## Mathematics Courses

Brief review of arithmetic operations and 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 pass/fail.

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).

This course is intended to give an overview of topics in finite mathematics with applications. This course covers mathematics of finance, logic, set theory, elementary probability and statistics. This course does not provide sufficient background for students who will need to take Precalculus Algebra or Calculus.

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). Degree credit will not be granted for both MATH 115 and (MATH 112 or MATH 113).

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). Degree credit will not be granted for both MATH 115 and (MATH 112 or MATH 113).

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).

A brief overview of calculus primarily for students in the Culverhouse College of Commerce and Business Administration. This course does not provide sufficient background for students who will need higher levels of Calculus. Note: This course does not satisfy the requirement for MATH 125 or 126. Degree credit will not be granted for both MATH 121 and MATH 125 or MATH 145.

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 or MATH 145.

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.

This course covers the same material as MATH 125 but in a depth appropriate for honors students. It is the first course in the three part honors calculus sequence for students majoring in mathematics, science or engineering. Topics include limits, continuity, differentiation, applications of differentiation, and integration. Applications of the derivative are covered in detail, including approximation of errors using differentials, maxima and minima problems, curve sketching, optimization problems, and Newton’s method. Topics on integration include Riemann sums, properties of definite integrals, integration by substitution and integrals involving logarithmic exponential and trigonometric functions.

This course covers the same material as MATH 126 but in a depth appropriate for honors students. It is the second course in the three part honors calculus sequence for students majoring in mathematics, science or engineering. Topics include vectors and the geometry of space, L'Hospital's Rule, applications of integration, integration techniques, improper integrals, infinite series, conic sections, plane curves, parametric equations, and polar coordinates.

This course is the first of a three-course sequence designed to develop deeper understanding of elementary school mathematics content needed for teaching. The course topics include whole numbers and integers, fractions, ratio, percent, decimals and arithmetic operations within these systems. The goal of the course is to develop conceptual understanding (instead of just procedural understanding) of the number systems and operations by focusing on basic concepts and principles, exploring multiple representations and strategies, and illuminating connections among concepts and procedures. The content knowledge needed for teaching will be reinforced by engaging in inquiry-based activities, analyzing children’s ways of thinking, focusing on explanation and communication of underlying mathematical principles when solving problems, and using appropriate manipulative and technology.

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.

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 related to elementary school teaching is integrated throughout the course.

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; and 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.

Fundamentals of linear algebra and matrix theory are covered. Topics include vectors in Euclidean spaces, solving systems of linear equations, matrix algebra, inverses, determinants, eigenvalues, and eigenvectors. Also vector spaces and the basic notions of span, subspace, linear independence, basis, dimension, linear transformation, kernel and range are considered. Computing proficiency is required for a passing grade in this course.

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 of parameters), applications to physical models; the Laplace transform (including intial value problems with discontinuous forcing functions). Use of mathematics software is an integral part of the course. Computing proficiency is required for a passing grade in this course.

This course covers the same material as MATH 227 but in a depth appropriate for honors students. It is the third course in the three part honors calculus sequence for students majoring in mathematics, science or engineering. Topics include analytic geometry in space, vector-valued functions and motion in space, functions of two or more variables and their partial derivatives, applications of partial differentiation (including Lagrangian multipliers), quadric and cylindrical surfaces, and multiple integration (including Jacobian) and applications, line integrals, Green's Theorem, curl and divergence, surface integrals, and Stokes’ Theorem.

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.

An introduction to mathematical logic and proof within the context of discrete structures. Topics include basic mathematical logic, elementary number theory, basic set theory, functions, and relations. Writing proficiency is required for a passing grade in this course. A student who does not write with the skill normally required of an upper-division student will not earn a passing grade, no matter how well the student performs in other areas of the course.

A supplemental course in discrete mathematics covering select topics of interest in computer science. Topics include graphs and trees, finite state automata and regular expressions, efficiency of algorithms.

Continuation of Appl Diff Equations I (MATH 238) and is designed to equip students with further methods of solving differential equations. Topics include initial value problems with variable coefficients, methods of infinite series, two-point boundary value problems, wave and heat equations, Fourier series, Sturm-Liouville theory, phase plane analysis, and Liapunov's second method.

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; and an introduction to combinatorics, discrete random variables, expectation of discrete random variables, Chebyshev's Inequality, continuous variables and their distribution functions, and special densities.

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.

Explore the interconnections between the algebraic, analytic, and geometric areas of mathematics with a focus on properties of various number systems, importance of functions, and the relationship of algebraic structures to solving analytic equations. This exploration will also include the development and sequential nature of each of these branches of mathematics and how it relates to the various levels within the algebra mathematics curriculum.

This is a seminar style course focusing on various mathematical topics related to the high school curriculum. Topics will vary depending upon instructor.

This course will give an overview of geometry from a modern point of view. Axiomatic, analytic, transformational, and algebraic approaches to geometry will be used. The relationship between Euclidean geometry, the geometry of complex numbers, and trigonometry will be emphasized.

Concepts and techniques of posing questions and collecting, analyzing, and interpreting data. Topics include: univariate and bivariate statistics, probability, simulation, confidence intervals and hypothesis testing.

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.

Credit will not be granted for both MATH 411 and MATH 300. An introduction to numerical methods. Topics include numerical methods for solving nonlinear equations; iterative methods for solving systems of equations; approximations and interpolations; numerical differentiation and integration; and numerical methods for solving initial value problems for ordinary differential equations.

This is the second course in the numerical analysis sequence for senior students in mathematics, science, or engineering. Topics include numerical methods for solving boundary value problems, ordinary differential equations, and partial differential equations, multistep methods for initial value problems, and approximation theory (least-squares problems,fast Fourier Transforms).

This course is an introduction to theory of linear programming (focused on development of theory and algorithms with only a limited coverage of examples and applications), a basic component of optimization theory. Topics include: basic theory (fundamental theorem of LP, equivalence of basic feasible solutions and extreme points, duality and sensitivity results), simplex algorithm and its variations, and special applications to transportation and network problems. Non-simplex methods are also briefly introduced.

This course is an introduction to nonlinear programming. Topics will include necessary and sufficient conditions for optimality, as well as basic theory and numerical algorithms for several traditional optimization methods, e.g., basic descent methods, conjugate direction methods, quasi-Newton methods, penalty and barrier methods, Lagrange multiplier methods. A brief introduction to selected modern topics may be added if time permits.

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.

Complex variable methods, integral transforms, asymptotic expansions, WKB method, Airy's equation, matched asymptotics, and boundary layers.

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.

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.

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.

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.

Basic notions in topology that can be used in other disciplines in mathematics. Topics include topological spaces, open sets, basis for a topology, continuous functions, seperation axioms, compactness, connectedness, product spaces, quotient spaces.

Homotopy, fundamental groups, covering spaces, covering maps, and basic homology theory, including the Eilenberg Steenrod axioms.

A first course in abstract algebra. Topics include groups, cyclic groups, non-abelian groups, Lagrange's theorem, subgroups, cosets, homomorphisms, isomorphisms, rings.

An introduction to ring theory. Topics include rings, polynomial rings, matrix rings, modules, fields and semi-simple rings. Usually offered in the fall semester.

Rigorous development of the calculus of real variables. Topics include the topology of the real line, sequences and series, limits, limit suprema and infima, continuity, and differentiation.

A continuation of MATH 486. Topics include Riemann integration, sequences and series of functions, uniform convergence, power series, Taylor series. Optional topics may include the Reimann-Stieltjes integration, Weierstrass Approximation Theorem and the Arzela-Ascoli Theorem, metric spaces, multi-variable calculus.

Offered as needed.

Independent or collaborative research experience in mathematics.