Semantical and syntactical developments of propositional logic, discussion of several propositional calculi, applications to Boolean algebra and related topics, semantics and syntax of first-order predicate logic including Godel’s completeness theorem, the compactness theorem.

866. Introduction to Mathematical Logic II (3 cr)

Prereq: MATH 865

Semantics and syntax of first-order predicate logic including Godel’s completeness theorem, decision problems, formalization of deductive theories, the structure of applied predicate calculi, the calculus of classes, introduction to higher order predicate logic.

*871. General Topology I (3 cr)

Prereq: 6 hrs MATH beyond MATH 208

Set theory, topological spaces, continuity, connectedness, coverings, separation axioms, product and quotient spaces, and sequences, nets, and filter bases.

*872. General Topology II (3 cr)

Prereq: 6 hrs MATH beyond MATH 208

For course description, see MATH *871.

*874M. Mathematics Integration (CHEM 874M) (2-3 cr)

May be counted towards the MAT and MScT degrees in mathematics and statistics, not the MA, MS, or PhD.

889. Stochastic Processes and Advanced Mathematical Finance (3 cr) Lec 3.

Prereq: MATH 221/821 and/or STAT/MATH 380

Properties of stochastic processes and solutions of stochastic differential equations as a means for understanding modern financial instruments. Derivation and modeling of financial instruments, advanced financial models, advanced stochastic processes, partial differential equations, and numerical methods from probabilistic point of view.

895. Honors Seminar (1-3 cr per sem, max 6)

Prereq: MATH 208 and permission

896. Seminar in Mathematics (1-3 cr per sem, max 6)

Prereq: Permission

897. Reading Course (1-4 cr)

Prereq: Permission