# Department of Mathematics

## Course Descriptions

### 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. Note that this course cannot be used to fulfill the Math/Natural sciences requirement in the School of Arts and Sciences.

### MATH 110: Finite Mathematics for Business and Economics

3.00 Credits

This course introduces students to mathematics used in business and economics. The main topic areas are: (1) Functions, particularly linear and quadratic functions and models. (2) Linear systems and Gauss-Jordan elimination; graphical and simplex methods of linear programming. (3) Exponential and logarithmic functions; fundamentals of financial mathematics including compound interest, annuities, and amortization. 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 MATH 121. Prerequisite:

School of Business and Economics: Placement score of 1 or 2, or, Grade of P in Online Review Math 11, or, Grade of C- or better in Math 110

All others: Placement score of 1 or 2, or, Grade of P in Online Review Math 21, or, Grade of C- or better in Math 108

### 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 MATH 122. Prerequisite: a grade of C- or better in MATH 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 MATH 111 except by Department permission. Prerequisite: Placement score of 1 or 2, or, Grade of P in Online Review Math 21, or, Grade of C- or better in Math 108.

### 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 MATH 112 except by Department permission. Prerequisite: a grade of C- or better in MATH 121.

### MATH 168: Mathematics in the Modern World

3.00 Credits

Intended for liberal arts students. 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 175: Mathematics in Politics

3.00 Credits

This course, intended for liberal arts students, explores the mathematics involved in political concepts and applications. Topics include social choice, voting procedures and their inherent paradoxes, contributions of Arrow and Codorcet; yes/no voting and the Banzhaf and Shapley-Shubik power indices; apportionment of the House of Representatives relating the procedures of Hamilton, Jefferson, Adams, Webster and Hill-Huntington and their inherent paradoxes; fair division, including cake-cutting and inheritance division procedures. Not open to students who took MATH 168 in Spring 2012 or prior

### 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: MATH 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: MATH 221 or permission of instructor.

### MATH 230: Mathematical Topics in the Social Sciences I (UH)

3.00 Credits

A rigorous mathematical treatment of the following topics: 1. Theory of social choice including a critical approach to different vote-aggregation procedures and a study of their vulnerability to manipulation; Condorcet paradox and the intransitivity of the pair-wise majority rule; other paradoxes of collective choice; May's theorem. 2. Yes-No voting: Banzhaf and Shapley-Shubik indices of political power, their paradoxes and the formal mathematical relation between them; swap and trade robustness; vector-weightedness and dimension of a yes-no voting system.

### MATH 248: 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: MATH 122 or department permission.

### MATH 301: 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: MATH 221 or MATH 248 or CSC 210.

### 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 122, or permission of department.

### MATH 314: Statistics II

3.00 Credits

This is a second course in statistics building off of Math 114. It will focus on design of experiments and analysis of variance, categorical data analysis, regression analysis, time series/forecasting, and data visualization. The main emphasis will be on regression. Prerequisite: Math 114 or equivalent.

### MATH 321: Abstract Algebra I

3.00 Credits

Groups; finite groups; permutations; cyclic groups; subgroups, Lagrange’s theorem; normal subgroups and factor groups; group homomorphisms; isomorphism theorems; external and internal direct products; Fundamental Theorem on finite Abelian groups; introduction to rings; integral domains; ideals and factor rings; ring homomorphisms. Prerequisite: MATH 248 or CSC 210

### MATH 322: Abstract Algebra II

3.00 Credits

A study of rings, integral domains and fields; polynomial rings; division and factorization in integral domains; unique factorization domains; Euclidean domains; finite and algebraic field extensions; finite fields; introduction to Galois theory; solvability of polynomials by radicals. Prerequisite: MATH 321.

### MATH 330: Mathematical Topics in the Social Sciences II (UH)

3.00 Credits

A rigorous mathematical treatment of the following topics: 1. Apportionment of the House of Representatives with focus on the mathematical theory and the paradoxes involved; history and development of apportionment procedures from the early contributions of Alexander Hamilton and Thomas Jefferson to the currently used Hill-Huntington procedure; the impossibility theory of Balinski and Young; different measures of inequity and the alternative approach to the apportionment problem through equity considerations. 2. Social Welfare theory including a thorough treatment of Arrow's impossibility theorem and Arrow's axioms of unrestricted domain, collective rationality, weak Pareto and independent of irrelevant alternatives; dictatorship, oligarchy and the weakening Arrow's axioms.

### MATH 403: 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: MATH 248 or CSC 210.

### MATH 407: 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 248 or CSC 210.

### MATH 408: 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 248 or CSC 210.

### MATH 409: 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: MATH 408.

### MATH 410: Introduction to Lie groups and Lie algebras

3.00 Credits

An introduction to the theory of Lie groups and Lie Algebras via matrix groups. We construct the prominent classical families of Lie groups by working with matrix algebras over the real numbers, complex numbers, and quaternions. We then define the Lie algebras as tangent spaces of these groups. Prerequisites: MATH 221, MATH 248, MATH 301

### MATH 415: 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; and applications. Prerequisite: MATH 248 or CSC 210.

### MATH 420: 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: MATH 221 and either MATH 321 or MATH 421.

### MATH 421: Introductory Analysis I

4.00 Credits

The real number system; limits of sequences; Bolzano-Weierstrass Theorem; numerical series; basic topology on the set of real numbers, compactness; functional limits; continuity and uniform continuity of function of one real variable; derivative, properties of differentiable functions; sequences and series of functions; pointwise and uniform convergence; power series; Taylor series; Riemann Integral; Fundamental Theorem of Calculus. Prerequisites: MATH 221 and MATH 248.

### MATH 422: Introductory Analysis II

3.00 Credits

The Riemann-Stieltjes integral; equicontinuous families of functions and Arzela-Ascoli theorem; introduction to Fourier analysis; topological spaces, normed vectors spaces, compact spaces and the Tychonoff theorem. Linear functionals, Banach and Hilbert spaces, orthonormal bases and orthogonal projections in Hilbert spaces. Hahn-Banach, Baire Category, and open mapping theorems and the uniform boundedness principle. Prerequisite: MATH 421.

### MATH 424: 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: MATH 221.

### MATH 427: 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: MATH 222.

### MATH 431: 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: MATH 221, or permission of department.

### MATH 432: 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: MATH 431.

### MATH 434: Introduction to Mathematical Finance

3.00 Credits

An introduction to elementary probability and finance, geometric Brownian motion, and martingales. Interest rate models, Modern portfolio theory, and Option pricing theory. Prerequisites: (MATH 112 or 122) and (MATH 309 or 431)

### MATH 436: Introduction to Game Theory

3.00 Credits

Strategic games, Nash equilibrium, two-person zero-sum games, two-person general-sum games, extensive games with perfect information, cooperative games, non-cooperative games; and applications. Prerequisites: MATH 301 and MATH 431 or permission of the instructor.

### MATH 441: Introduction to Partial Differential Equations

3.00 Credits

Survey of linear equation types: dispersion, diffusion, wave, transport. Exposition of techniques to find solutions for initial value problems, including separation of variables, characteristics and transform method. Introduction to boundary value problems and the modern unified transform method. Prerequisites: MATH 221, 222 or permission of the instructor.

Graduate Level Version of the course will include the above but will also require: Analysis of nonlinear equations including the nonlinear Schrodinger equation and Korteweg de-Vries equation. Prerequisites: MATH 221, 222 or permission of the instructor.

The real number system; limits of sequences; Bolzano-Weierstrass Theorem; numerical series; basic topology on the set of real numbers, compactness; functional limits; continuity and uniform continuity of function of one real variable; derivative, properties of differentiable functions; sequences and series of functions; pointwise and uniform convergence; power series; Taylor series; Riemann Integral; Fundamental Theorem of Calculus. Prerequisites: MATH 221 and MATH 248.

### MATH 442: Introduction to Difference Equations

3.00 Credits

First order difference equations,higher order difference equations, stability analysis, z-transforms and applications. Prerequisites: MATH 222, MATH 301 or permission of the instructor.

### MATH 450: Foundations of Mathematics

3.00 Credits

Sets, logic, and axiomatic and constructive treatment of real numbers. Prerequisite: MATH 248.

### MATH 451: 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: MATH 248 and MATH 321.

### MATH 461: 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: MATH 222 and MATH 301.

### MATH 462: 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: MATH 461.

3.00 Credits

3.00 Credits

### MATH 498: Undergraduate Comprehensive Examination

0.00 Credits

no description available

### 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: MATH 221 or MATH 248 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: MATH 248 or CSC 210.

### MATH 505: Abstract Algebra I

3.00 Credits

Groups; finite groups; permutations; cyclic groups; subgroups, Lagrange’s theorem; normal subgroups and factor groups; group homomorphisms; isomorphism theorems; external and internal direct products; Fundamental Theorem on finite Abelian groups; conjugacy classes; Sylow theorems; introduction to rings; integral domains; ideals and factor rings; ring homomorphisms. Prime and maximal ideals of commutative rings. Prerequisite: MATH 248 or CSC 210

### MATH 506: Abstract Algebra II

3.00 Credits

A study of rings, integral domains and fields; polynomial rings; division and factorization in integral domains; unique factorization domains; Euclidean domains; finite and algebraic field extensions; finite fields; separable and normal field extensions; introduction to Galois theory; solvability of polynomials by radicals; ruler and compass constructions. Prerequisite: MATH 321.

### MATH 507: Graph Theory

3.00 Credits

In-depth coverage of topics: Directed and undirected graphs, trees, connectivity; cut edges, cut vertices, and blocks; Eulerian and Hamiltonian graphs; planarity; coloring problems; Domination in Graphs, Graph Parameters; graph-theoretic algorithms and applications. Prerequisite: MATH 248 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 248 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: MATH 508.

### 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: MATH 322.

### MATH 514: Statistics II

3.00 Credits

The course has an in-depth focus on design of experiments and analysis of variance, categorical data analysis, regression analysis, time series/forecasting, and data visualization. Additional topics include logistic regression, survival analysis, and data analysis using R. Prerequisite: Math 114 or equivalent.

### MATH 515: Combinatorics

3.00 Credits

In-depth coverage of topics: 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; Special counting sequences; Experimental design; and applications. Prerequisite: MATH 248 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 431 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: MATH 221 and either MATH 321 or MATH 421.

### MATH 521: Introductory Analysis I

4.00 Credits

The real number system; limits of sequences; Bolzano-Weierstrass Theorem; numerical series; basic topology on the set of real numbers, compactness; functional limits; continuity and uniform continuity of function of one real variable; derivative, properties of differentiable functions; sequences and series of functions; pointwise and uniform convergence; power series; Taylor series; Riemann Integral; Fundamental Theorem of Calculus; basic concepts of topology; open and closed sets; compact sets; metric spaces. Prerequisites: MATH 221 and MATH 248.

### MATH 522: Introductory Analysis II

3.00 Credits

The Riemann-Stieltjes integral; measure theory in R; Lebesgue integration in R; Lp-spaces; equicontinuous families of functions and Arzela-Ascoli theorem; introduction to Fourier analysis; topological spaces, normed vectors spaces, compact spaces and the Tychonoff theorem. Linear functionals, Banach and Hilbert spaces, orthonormal bases and orthogonal projections in Hilbert spaces. Hahn-Banach, Baire Category, and open mapping theorems and the uniform boundedness principle. Prerequisite: MATH 421.

### 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: MATH 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: MATH 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: MATH 420 and MATH 421.

### 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: MATH 221, or permission of department.

### 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, and special topics which may include but not limited to Bayesian methods, Nonparametric statistics. Prerequisite: MATH 431

### MATH 533: Stochastic Processes

3.00 Credits

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

### MATH 536: Introduction to Game Theory

3.00 Credits

In-depth coverage of topics: Strategic games, Nash equilibrium, two-person zero-sum games, two-person general-sum games, extensive games with perfect information, cooperative games, non-cooperative games, N-Person Cooperative Games; and applications. Prerequisites: MATH 301 and MATH 431 or permission of the instructor.

### MATH 537: Introduction to Fuzzy Sets and Fuzzy Logic

3.00 Credits

Fuzzy sets; fuzzy sets versus crisp sets; operations on fuzzy sets;, fuzzy relations; fuzzy theory versus probability theory and applications. Prerequisites: MATH 221 and MATH 431.

### 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: MATH 222 and MATH 301.

### MATH 541: Introduction to Partial Differential Equations

3.00 Credits

Fourier Series, Fourier Transforms, 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, MATH 301 or permission of the instructor.

### MATH 550: Foundations of Mathematics

3.00 Credits

Sets, logic, and axiomatic and constructive treatment of real numbers. Prerequisite: MATH 248.

### 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: MATH 248 and MATH 321.

### 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: MATH 451.

### 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: MATH 222 and MATH 301.

### 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: MATH 461.

### MATH 570: Algebraic Topology

3.00 Credits

Brief introduction to category theory: categories and their functors, examples. Homology theory: chain complexes, homology groups of a simplicial complex. Degrees of maps between manifolds and applications. Homotopy theory: definition of the fundamental group, presentations and calculations of such groups.

### 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: MATH 222 and MATH 301.

### MATH 600: Lattice Theory

3.00 Credits

Partially ordered sets; lattices; complete lattices; modular and distributive lattices; Boolean algebras; Geometric lattices; and applications. Prerequisite: MATH 248 or CSC 210 or permission of instructor.

### MATH 601: Algebraic Categories I

3.00 Credits

Categories; Functors: Subcategories: Equivalent of Categories, Dual equivalence, Special objects, Generator, Co-generator, projective and injective objects; products of categories, Hom functor, internal Hom functor, natural transformations, limit and co-limits, completeness and cocompleteness, left adjoints, and related topics. Prerequisite: MATH 501 or permission of instructor.

### MATH 602: Algebraic Categories I

3.00 Credits

Emphasis on Functors, Adjoint Functor Theorem, Reflective, Additive Categories, Abelian Categories and related topics. Prerequisite: MATH 601.