## MATHEMATICS (MATH)

Professor Zhijian Wu, Chairperson

Office: 345 Gordon Palmer Hall

Professor Jon Corson, Undergraduate Advisor

Office: 333-C Gordon Palmer Hall

*Each student entering the University takes a mathematics placement examination. Students placed in MATH 005 must complete MATH 005 as a prerequisite for MATH 100. Students placed in MATH 100 must complete MATH 100 as a prerequisite for MATH 110, MATH 112, MATH 115, or any other **MA**-designated course. A grade of "C-" or higher is required in all prerequisite mathematics courses.*

**MATH 005 Remedial Mathematics.** No credit awarded.

Prerequisite: One unit of high-school mathematics.

Brief review of arithmetic operations followed by intensive drill in basic algebraic concepts: factoring, operations with polynomials and rational expressions, linear equations and word problems, graphing linear equations, simplification of expressions involving radicals or negative exponents, and elementary work with quadratic equations. Grades are reported as "A," "B," "C," or "NC" (No credit).

**MATH 100 Intermediate Algebra.** Three hours.

Prerequisites: Placement and two units of college-preparatory mathematics; if a student has previously been placed in MATH 005, a grade of "C" or higher in MATH 005 is required.

Intermediate-level course including work on functions, graphs, linear equations and inequalities, quadratic equations, systems of equations, and operations with exponents and radicals. The solution of word problems is stressed. NOT APPLICABLE to UA Core Curriculum mathematics requirement. Grades are reported as "A," "B," "C," or "NC" (No credit).

**MATH 110 Finite Mathematics.** Three hours.

Prerequisites: Placement and two units of college-preparatory mathematics; if a student has previously been placed in MATH 100, a grade of "C-" or higher in MATH 100 is required.

Sets and counting, permutations and combinations, basic probability, conditional probability, matrices and their application to Markov chains, and a brief introduction to statistics. Grades are reported as "A," "B," "C," or "NC" (No credit).

**MATH 112 Precalculus Algebra.** Three hours.

Prerequisites: Placement and three units of college-preparatory mathematics; if a student has previously been placed in MATH 100, a grade of "C-" or higher in MATH 100 is required.

A higher-level course emphasizing functions including polynomial functions, rational functions, and the exponential and logarithmic functions. Graphs of these functions are stressed. The course also includes work on equations, inequalities, systems of equations, the binomial theorem, and the complex and rational roots of polynomials. Applications are stressed. Grades are reported as "A," "B," "C," or "NC" (No credit).

**MATH 113 Precalculus Trigonometry.** Three hours.

Prerequisite: MATH 112.

Continuation of MATH 112. The course includes study of trigonometric functions, inverse trigonometric functions, trigonometric identities, and trigonometric equations. Complex numbers, De Moivre's Theorem, polar coordinates, vectors, and other topics in algebra are also addressed, including conic sections, sequences, and series. Grades are reported as "A," "B," "C," or "NC" (No credit).

**MATH 115 Precalculus Algebra and Trigonometry.** Three hours.

Prerequisite: Placement and a strong background in college-preparatory mathematics, including one-half unit in trigonometry.

Properties and graphs of exponential, logarithmic, and trigonometric functions are emphasized. Also includes trigonometric identities, polynomial and rational functions, inequalities, systems of equations, vectors, and polar coordinates. Grades are reported as "A," "B," "C," or "NC" (No credit). Degree credit will not be granted for both MATH 115 and MATH 112 or MATH 113.

**MATH 121 Calculus and Its Applications.** Three hours.

Prerequisite: MATH 112 or equivalent.

A brief overview of calculus primarily for students in the Culverhouse College of Commerce and Business Administration. **Warning: This course is not satisfactory preparation for curricula requiring standard calculus or higher mathematics,** *and*** it is not a prerequisite to calculus or higher mathematics.** Includes differentiation and integration of algebraic, exponential, and logarithmic functions, and applications in business and economics. Some work on functions of several variables and Lagrange multipliers is done. L'Hopital's Rule and multiple integration are included. Only business-related applications are covered. Degree credit will not be granted for both MATH 121 and MATH 125.

**MATH 125 Calculus I.** Four hours.

Prerequisites: MATH 112 and MATH 113, MATH 115, or placement.

This is the first of three courses in the basic calculus sequence. Topics include the limit of a function; the derivative of algebraic, trigonometric, exponential, and logarithmic functions; and the definite integral. Applications of the derivative are covered in detail, including approximations of error using differentials, maxima and minima problems, and curve sketching using calculus. There is also a brief review of selected precalculus topics at the beginning of the course. Degree credit will not be granted for both MATH 121 and MATH 125.

**MATH 126 Calculus II.** Four hours.

Prerequisites: MATH 125.

This is the second of three courses in the basic calculus sequence. Topics include vectors and the geometry of space, applications of integration, integration techniques, L'Hopital's Rule, improper integrals, parametric equations, polar coordinates, conic sections, and infinite series.

**MATH 145 Honors Calculus I.** Four hours.

Honors sections of MATH 125.

**MATH 146 Honors Calculus II.** Four hours.

Honors sections of MATH 126.

**MATH 208 Mathematics for Elementary School Teachers: Numbers and Operations.** Three hours.

Prerequisites: Elementary education major and grade of "C-" or higher in MATH 100.

Arithmetic of whole numbers and integers, fractions, proportion and ratio, and place value. Class activities initiate investigations underlying mathematical structure in arithmetic processes and include hands-on manipulatives for modeling solutions. Emphasis is on the explanation of the mathematical thought process. Students are required to verbalize explanations and thought processes and to write reflections on assigned readings on the teaching and learning of mathematics.

**MATH 209 Mathematics for Elementary School Teachers: Geometry and Measurement.** Three hours.

Prerequisites: Elementary education major and grade of "C-" or higher in MATH 100.

Properties of two- and three-dimensional shapes, rigid motion transformations, similarity, spatial reasoning, and the process and techniques of measurement. Class activities initiate investigations of underlying mathematical structure in the exploration of shape and space. Emphasis is on the explanation of the mathematical thought process. Technology specifically designed to facilitate geometric explorations is integrated throughout the course.

Prerequisites: Elementary education major and grade of "C-" or higher in MATH 208 or MATH 209.

Data analysis, statistics, and probability, including collecting, displaying/representing, exploring, and interpreting data, probability models, and applications. Focus is on statistics for problem solving and decision making, rather than calculation. Class activities deepen the understanding of fundamental issues in learning to work with data Technology specifically designed for data-driven investigations and statistical analysis is integrated throughout the course.

**MATH 227 Calculus III.** Four hours.

Prerequisite: MATH 126.

This is the third of three courses in the basic calculus sequence. Topics include vector functions and motion in space, functions of two or more variables and their partial derivatives, applications of partial derivatives (including Lagrange multipliers), quadric surfaces, multiple integration (including Jacobian), line integrals, Green's Theorem, vector analysis, surface integrals, and Stokes' Theorem.

**MATH 237 Applied Matrix Theory.** Three hours.

Prerequisite: MATH 126.

Corequisite: MATH 227.

Fundamentals of matrices and vectors in Euclidean space. Topics include solving linear systems of equations, matrix algebra, inverses, determinants, eigenvalues and eigenvectors. Also covers the basic notions of span, subspace, linear independence, basis, dimension, linear transformation, range, and null-space. Use of mathematics software is an integral part of the course.

**MATH 238 Applied Differential Equations I.** Three hours.

Prerequisite: MATH 227.

Introduction to analytic and numerical methods for solving differential equations. Topics include numerical methods and qualitative behavior of first order equations, analytic techniques for separable and linear equations, applications to population models and motion problems; techniques for solving higher-order linear differential equations with constant coefficients (including undetermined coefficients, reduction of order, and variation parameters), applications to physical models; the Laplace transform (including initial value problems with discontinuous forcing functions). Use of mathematics software is an integral part of the course.

**MATH 247 Honors Calculus III.** Four hours.

Honors sections of MATH 227.

**MATH 257 Linear Algebra.** Three hours.

Prerequisite: MATH 126.

Corequisite: MATH 227.

A theory-oriented course in which students are expected to understand and prove theorems. Topics include vector spaces and subspaces, linear independence, bases and dimension of vector spaces, solving systems of linear equations, matrices, determinants, linear transformations, eigenvalues, eigenvectors, and diagonalization.

**MATH 300 Introduction to Numerical Analysis.** Three hours.

Prerequisites: MATH 227, CS 114 or GES 126, and ability to program in a high-level programming language.

Credit will not be granted for both MATH 300 and MATH 411. A beginning course in numerical analysis. Topics include number representation in various bases, error analysis, location of roots of equations, numerical integration, interpolation and numerical differentiation, systems of linear equations, approximations by spline functions, and approximation methods for first-order ordinary differential equations and for systems of such equations.

**MATH 301 Discrete Mathematics.** Three hours.

Prerequisite: MATH 125.

An introductory course that primarily covers logic, recursion, induction, modeling, algorithmic thinking, counting techniques, combinatorics, and graph theory.

**MATH 303 Contemporary Applied Mathematics.** Three hours.

Prerequisites: CS 110 or CS 114, and MATH 125.

The course is primarily concerned with mathematical models of real-world situations in the physical and social sciences and the professions. It provides excellent background material for middle-school and secondary-school mathematics teachers. Usually offered in the fall semester.

**MATH 307 Introduction to the Theory of Numbers.** Three hours.

Prerequisite: MATH 227.

Divisibility theory in the integers, the theory of congruencies, Diophantine equations, Fermat's theorem and generalizations, and other topics. Usually offered in the spring semester.

**MATH 309 Foundations of Mathematics.** Three hours.

Prerequisite: MATH 126.

Provides background material for middle school and secondary school mathematics teachers. Topics include logic and proof, set theory, mathematical induction, Cartesian products, relations, functions, cardinality, basic concepts of higher algebra, and field properties of real numbers. Usually offered in the fall semester.

**MATH 343 Applied Differential Equations II.** Three hours.

Prerequisite: MATH 238.

Continuation of MATH 238. Topics include Laplace Transform methods, series solutions of second-order differential equations, the method of Frobenius, Bessel equations and functions, Fourier series, separation of variable method, elementary boundary valve problem for the Laplace, heat and wave equations, an introduction to Sturm-Liouville boundary valve problems, and phase plane analysis. Usually offered in the fall semester.

**MATH 355 Theory of Probability.** Three hours.

Prerequisite: MATH 227.

The foundations of the theory of probability, laws governing random phenomena, and their practical applications in other fields. Topics include probability spaces, properties of probability set functions, conditional probability, an introduction to combinatorics, discrete random variables, expectation of discrete random variables, Chebyshev's Inequality, continuous variables and their distribution functions, and special densities.

**MATH 371 Advanced Linear Algebra.** Three hours.

Prerequisite: MATH 237 or MATH 257.

Topics include inner product spaces, norms, self adjoint and normal operators, orthogonal and unitary operators, orthogonal projections and the spectral theorem, bilinear and quadratic forms, generalized eigenvectors, and Jordan canonical form. Usually offered in the spring semester.

**MATH 382 Advanced Calculus.** Three hours.

Prerequisites: MATH 227 and MATH 257; with permission of the department.

Further study of calculus with emphasis on theory. Topics include limits and continuity of functions of several variables; partial derivatives; transformations and mappings; vector functions and fields; vector differential operators; the derivation of a function of several variables as a linear transformation; Jacobians; orthogonal curvilinear coordinates; multiple integrals; change of variables; line integrals; and Green's, Stokes', and Divergence Theorems.

**MATH 402 History of Mathematics.** Three hours.

Prerequisite: Permission of the department; background in traditional high-school geometry, algebra, or calculus is recommended.

Survey of the development of some of the central ideas of modern mathematics, with emphasis on the cultural context.

**MATH 405 Geometry for Teachers.** Three hours.

Prerequisite: MATH 125 or permission of the instructor.

Topics include advanced Euclidean and analytic geometry. The course provides excellent background material for middle school and secondary school mathematics teachers. Usually offered in the fall semester.

**MATH 406 Curriculum in Secondary Mathematics.** Three hours.

Prerequisites: Admission to the teacher education program in secondary mathematics, BCT 300, and MATH 227; or permission of the instructor.

Future secondary mathematics teachers examine advanced concepts, structures, and procedures that comprise secondary mathematics.

**MATH 410 Numerical Linear Algebra.** Three hours.

Prerequisite: MATH 237 or MATH 257.

Further study of matrix theory, emphasizing computational aspects. Topics include direct solution of linear systems, analysis of errors in numerical methods for solving linear systems, least-squares problems, orthogonal and unitary transformations, eigenvalues and eigenvectors, and singular value decomposition. Usually offered in the spring semester.

**MATH 411 Introduction to Numerical Analysis** (previously MATH 311). Three hours.

Prerequisites: MATH 238; MATH 237 or MATH 257; CS 114 or GES 126; and ability to program in a high-level programming language.

Credit will not be granted for both MATH 411 and MATH 300. A rigorous introduction to numerical methods, formal definition of algorithms, and error analysis and their implementation on a digital computer. Topics include interpolation, roots, linear equations, integration and differential equations, and orthogonal function approximation. Usually offered in the fall semester.

**MATH 413 Finite-Element Methods.** Three hours.

Prerequisites: MATH 343 and MATH 382.

Corequisite: MATH 410.

Quadratic functionals on finite dimensional vector spaces, variational formulation of boundary value problems, the Ritz Galerkin method, the finite-element method, and direct and iterative methods for solving finite-element equations. Usually offered in the fall semester.

**MATH 419 Introduction to Optimization.** Three hours.

Prerequisite: MATH 237 or MATH 257.

A one-semester introduction to both linear and nonlinear programming for undergraduate students and non-math graduate students. Emphasis is on basic concepts and algorithms and the mathematical ideas behind them. Major topics in linear programming include the simplex method, duality, sensitivity, and network problems; major topics in nonlinear programming include optimality conditions, several search algorithms for unconstrained problems, and a brief discussion of constrained problems. In-depth theoretical development and analysis are not included.

**MATH 420 Linear Optimization Theory.** Three hours.

Prerequisite: MATH 237 or MATH 257

In-depth theoretical development and analysis of linear programming. Topics include formulation of linear programs, various simplex methods, duality, sensitivity analysis, transportation and networks, and various geometric concepts.

**MATH 421 Nonlinear Optimization Theory.** Three hours.

Prerequisite: MATH 237 or MATH 257.

In-depth theoretical development and analysis of nonlinear programming with emphasis on traditional constrained and unconstrained nonlinear programming methods and an introduction to modern search algorithms.

**MATH 422 Mathematics for Finance.** Three hours.

Prerequisites: MATH 227 and MATH 355, or consent of the department.

Topics include the basic no-arbitrage principle, binomial model, time value of money, money market, risky assets such as stocks, portfolio management, forward and future contracts, and interest rates.

**MATH 428 Introduction to Optimal Control.** Three hours.

Prerequisite: MATH 238.

Corequisite: MATH 410 or permission of the instructor.

Introduction to the theory and applications of deterministic systems and their controls. Major topics include calculus of variations, the Pontryagin's maximum principle, dynamic programming, stability, controllability, and numerical aspects of control problems. Usually offered in the fall semester.

**MATH 432 Graph Theory and Applications.** Three hours.

Prerequisite: MATH 237 or MATH 257.

Survey of several of the main ideas of general theory with applications to network theory. Topics include oriented and nonoriented linear graphs, spanning trees, branching and connectivity, accessibility, planar graphs, networks and flows, matching, and applications. Usually offered in the fall semester.

**MATH 441 Boundary Value Problems.** Three hours.

Prerequisite: MATH 238.

Methods of solving the classical second-order linear partial differential equations: Laplace's equation, the heat equation, and the wave equation, together with appropriate boundary or initial conditions. Usually offered in the fall semester.

**MATH 442 Integral Transforms and Asymptotics.** Three hours.

Prerequisite: MATH 441.

Complex variable methods, integral transforms, asymptotic expansions, WBK method, Airy's equation, matched asymptotics, and boundary layers. Usually offered in the spring semester.

**MATH 445 Theoretical Foundations of Fluid Dynamics I.** Three hours.

Prerequisite: MATH 343, AEM 264 or equivalent, or permission of the department.

Introduction to continuum mechanics and tensors. Local fluid motion. Equations governing fluid flow and boundary conditions. Some exact solutions of the Navier-Stokes equations. Vortex motion. Potential flow and aerofoil theory.

**MATH 451 Mathematical Statistics with Applications I.** Three hours.

Prerequisites: MATH 237, or MATH 257 and MATH 355.

Introduction to mathematical statistics. Topics include bivariate and multivariate probability distributions, functions of random variables, sampling distributions and the central limit theorem, concepts and properties of point estimators, various methods of point estimation, interval estimation, tests of hypotheses, and Neyman-Pearson Lemma, with some applications. Usually offered in the fall semester.

**MATH 452 Mathematical Statistics with Applications II.** Three hours.

Prerequisite: MATH 451.

Further applications of the Neyman-Pearson Lemma, Likelihood Ratio tests, Chi-square test for goodness of fit, estimation and test of hypotheses for linear statistical models, analysis of variance, analysis of enumerative data, and some topics in nonparametric statistics. Usually offered in the spring semester.

**MATH 457 Stochastic Processes with Applications I.** Three hours.

Prerequisite: MATH 355 or equivalent.

Introduction to the basic concepts and applications of stochastic processes. Markov chains, continuous-time Markov processes, Poisson and renewal processes, and Brownian motion. Applications of stochastic processes including queueing theory and probabilistic analysis of computational algorithms.

**MATH 459 Stochastic Processes with Applications II.** Three hours.

Prerequisite: MATH 457 or equivalent.

Continuation of MATH 457. Advanced topics of stochastic processes including Martingales, Brownian motion and diffusion processes, advanced queueing theory, stochastic simulation, and probabilistic search algorithms (simulated annealing). Usually offered in the fall semester.

**MATH 460 Introduction to Differential Geometry.** Three hours.

Prerequisite: MATH 486, or MATH 382 and permission of the department.

Introduction to basic classical notions in differential geometry: curvature, torsion, geodesic curves, geodesic parallelism, differential manifold, tangent space, vector field, Lie derivative, Lie algebra, Lie group, exponential map, and representation of a Lie group. Usually offered in the spring semester.

**MATH 465 Introduction to General Topology.** Three hours.

Prerequisite: MATH 486.

Basic notions in topology that can be used in other disciplines in mathematics. Topics include topological spaces, open sets, closed sets, basis for a topology, continuous functions, separation axioms, compactness, connectedness, product spaces, quotient spaces, and metric spaces. Usually offered in the fall semester.

**MATH 466 Introduction to Algebraic Topology.** Three hours.

Prerequisites: MATH 465 and MATH 470.

Homotopy, fundamental groups, covering spaces, covering maps, and basic homology theory, including the Eilenberg Steenrod axioms. Usually offered in the spring semester.

**MATH 467 Advanced Geometry.** Three hours.

Prerequisite: MATH 405 or permission of the instructor.

This is a second course in axiomatic geometry. Topics include Euclidean and non-Euclidean geometry, studied from an analytic point of view and from the point of view of transformation geometry. Some topics in projective geometry may also be treated. Usually offered in the spring semester.

**MATH 470 Principles of Modern Algebra I.** Three hours.

Prerequisite: MATH 237 or MATH 257.

A first course in abstract algebra. Topics include groups, permutation groups, Cayley's theorem, finite abelian groups, isomorphism theorems, rings, polynomial rings, ideals, integral domains, and unique factorization domains. Usually offered in the fall semester.

**MATH 471 Principles of Modern Algebra II.** Three hours.

Prerequisite: MATH 470.

Introduction to the basic principles of Galois Theory. Topics include rings, polynomial rings, fields, algebraic extensions, normal extensions, and the fundamental theorem of Galois Theory. Usually offered in the spring semester.

**MATH 474 Cryptography.** Three hours.

Prerequisites: MATH 307, MATH 470, or permission of the department.

Introduction to the rapidly growing area of cryptography, an application of algebra, especially number theory. Usually offered in the fall semester.

**MATH 485 Introduction to Complex Calculus.** Three hours.

Prerequisite: MATH 227.

Some basic notions in complex analysis. Topics include analytic functions, complex integration, infinite series, contour integration, and conformal mappings. Usually offered in the spring semester.

**MATH 486 Introduction to Real Analysis I** (previously MATH 380)**.** Three hours.

Prerequisite: MATH 237 or MATH 257.

Rigorous development of the calculus of real variables. Topics include topology of the real line, sequences, limits, continuity, and differentiation. Usually offered in the fall semester.

**MATH 487 Introduction to Real Analysis II** (previously MATH 481). Three hours.

Prerequisite: MATH 486.

Riemann integration, introduction to Reimann-Stieltjes integration, series of constants and convergence tests, sequences and series of functions, uniform convergence, power series, Taylor series, and the Weierstrass Approximation Theorem. Usually offered in the spring semester.

**MATH 495 Seminar/Directed Reading.** One to three hours.

Offered as needed.