Course Descriptions
Mathematics (MATH)

To view the complete schedule of courses for
each semester, go to Cardinal Students.

MATH 101: Review of Basic Mathematics
0.00 Credits

Real numbers, inequalities, functions; polynomial, rational, exponential, and logarithmic functions; systems of equations and inequalities. This is a NON-CREDIT remedial course. Placement test required.

MATH 108: Elementary Functions
3.00 Credits

Real numbers, inequalities, functions; polynomial, exponential, logarithmic, trigonometric functions and their inverses; systems of equations; sequences. Prerequisite: Placement.

MATH 110: Finite Mathematics for Business and Economics
3.00 Credits

This course introduces students to the fundamental ideas of finite mathematics with examples from and emphasis on business and economics. The main topics are (1) linear functions and models, linear systems and solution by geometric and algebraic methods, matrix methods and linear programming problems in two variables. (2) Fundamentals of financial mathematics including simple and compound interest, annuities, amortization. (3) Sets and basic combinatorial methods for counting: permutations and combinations. Prerequisite: Open to majors in Business/Economics by placement. Not open to students who have taken Math 111.

MATH 111: Calculus for Social-Life Sciences I
3.00 Credits

Functions and their graphs; linear functions; functional models; derivative, rate of change and marginal analysis; approximation by differentials; chain rule, implicit differentiation and higher-order derivatives; curve sketching: relative extrema, concavity; absolute extrema; exponential functions and natural logarithms and their derivatives; compound interest. Not open to students who have had 121. Prerequisite: Placement. For majors in Business and Economics, Placement, or a grade of C- or better in Math 110.

MATH 112: Calculus for Social-Life Sciences II
3.00 Credits

The concept of antiderivative; integration by substitution, by parts, and by use of tables; definite integral; area under a curve; applications to business and economics; definite integral as the limit of a sum; improper integrals; probability density; numerical integration; linear and separable differential equations; functions of several variables, partial derivatives, chain rule and total differential; relative extrema; Lagrange multiplier methods; the least square approximation. Not open to students who have had 122. Prerequisite: a grade of C- or better in 111.

MATH 114: Probability and Statistics
3.00 Credits

Designed for students in the social sciences, to acquaint them with the techniques of elementary statistics. Emphasizes computation and interpretation of data. Topics include calculation and graphing methods, measures of central tendency, measures of variation, measures of association and correlation; sampling and hypothesis testing.

MATH 121: Analytic Geometry and Calculus I
4.00 Credits

Coordinate systems, functions, graphs, one-to-one and inverse functions; composition of functions; lines and slopes; limits, continuity, maximum and minimum, derivative of a function of one variable; differentiation of polynomials, chain rule; derivatives of trigonometric functions and their inverses; implicit differentiation; antiderivative and definite integral; fundamental theorem of calculus. Not open to students who have had 111. Prerequisite: Placement.

MATH 122: Analytic Geometry & Calculus II
4.00 Credits

Antiderivative and definite integral; integration by parts and by substitution; integration of rational functions, powers of trigonometric functions, and rational functions of sin and cos; logarithms and exponential functions and their derivatives: application to computing area, volume; center of gravity and work; polar coordinates, parametric equations; arc length and speed on a curve; area of a surface; curvature; sequences and series; convergence tests; Taylor's formula. Not open to students who have had 112. Prerequisite: a grade of C- or better in 121.

MATH 168: Mathematics in the Modern World
3.00 Credits

Explores mathematical ideas and current applications of these ideas. Topics include mathematical applications in the management sciences and social sciences and applications of geometry to physics, astronomy, chemistry, and biology.

MATH 187: Introduction to Mathematical Thought
3.00 Credits

Intended for liberal arts students. Topics chosen from among: basic logic, number theory, infinite sets and cardinal numbers, symmetry and finite groups, graph theory and polyhedra, Euclidean and non-Euclidean geometry, and others.

MATH 221: Analytic Geometry & Calculus III
4.00 Credits

Partial derivatives and differential; the chain rule: gradient and directional derivative; derivative of a vector function; tangent planes; critical points and local extrema; Lagrange multipliers; integration over regions in R2; use of polar coordinates; integration over regions in R3; use of cylindrical and spherical coordinates; line integrals; conservative vector fields; Green's theorem; surface integrals and divergence theorem; Stokes' theorem. Prerequisite: 122.

MATH 222: Calculus IV Differential Eqns
4.00 Credits

Separable and linear differential equations with one unknown function, method of integrating factors; second order linear differential equations with variable coefficients; Wronski and Cauchy methods; systems of linear differential equations with variable and constant coefficients; Euler and Runge-Kutta methods; Laplace transform methods; series solutions of differential equations and special functions. Prerequisite: 221 or permission of instructor.

MATH 230: The Mathematics of Politics (UH)
3.00 Credits

A rigorous mathematical treatment of the following topics: models of conflict as represented by ordinal games; the theory of moves; escalation; voting systems; vector-weightedness and dimension of a voting system; the Shapley-Shubik, Banzhaf, and other indices of political power; social choice and Arrow's impossibility theorem.

MATH 305: Fundamentals of Advanced Mathematics
3.00 Credits

Prepares sophomore-level mathematics majors and minors for upper-level mathematics. Emphasis on problem-solving techniques and practice, covering topics and concepts common to all upper-level courses, including: sets, functions, relations, cardinality, first-order logic and proof-techniques, core material in elementary number theory, combinatorics, rational and irrational numbers, and the real number field. Additional topics selected by instructor, as time permits. Prerequisite: 122.

MATH 309: Probability and Statistics for Engineers
3.00 Credits

Course intended for engineering students and students working towards obtaining a minor in mathematics. Introduction to probability, random variables and probability distributions, mathematical expectations, discrete and continuous probability distributions, sampling distributions, point and interval estimation, hypothesis testing. Time permitting - simple linear regression and correlation will also be covered. Prerequisite: MATH 221

MATH 330: Topics in Mathematical Social Sciences (UH)
3.00 Credits

A more advanced study of the topics covered in MATH 230 (UH) with special emphasis on social welfare functions and Arrow's impossibility Theorem; Topics in game theory including the extensive and normal forms; Bargaining; Fari division; Apportionment.

MATH 501: Linear Algebra
3.00 Credits

Matrices and systems of equations; Gaussian elimination, matrix algebra; determinants; vector spaces; linear transformations; orthogonality, inner product spaces, normed spaces, least square technique; Gram-Schmidt orthogonalization; eigenvalues and eigenvectors, systems of linear differential equations, diagonalization method; numerical techniques. Prerequisite: 221 or 305 or CSC 210.

MATH 503: Euclidean & Noneuclidean Geometry
3.00 Credits

Study of various geometries from a modern viewpoint in which Euclidean and non-Euclidean geometries are treated synthetically and analytically. Prerequisite: 305 or CSC 210.

MATH 505: Abstract Algebra I
3.00 Credits

Groups, group homomorphisms and isomorphisms, and the Fundamental Theorem of Group Homomorphisms. Finite groups. Prerequisite: 305 or CSC 210.

MATH 506: Abstract Algebra II
3.00 Credits

A study of rings, integral domains and fields. Division and factorization in polynomial rings and Euclidean domains. Ideals in rings. Ring homomorphisms and isomorphisms, and the Fundamental Theorem of Ring Homomorphisms. Introduction to Galois Theory. Prerequisite: 505.

MATH 507: Graph Theory
3.00 Credits

Directed and undirected graphs, trees, connectivity; cut edges, cut vertices, and blocks; Eulerian and Hamiltonian graphs; planarity coloring problems; graph-theoretic algorithms and applications. Prerequisite: Math 305 or CSC 210.

MATH 508: Elementary Number Theory
3.00 Credits

A study of the basic properties of integers. Topics include properties of primes, factorization, congruences, Fermat's little theorem, diophantine equations, number theoretic functions, and quadratic reciprocity. Prerequisite: Math 305 or CSC 210.

MATH 509: Algebraic Number Theory
3.00 Credits

The study of number theory using algebraic techniques. Topics include extension fields of rational numbers, algebraic integers, non-uniqueness of factorization of algebraic integers, quadratic and cyclotomic number fields, primes and units in algebraic number fields, integral bases, uniqueness of factorization into prime ideals, class numbers and their relationship to Fermat's last theorem. Prerequisite: 508.

MATH 511: Mathematical Methods in Physics & Engineering
3.00 Credits

A selection of topics in applied mathematics of interest to scientists and engineers. Topics include vectors, matrices, complex numbers and functions, ordinary and partial differential equations, Fourier and Laplace transforms, and special functions. Same as PHYS 511, 512. Prerequisite: 221 and 222.

MATH 512: Mathematical Methods in Physics & Engineering
3.00 Credits

A selection of topics in applied mathematics of interest to scientists and engineers. Topics include vectors, matrices, complex numbers and functions, ordinary and partial differential equations, Fourier and Laplace transforms, and special functions. Same as PHYS 511, 512. Prerequisite: 511.

MATH 513: Rings and Modules
3.00 Credits

Topics include rings, modules, ideal theory, Artinian rings, Noetherian rings, indecomposable modules, projective and injective modules, localizations. Prerequisite: 506.

MATH 515: Combinatorics
3.00 Credits

Basic counting rules. Principle of inclusion and exclusion. Polya Enumeration Theorem. The Pigeonhole Principle and its generalizations. Generating functions; recurrence relations; elements of graph theory. Optimization for graphs and networks. Experimental design. Several applications given. Prerequisite: 305 or CSC 210.

MATH 516: Coding and Information Theory
3.00 Credits

Uniquely Decodable Codes, Instantaneous codes and their construction, Optimal Codes, Binary Huffman Codes, Average Word Length and Optimality of Binary Huffman codes, r-ary Huffman Codes, Information and Entropy, Average Word-Length, Shannon's First Theorem, Information Channels, Decision Rules, Hamming Distance, Comments on Shannon's Theorem, Error-Correcting Codes, Minimum Distance, Hadamard Matrices, Linear Codes. Prerequisites: Math 221, Math 531 or permission of the instructor.

MATH 520: Topology
3.00 Credits

Set theoretic background, in particular different forms of Axiom of Choice, basic concepts of point set topology, separation axioms, compact and locally compact spaces, compactifications, product spaces, Tychonoff theorem, metric and metrizable spaces. Baire category theorem. Prerequisites: 221 and either 505 or 521.

MATH 521: Introductory Analysis I
4.00 Credits

Infinite sets and cardinality; the real number system; introduction to metric space topology including convergence of sequences, compactness, continuity and uniform continuity of functions on metric spaces; Bolzano-Weierstrass theorem; Heine-Borel theorem; differentiation and Taylor's theorem for a function of one real variable; Riemann integral of a function of one real variable; sequences and series of real numbers and functions; interchange of limit operations. Prerequisites: 221 and 305.

MATH 522: Introductory Analysis II
3.00 Credits

Riemann-Stieltjes integral; equicontinuous families of functions and Arzela-Ascoli theorem; Tietze's extension theorem; Baire category theorem; differentiation and integration of a function of several variables; fixed point theorem; implicit function theorem; inverse function theorem; existence and uniqueness theorems for ordinary differential equations. Prerequisite: 521.

MATH 524: Complex Variables
3.00 Credits

Field of complex numbers. Elementary functions in complex variables: polynomials, rational, trigonometric and exponential functions. Limits and continuity. The complex derivative, Cauchy-Riemann equations. Analytic and harmonic functions. Complex integration, Cauchy's integral formula. Taylor and Laurent series. Residue theory. Uniform convergence. Analytic continuation. Conformal mapping. Prerequisite: 221.

MATH 527: Chaotic Dynamics
3.00 Credits

Periodic points, fixed points, bifurcation, 1-dimensional chaos, Cantor sets, 2-dimensional chaos, dynamics of linear functions, nonlinear maps, fractals, capacity dimension, Lyapunov dimension, Julia sets and the Mandelbrot set, iterated function systems, systems of differential equations, the Lorenz system. Prerequisite: 222.

MATH 528: Fractal Geometry
3.00 Credits

Examples of fractals, the triadic Cantor set, Sierpinski Gasket, metric topology, separable and compact spaces, uniform convergence, Hausdorff metric, topological dimension, self-similarity, Lebesgue measure, Hausdorff measure, Hausdorff dimension, other fractal dimensions. Prerequisites: 520 and 521.

MATH 531: Probability and Statistics with Applications I
3.00 Credits

An introduction to probability and statistics, with applications in the natural and social sciences. Axiomatic probability, independence and conditional probability. Random variables, their distributions, expectation, variance and moment-generating functions. Probability models and estimation of parameters. Prerequisite: 221.

MATH 532: Probability and Statistics with Applications II
3.00 Credits

Methods of statistical inference. Hypothesis testing: one and two sample problems, goodness-of-fit tests. Confidence intervals, regression and correlation, nonparametric statistics. Prerequisite: 531.

MATH 533: Stochastic Processes
3.00 Credits

Applied probabilistic modelling. Topics include Markov chains, continuous-time Markov processes, birth and death processes, Brownian motion, applications to economics, computer science, biology and physics. Prerequisite: 531.

MATH 537: Introduction to Fuzzy Sets and Fuzzy Logic
3.00 Credits

Introduces the concept of fuzzy set: a mathematical object modeling the vagueness present in our natural language when we describe phenomena that do not have sharply defined boundaries. Covers basic concepts of fuzzy sets, fuzzy sets versus crisp sets, operations on fuzzy sets, fuzzy relations, fuzzy theory versus probability theory and applications. Prerequisites: 221 and 531.

MATH 540: Ordinary Differential Equations
3.00 Credits

Existence and uniqueness of solutions; continuity and differentiability of solutions with respect to initial conditions and other parameters. Linear systems with constant and variable coefficients; resolvent matrix for a linear system; finite difference methods with error estimates; k-th order Euler's method and Runge-Kutta methods. Prerequisites: 222 and 501.

MATH 541: Introduction to Partial Difference Equations
3.00 Credits

Fourier Series, FourierTransforms, Solutions of the heat, wave and potential equations using separation of variables. Prerequisites: Math 221, 222 or permission of the instructor.

MATH 542: Introduction to Difference Equations
3.00 Credits

First order difference equations,higher order difference equations, stability analysis, z-transforms and applications. Prerequisites: Math 222, 501 or permission of the instructor.

MATH 550: Foundations of Mathematics
3.00 Credits

Sets, logic, and axiomatic and constructive treatment of real numbers. Prerequisite: 305.

MATH 551: Introduction to Mathematical Logic
3.00 Credits

Classical propositional and first-order predicate logic; syntax, semantics, basic metamathematical theorems including the Goedel-Henkin completeness theorem and the Skolem-Lowenheim theorem. Other possible topics: first-order recursive arithmetic, Goedel's incompleteness theorems, intuitionistic systems, Church's theorem, Tarski's theorem. Prerequisites: 305 and 505.

MATH 552: Formal Languages and the Theory of Computation
3.00 Credits

Languages and their representation, finite automata and regular grammars, pushdown automata and context-free languages. Turing machines; the halting problem, linear bounded automata and context-sensitive languages; relations between formal languages and recursive sets, time and tape bounds on Turing machines, deterministic pushdown automata. Prerequisite: 551.

MATH 561: Numerical Analysis I
3.00 Credits

Numerical integration: the rectangle, trapezoid, and spline quadrature. Linear systems of equations; matrix notation, properties of matrices, iterative determination of eigenvalues and eigenvectors, elimination methods. Solution of nonlinear equations; real and complex roots, zeroes of polynomials. Interpolation: polynomial, Hermite, and spline interpolations. Prerequisites: 222 and 506.

MATH 562: Numerical Analysis II
3.00 Credits

Approximation theory: introduction to approximation, orthogonal polynomials and least square approximation methods, numerical quadrature. Solution of ordinary differential equations: initial-value problems for ordinary differential equations, error propagation; higher-order Taylor, Runge-Kutta, multistep and extrapolation methods; control of stepsize; stiff equations; stability. Boundary-value problems for ordinary differential equations; shooting method for linear and nonlinear problems. Prerequisite: 561.

MATH 584: Numerical Linear Algebra
3.00 Credits

Numerical solution of linear systems by direct and iterative methods. Computation of eigenvalues and eigenvectors. Introduction to numerical methods for partial differential equations. Prerequisites: 222 and 501.

MATH 595: Directed Reading
3.00 Credits

Prerequisite: Permission of Undergraduate Committee.

MATH 596: Independent Study
3.00 Credits

Prerequisite: Permission of Undergraduate Committee.

MATH 600: Lattice Theory
3.00 Credits

Topics include partially ordered sets, lattices, complete lattices, ideals, modular and distributive lattices, Boolean algebras, geometric lattices, topological representation and applications. Prerequisite: 505.

MATH 601: Algebraic Categories I
3.00 Credits

Topics include categories, functors, subcategories, equivalence of categories, dual equivalence, special objects, generator, cogenerator, projective and injective objects, projective generator, and injective cogenerator, products of categories, Hom functor, internal Hom functor, functor categories, natural transformations, limit and co-limits, completeness and co-completeness, adjoint functors, and related topics. Prerequisite: 506.

MATH 602: Algebraic Categories II
3.00 Credits

Emphasis on functors, adjoint functor theorem, reflective subcategories, additive categories, Albelian categories and their applications. Prerequisite: 601.

MATH 623: Analytic Functions
3.00 Credits

Review of the classical theory of analytic functions and introduction to Banach spaces and Banach algebras; weak and strong analyticity of Banach space-valued functions, generalizations of classical theorems about analytic functions (e.g., Cauchy, Liouville, etc.); analytic functions in Banach algebras; application to Gelfand theory and Jacobson's theory of radicals; related topics. Prerequisites: 522 and 524.

MATH 624: Measure & Integration Theory
3.00 Credits

Basic measure theory, a development of the Lebesgue integral, the basic convergence theorems, Randon-Nikodym theorem, Fubini's theorem, Lp spaces. Prerequisite: 522.

MATH 625: Introduction to Functional Analysis
3.00 Credits

Banach and Hilbert spaces. Linear operators. Hahn-Banach theorem, open mapping and closed graph theorem, uniform boundedness theorem. Dual spaces and their representations. Prerequisite: 521.

MATH 626: Nonlinear Functional Analysis
3.00 Credits

Banach fixed point theorem with applications. Green's functions with boundary value problems. Semigroups of nonlinear contraction, and existence theorem for an initial value problem associated with a maximal monotone set. Prerequisite: 625.

MATH 627: Differential Equations in Banach Spaces
3.00 Credits

Mean value theorem for Banach space-valued piecewise differentiable functions, differentiable and Lipschitzian operators, construction of epsilon-approximate solution to a differential equation, existence and uniqueness of local solutions, extensions to a maximal solution. Continuity and differentiability of a solution with respect to initial data and parameters. Linear differential equations in the class of continuous operators, existence and properties of resolvent. Relations with PDEs of first order. Application of the theory to ODEs in concrete Banach spaces. Prerequisites: 540 and 625.

MATH 630: Theory of Probability
3.00 Credits

Probability and expectation spaces. Real random variables, almost sure convergence, convergence in probability. Product spaces and independent random variables. Conditional expectation and martingales. Strong and pointwise ergodic theorems. Kolmogorov's strong law of large numbers. Prerequisite: 624.

MATH 631: Computer Simulation Random Processes
3.00 Credits

Review of probability theory. Ergodic theorem and ergodic transformations as random number generators. Pseudo-random number generators on digital computers and their characteristics. Simulation of Bernoulli, Poison, and Gaussian random processes. Simulation of systems of independent random variables with arbitrary densities or distribution laws. Simulation of random variables on differentiable manifolds. A priori estimates of efficiency of stochastic methods for optimization of linear functions. Application to operations research and performance analysis of computer algorithms. Prerequisites: 531 and CSC 511.

MATH 633: Functional Analysis
0.00 Credits

Topics selected to serve research interests of instructor and students. Qualified students encouraged to work on research projects.

MATH 634: Functional Analysis
0.00 Credits

Topics selected to serve research interests of instructor and students. Qualified students encouraged to work on research projects.

MATH 638: Introduction to Finite Element Methods
3.00 Credits

Topics examined: steps in the method, such as domain discretization, choice of element configuration, selection of approximation model, derivation of element equation via a stationary principle, assemblage of global matrix equation, imposition of boundary conditions and solving for primary and secondary variables. Presents applications in elasticity, heat transfer and fluid mechanics. Prerequisites: 522 and CSC 113.

MATH 640: Partial Differential Equations
3.00 Credits

Preliminaries from ordinary differential equations, calculus and Lebesgue Integration Theory. Methods of solution of partial differential equations of the first order. Classification of the second order linear partial differential operators. Examples of applied problems leading to hyperbolic, parabolic, and elliptic equations. Cauchy problem. Fredholm alternative in Banach and Hilbert spaces. Properties of potentials. Dirichlet and Neumann problems. Finite difference method. Green's function, separation of variables, and expansion of solutions into eigenfunctions. Prerequisites: 540 and 624.

MATH 641: Optimal Control Theory
3.00 Credits

Examples of problems: electro-dynamical systems leading to time optimal control problems, spacecraft navigation leading to fuel optimal control problems. Proof of Pontryagin's maximum principle. Existence of optimal controls for linear systems with convex constraints on the control function. Derivation of Lagrange, Euler-Lagrange, and Hamilton equations of classical variational calculus from Pontryagin's principle. Method of first integrals and Hamilton-Jacobi generating function. Application to problems in celestial mechanics. Prerequisites: 540 and 624.

MATH 646: Banach Algebra
3.00 Credits

Develops Gelfand's theory of commutative Banach algebras. Presents applications of the theory to the special theory of operators, Stone-cech compactification, theory of C*-algebras, H*-algebras, and group algebras. Prerequisite: 625

MATH 648: Harmonic Analysis on Locally Compact Groups
3.00 Credits

Topological groups. Theorems of Urysohn's type. Haar integral over a locally compact group, commutative group and its dual. Pontryagin's duality theorem, group algebra. Unitary representation of a group and corresponding representation of group algebra. Stone's theorem. Positive definite functions and Bochner's theorem. Applications of Gelfand's theory to the group algebra. Prerequisite: 646.

MATH 653: Topological Vector Spaces
3.00 Credits

Vector space topologies, metrizability, locally convex topological vector spaces, Hahn-Banach theorem, projective and inductive topologies, barreled and bornological spaces. Linear mappings, Banach's homomorphism theorem, uniform boundedness and the Banach-Steinhaus theorem, duality, dual systems and weak topologies; strong dual, bi-dual, and reflexive spaces; theorems of Grothendieck, weak compactness, open mappings, and closed graph theorems; linear manifolds and applications. Prerequisites: 620 and 625 .

MATH 654: Generalized Functions and PDE
3.00 Credits

L. Schwartz spaces of infinitely differentiable functions, convolution operation, partition of unity. Dual spaces and differential operators on generalized functions. Fourier transforms on generalized functions, fundamental solution for a linear differential equation, existence of a general solution, application to Laplace equations. Approximation of generalized functions by infinitely differentiable functions. Kernel theorem and its applications. Prerequisite: 625 or 653.

MATH 666: Mathematical Foundations of Quantum Mechanics
3.00 Credits

Banach and Hilbert spaces, contraction mapping theorem, Lorentzian time delays, motion of n-charges in their electromagnetic field. Hamiltonian mechanics and category of Hamiltonian flows. Randomness due to initial trajectory. Direct integral of Hilbert spaces and probability amplitude. Description of position and momenta by self-adjoint operators. Functor of classical analogy. Heisenberg's and Schroedinger's representations. C*-algebras and operational calculus of operators based on Baire functions. Existence of spectral representations. Prerequisites: 624, 625, and 646.

MATH 991: Directed Reading
3.00 Credits

Staff.

MATH 992: Directed Reading
3.00 Credits

Staff.

MATH 993: Directed Reading
3.00 Credits

Staff.

MATH 994: Directed Reading
3.00 Credits

Staff.

MATH 995: Thesis - Masters
0.00 Credits

no description available

MATH 996: Thesis - Masters
0.00 Credits

no description available

MATH 997: Doctoral Dissertation Guidance
0.00 Credits

no description available

MATH 998: Doctoral Dissertation Guidance
0.00 Credits

no description available