# Math courses

## Fiqws 12201: The Statistical Study

We look at mathematical issues involved in planning and carrying out a truthful statisical study. Among them are sampling and data analysis. We cover the following topics from elementary probability and descriptive statistics: Frequency histograms; the multiplication rule; conditional probability; probability distributions; independent events; Bayes' theorem; random variables, expected values; testing statistical hypotheses; correlation; linear regression and least squares. Sections

## Math 15000: Mathematics for the Contemporary World

Bombarded by statistics, assailed by advertisers and advocates of all persuasions, the average person needs mathematics to make sense of the world. This course aims to give students the tools needed to critically examine the quantitative issues of our times. Students will learn the basics of logical reasoning, the use of graphs and algebra to create quantitative models, and the role of statistics and probability in analyzing data. We will apply these ideas to assess the quantitative claims raised in contemporary case studies commonly discussed in the media. 3 hr./wk; 3 cr.

## Math 17300: Introduction to Probability and Statistics

Descriptive statistics and frequency histograms; measures of location and dispersion; elementary probability; permutations and combinations; multiplication rule and conditional probability; Bayes' Theorem; independent events; random variables, expected values; applications to binomial, hypergeometric, uniform and normal distributions; the Central Limit Theorem; testing statistical hypotheses; correlation; linear regression and least squares. Prereq.: placement by the Department. Credit will be given for only one of the following courses: Math 17300, Eco 29500, Psy 21500, Soc 23100.4 hr./wk.; 4 cr.

## Math 17700: Introduction to Biostatistics

Designed for students with little previous coursework in mathematics or statistics. Topics include research ethics, study design, data description, elements of probability, distribution of random variables, applications of the binomial and normal distributions, estimation and confidence intervals, hypothesis testing, contingency tables, regression, and analysis of variance. Additional topics include introduction to statistical computing and data management, distribution free statistical methods, demographic measures, and life tables.

## Math 18000: Quantitative Reasoning

Investigation of the basis for elementary operations in concrete situations, diagrams, and symbolic representation. Understanding of, and problem-solving in, the following areas: numerical operations, ratios and percents, linear and exponential growth in situations, formulas, and graphs; rate of change; mensurational geometry; units, dimension, and scaling. Prereq.: placement by the Department. 4 hr./wk.; 3 cr.

## Math 18500: Basic Ideas in Mathematics

Problem solving, sets, operations with sets, functions, numerical systems with different bases, topics in number theory, probability and geometry. Includes writing exercises and collaborative work. This course is for potential education majors only. Prereq.: C or better in Math 18000 or placement by the department.4 hr./wk.; 3 cr.

## Math 19000: College Algebra and Trigonometry

Introduction to functions, rational expressions and their applications, rational exponents, conic sections, Gaussian elimination and determinants, nonlinear systems of equations, introductions to trigonometric functions. Prereq.: placement at college entry or by subsequent examination. 4 hr./wk.; 2 cr.

## Math 19500: Precalculus

Intervals, inequalities, operations on functions, inverse functions, graphing polynomial functions, exponential and logarithmic functions, trigonometric functions and formulas. Prereq.: Math 19000 with an A, B, or C or placement. 4 hr./wk.; 3 cr.

## Math 20100: Calculus I

Limits, derivatives, rules of differentiation, trigonometric functions and their derivatives, differentials, graph sketching, maximum and minimum problems, related rates, antidifferentiation, conic sections, introduction to vectors.: grade of C or higher in Math 19500 or placement by the Department. Credit will be given for only one of the following courses: Math 20100 or 20500. (Part of sequence 20100, 20200, 20300.) 4 hr./wk.; 3 cr.

## Math 20200: Calculus II

Areas between curves; volumes of solids of revolution; integration of trigonometric, exponential and logarithmic functions, analytical and numerical methods of integration, improper and infinite integrals, conic sections, polar coordinates; parametric representation of curves, vectors in the plane. Prereq.: grade of C or higher in Math 20100 or placement by the Department. After completion of Math 20900, only 3 credits will be given for Math 20200. (Part of sequence 20100, 20200, 20300.)4 HR. LECT./WK.; 3 CR.

## Math 20300: Calculus III

Vectors, infinite series, Taylor's theorem, solid analytic geometry, partial derivatives, multiple integrals with applications. Interpretations and calculations using Matlab software. Prereq.: Grade of C or higher in Math 20200 or placement by the Department. 4 HR./WK.; 1 HR LAB./WK.; 4 CR.

## Math 20500: Elements of Calculus

Limits, derivatives, rules of differentiation, differentials, graph sketching, maximum and minimum problems, related rates, exponential and logarithmic functions, differential equations, anti-derivatives, area, volume, applications to economics. Prereq.: grade of C or higher in Math 19500 or placement by the Department. Credit will be given for only one of the following courses: Math 20100 or 20500. 4 hr./wk.; 4 cr. 20500 is for Biology and Economics majors, as well as students in the School of Architecture.

## Math 20900: Elements of Calculus and Statistics

Introduction to differential equations including numerical method; qualitative analysis of solutions; phase plane analysis for systems; biological applications; analysis of univariate and bivariate data; regression and correlation; random variables; the normal, Poisson and binomial distributions; statistical inference. A spreadsheet program such as Excel is used throughout the course. Prereq.: Math 20500 or placement� by the Department. (Part of sequence 20500, 20900 for Biology majors.)� 4 HR./WK.; 4 CR.

## Math 30100: Honors I

Approval of Department Honors Advisor required. Credit flexible but usually 3 credits per term.

## Math 30200: Honors II

Approval of Department Honors Advisor required. Credit flexible but usually 3 credits per term.

## Math 30300: Honors III

Approval of Department Honors Advisor required. Credit flexible but usually 3 credits per term.

## Math 30400: Honors IV

Approval of Department Honors Advisor required. Credit flexible but usually 3 credits per term.

## Math 30800: Bridge to Advanced Mathematics

This course explores the logical and foundational structures of mathematics, with an emphasis on understanding and writing proofs. Topics include set theory, logic, mathematical induction, relations and orders, functions, Cantor's theory of countability, and development of the real number system. 3 HR./WK.; 3 CR.

## Math 31000: Independent Study

A program of independent study under the direction of a member of the Department with the approval of the Assistant Chair. Credit may be from 1-4 credits, as determined before registration by the instructor with the approval of the Assistant Chair.

## Math 31100: Selected Topics in Mathematics

Topics not covered in the usual department offerings. Topics vary from semester to semester, depending on student and instructor interest. Prerequisites as determined by the instructor.� Credits and hours will be determined by the instructor and the department, with a maximum of 4 credits per course.

## Math 31200: Selected Topics in Mathematics

Topics not covered in the usual department offerings. Topics vary from semester to semester, depending on student and instructor interest. Prerequisites as determined by the instructor.� Credits and hours will be determined by the instructor and the department, with a maximum of 4 credits per course.

## Math 31300: Selected Topics in Mathematics

Topics not covered in the usual department offerings. Topics vary from semester to semester, depending on student and instructor interest. Prerequisites as determined by the instructor.� Credits and hours will be determined by the instructor and the department, with a maximum of 4 credits per course.

## Math 31400: Selected Topics in Mathematics

## Math 31500: Selected Topics in Mathematics

## Math 31600: Selected Topics in Mathematics

## Math 31700: Selected Topics in Mathematics

## Math 31800: Multivariable Advanced Calculus

A second semester of advanced calculus, combining Math 32400 and 32500 (to be taken instead of them), with emphasis on the topics in multivariable calculus. Prereq.: Math 32300 and 34600, or departmental permission. 4 HR./WK.; 4 CR.

## Math 31900: Selected Topics in Mathematics

## Math 32000: Selected Topics in Mathematics

## Math 32300: Advanced Calculus I

Sequences, properties of continuous functions, derivatives and differentials, functions defined by series, integrability and integrals, convergence of function sequences. Prereq.: Math 30800 or departmental permission. 4 HR./WK.; 4 CR.

## Math 32400: Advanced Calculus II

Sequences, continuity, and completeness metric spaces, contraction mappings and fixed point theorems, applications to differential equations; Fourier analysis, differentiation in n-space. Prereq.: Math 32300� and 34600. (Part of sequence 32300, 32400, 32500.) 3 HR./WK.; 3 CR.

## Math 32500: Advanced Calculus III

Integration in n-space, implicit and inverse function theorems change of variables in multiple integrals, vector fields, line and surface integrals, theorems of Green, Stokes, and Gauss. Prereq.: Math 32400 and 34600. (Part of sequence 32300, 32400, 32500.) 3 HR./WK.; 3 CR.

## Math 32800: Methods of Numerical Analysis

Solution of equations by iteration techniques; Lagrange and Newton interpolation, Neville's method, divided differences, cubic splines; numerical integration, Romberg integration; systems of linear equations and pivoting techniques; Runge-Kutta methods for initial value problems. Prereq.: Math 34600, or 39200, and knowledge of MATLAB or other high level programming language. Pre- or Coreq.: Math 39100. HR./WK.; 3 CR.

## Math 34200: History of Mathematics

Historical development of mathematical ideas and methods in geometry, theory of numbers, algebra, and analysis. Prereq.: Math 30800. (W) 3 HR./WK.; 3 CR.

## Math 34500: Theory of Numbers

Divisibility, primes, fundamental theorem of arithmetic, congruences, number theory from an algebraic viewpoint, quadratic reciprocity, number theoretic functions, diophantine equations. Prereq.: Math 30800 or departmental permission. 3 HR./WK.; 3 CR.

## Math 34600: Elements of Linear Algebra

Vector spaces, basis and dimension, matrices, linear transformations, determinants, solution of systems of linear equations, eigenvalues, and eigenvectors. Prereq.: Math 20300; coreq.: Math 20300 and departmental permission. (After completion of Math 39200 only 2 credits will be given for Math 34600.) 3 HR./WK.; 3 CR.

## Math 34700: Elements of Modern Algebra

Sets, mappings, rings, isomorphisms, integral domains, properties of integers, fields, rational numbers, complex numbers, polynomials, groups. Prereq.: Math 30800 and 34600. With departmental permission, partial credit may be given for Math 44900 after completion of Math 34700. Recommended for prospective teachers and others who want a basic course in abstract algebra. 4 HR./WK.; 4 CR. Documents

## Math 36000: Introduction to Modern Geometry

Logical deficiencies in Euclidean geometry, Euclid's parallel postulate, introduction to non-Euclidean geometry, the logical consistency of the non-Euclidean geometries, Hilbert's Axioms. Prereq.: Math 30800. 3 HR./WK.; 3 CR.

## Math 36500: Elements of Combinatorics

The three problems of combinatorics (existence, counting, optimization), basic counting rules, graph theory, generating functions, principles of inclusion and exclusion, pigeonhole principle, selected additional topics. Prereq.: Math 20300. 4 HR./WK.; 4 CR.

## Math 36600: Introduction to Applied Mathematical Computation

Calculus, linear algebra, elements of probability theory and combinatorics are examined through use of Matlab. Topics selected from symbolic and numerical problems in analysis; matrices, linear mappings, eigenvalues and applications; queueing theory; random numbers and simulations; graphics. Prereq.: Math 34600. 3 HR./WK.; 2 CR.

## Math 37500: Elements of Probability Theory

Permutations and combinations, conditional probability, independent events, random variables, probability distributions and densities, expectation, moments, moment generating functions, functions of random variables, Central Limit Theorem, sampling, confidence intervals. Prereq.: Math 20300. 3 HR./WK.; 3 CR.

## Math 37600: Mathematical Statistics

The gamma, chi-square, T, F, and bivariate normal distributions; Central Limit Theorem; confidence intervals and tests of hypothesis; the Neymen-Pearson Theorem; likelihood ratio test; estimation; sufficiency, unbiasedness, completeness; the Rao- Blackwell Theorem; the Rao-Cramer inequality; the method of maximum likelihood; the chi-square test; introduction to the analysis of variance and regression. Prereq.: Math 37500. 4 HR./WK.; 4 CR.

## Math 37700: Applied Statistics and Probability

Introduction to SPSS; organization of data; various descriptive statistics such as measures of variability and location; categorical variables; sampling distributions with SPSS; statistical inference, linear regression models; regression analysis; analysis of variance; the jackknife methodology of computer based estimation, discriminant analysis, factor analysis, cluster analysis. Prereq.: Math 37600 or departmental permission. 3 HR./WK.; 2 CR.

## Math 38100: Discrete Models of Financial Mathematics

Definitions of options and exotic options on stocks, interests rates and indices; binomial trees; volatility and methods to estimate volatility; continuous models and Black-Scholes; hedging; bond models and interest rate options; spreadsheet methods and computational methods including difference methods and Monte Carlo simulations. Prereq.: Math 20200. 3 HR./WK.; 3 CR.

## Math 38200: Continuous Time Models in Financial Mathematics

Review of discrete time models and binomial trees. Cox, Ross, Rubinstein approach to the Black-Scholes model; Black-Scholes equation and option pricing formulae; Brownian motion and stochastic differential equations; Ito's calculus and Ito's lemma; stopping times; the heat equation; option pricing and the heat equation; numerical solution of parabolic partial differential equations; interest rate models; simulation and financial models. Prereq.: Math 38100 or departmental permission. 3 HR./WK.; 3 CR.

## Math 39100: Methods of Differential Equations

First order equations; higher order linear equations with constant coefficients, undetermined coefficients, variation of parameters, applications; Euler's equation, series solutions, special functions; linear systems; elementary partial differential equations and separation of variables; Fourier series. Prereq.: Math 20300. 3 HR./WK.; 3 CR.

## Math 39200: Linear Algebra and Vector Analysis for Engineers

Matrix theory, linear equations, Gauss elimination, determinants, eigenvalue problems and first order systems of ordinary differential equations, vector field theory, theorems of Green, Stokes, and Gauss. Prereq.: Math 20300; Pre- or coreq.: Math 39100. No specialization credit will be given for both Math 32500 and 39200. (After completion of Math 34600 only 2 credits will be given for Math 39200.) 3 HR./WK.; 3 CR.

## Math 43200: Theory of Functions of a Complex Variable

Cauchy-Riemann equations, conformal mapping, elementary, entire, meromorphic, multiple-valued functions, Cauchy integral theorems, series expansion. Prereq.: Math 32500. 4 HR./WK.; 4 CR.

## Math 43400: Theory of Functions of a Real Variable

Lebesgue measure and integration on the real line, differentiation of real functions and the relation with integration, classical Lp spaces. Prereq.: Math 32500. 4 HR./WK.; 4 CR.

## Math 43500: Partial Differential Equations, Integral Equations, Boundary Value Problems

First order equations, shock waves; classification and canonical forms of higher order equations, characteristics, the Cauchy problem for the wave equation: Huygens' principle; the heat equation; Laplace's equation; the Dirichlet and Neuman problems; harmonic functions; eigenvalue expansions; Green's functions. Prereq.: Math 32500 and 39100 or permission of the instructor. 4 HR./WK.; 4 CR.

## Math 44300: Set Theory

Axioms of Zermelo-Fraenkel set theory; relations functions, equivalences and orderings; cardinal numbers and cardinal arithmetic; well-ordered sets; ordinal numbers, transfinite induction and recursion; The Axiom of Choice and the Continuum Hypothesis. Prereq.: Math 32300 or permission of the instructor. 4 HR./WK.; 4 CR.

## Math 44400: Mathematical Logic

The propositional calculus, the sentential calculus, normal forms, first order theories, consistency, categoricity, decidability, Godel's incompleteness theorem, the Loewenheim-Skolem theorem. Prereq.: Math 32300, or permission of the instructor. 4 HR./WK.; 4 CR.

## Math 44900: Introduction to Modern Algebra

Groups, rings, fields. Prereq.: Math 32300 and 34600. With departmental permission, partial credit may be given for Math 44900 after completion of Math 34700. 4 HR./WK.; 4 CR.

## Math 46100: Differential Geometry

The theory of curves and surfaces in threedimensional space: frames, fundamental forms, geodesics; curvature of surfaces; surface area; surfaces with boundary, the Gauss-Bonnet Theorem; introduction to Riemannian metrics. Prereq.: Math 32500 or permission of the instructor. 4 HR./WK.; 4 CR.

## Math 46300: Topology

A course in general topology. Sets of points on the real line and in general abstract spaces, relations between sets of points and between a set and the space containing it, operations with sets, open sets, countability, compactness, connectedness, maps, continuity, metric spaces, general topological spaces. Prereq.: Math 32500 or permission of the instructor. 4 HR./WK.; 4 CR.

## Math 46700: Mathematical Modeling

Problems from industry, mathematical models, process of mathematical abstraction, problem-solving techniques, application of solutions. Prereq.: Math 34600, 36600, 37500, 39100. 3 HR./WK.; 3 CR.

## Math 47700: Probability Theory II

Special topics in probability such as stochastic processes, Markov chains. Prereq.: Math 34600, 37500; pre- or coreq: Math 32500. 4 HR./WK.; 4 CR .

## Math 47800: Mathematical Statistics II

The multivariate normal distribution, multiple and partial correlation, regression and least squares, the analysis of variance. Prereq.: Math 34600 and 37600. 4 HR./WK.; 4 CR.

## Math 51100: Selected Topics in Pure Mathematics

Topics to be chosen from the areas of algebra, analysis, topology, geometry, and logic. Prereq.: to be determined by the instructor. 3 HR./WK.; 4 CR

## Math 51200: Selected Topics in Classical Analysis

Topics to be chosen from applied mathematics and related fields. Typical subjects are: asymptotic methods, wave propagation, mathematical biology. Prereq.: Math 34600, 39100, and 32500, and other requirements to be determined by the instructor. 3 HR./WK.; 4 CR.

## Math 51300: Selected Topics in Probability, Statistics, and Operations Research

Topics to be chosen from the areas of probability, statistics, game theory, combinatorial analysis, etc. Prereq.: to be determined by the instructor. 3 HR./WK.; 4 CR.

## Math 39300: Four and Lap Trnsfrms

Basic properties of and calculations with Laplace transforms, Fourier series and Fourier transforms; introduction to impulse (delta) functions; applications to electric circuits, diffusion and steady-state phenomena, standing waves, Huygen's principle, elementary diffraction, CAT scan, Gibb's phenomenon. 4 HR./WK.; 4 CR. prereq: MATH 20300 or MATH 39100.