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Mathematical Sciences Course Catalog

Please see the Schedule Planner for class schedules. Not all courses listed here are offered every term.

Course No.

Course Title

Description

MATH 1009
(3 credit hrs)

Computer-Based Algebraic Problem Solving.

A laboratory-based problem solving course focused on personal computing applications. Topics include general problem solving techniques, deductive reasoning, elementary probability, computer algebraic software, optimization, graphical analysis, systems of equations, spreadsheets, functions, descriptive statistics, linear programming and elementary programming logic.
Prereq: basic high school algebra and some familiarity with Microsoft Windows.

MATH 1010
(3 credit hrs)

Mathematics for the Liberal Arts: GT-MA1.

Designed to give liberal arts students the skills required to understand and interpret quantitative information that they encounter in the news and in their studies, and to make quantitatively-based decisions in their lives. Topics include a survey of logic and analysis of arguments, identifying fallacies in reasoning, working with numbers and units, linear and exponential relations and essentials of probability and statistics. The emphasis is on applications with case studies in economics, finance, environmental sciences, health, music and science.
Colorado state-wide Guaranteed Transfer course. Prereq: three years of high school mathematics. Approved Core Course.

MATH 1070
(3 credit hrs)

Algebra for Social Sciences and Business: GT-MA1.

Topics in algebra designed for students who intend to take business calculus. Functions, graphs, scatter plots, curve-fitting, solving systems of equations, polynomial and rational functions, and selected other topics.
Colorado state-wide Guaranteed Transfer course. NOTE: Graphics calculator required. No co-credit with MATH 1110 or MATH 1130. Prereq: intermediate algebra and satisfactory score on the placement exam. Approved Core Course.

MATH 1075
(1 credit hrs)

Linear Programming and Probability.

Designed to fulfill College of Business requirements for students who have had college algebra but not MATH 1070. Emphasis is on applications of linear programming and probability.
Prereq: MATH 1110. No co-credit with MATH 1070.

MATH 1080
(3 credit hrs)

Polynomial Calculus: GT-MA1.

A one-semester course in single-variable calculus. Topics include limits, derivatives, differentiation rules, integration and integration rules. Emphasis is on applications to business and social sciences.
Colorado state-wide Guaranteed Transfer course. Note: No knowledge of trigonometry is required. Those planning to take more than one semester of calculus should take MATH 1401 instead of MATH 1080. Prereq: MATH 1070 or 1110. No co-credit with MATH 1401. Approved Core Course.

MATH 1110
(3 credit hrs)

College Algebra: GT-MA1.

Topics in algebra designed for students who intend to take the calculus sequence. Functions, domains, ranges, graphs, data scatter plots and curve fitting, solving equations and systems of equations, polynomial functions, rational functions, and selected other topics. Graphic calculators and/or computer algebra systems are used extensively. Applications are emphasized.
Colorado state-wide Guaranteed Transfer course. Note: No co-credit with either MATH 1070 or 1130. Prereq: intermediate algebra and satisfactory score on the placement exam. Approved Core Course.

MATH 1111
(1 to 3 credit hrs)

Freshman Seminar.

MATH 1120
(3 credit hrs)

College Trigonometry: GT-MA1.

Topics in trigonometry, analytic geometry, and elementary functions designed for students who intend to take the calculus sequence. Angles and trigonometry functions of acute angles, analytic trigonometry, fundamental trigonometric functions and identities including hyperbolic trigonometry, parametric equations, and polar coordinate system. Graphic calculators and/or computer algebra systems are used extensively. Applications are emphasized.
Colorado state-wide Guaranteed Transfer course. Prereq: MATH 1110 and placement test. No joint credit with MATH 1130. Approved Core Course.

MATH 1130
(4 credit hrs)

Precalculus Mathematics: GT-MA1.

Condensed treatment of the topics in MATH 1110 and 1120.
Colorado state-wide Guaranteed Transfer course. Prereq: satisfactory score on the placement exam. No co-credit with MATH 1070, 1110 or 1120. Approved Core Course.

MATH 1401
(4 credit hrs)

Calculus I: GT-MA1.

First course of a three-semester sequence (MATH 1401, 2411, 2421) in calculus. Topics covered include limits, derivatives, applications of derivatives, and the definite integral.
Colorado state-wide Guaranteed Transfer course. Note: Students cannot receive credit for both MATH 1080 and 1401. Prereq: MATH 1120 or 1130 and satisfactory score on the placement exam. Approved Core Course.

MATH 2411
(4 credit hrs)

Calculus II: GT-MA1.

The second of a three-semester sequence (MATH 1401, 2411, 2421) in calculus. Topics covered include exponential, logarithmic, and trigonometric functions, techniques of integration, indeterminate forms, improper integrals and infinite series.
Colorado state-wide Guaranteed Transfer course. Prereq: MATH 1401. Approved Core Course.

MATH 2421
(4 credit hrs)

Calculus III: GT-MA1.

The third of a three-semester sequence in calculus (MATH 1401, 2411 and 2421). Topics include vectors, vector-valued functions, partial differentiation, differentiation, multiple integration, and vector calculus.
Colorado state-wide Guaranteed Transfer course. Prereq: MATH 2411. Approved Core Course.

MATH 2511
(3 credit hrs)

Discrete Structures.

Covers the fundamentals of discrete mathematics, including: logic, sets, functions, growth of functions, algorithms, matrices, mathematical reasoning, proofs, induction, relations, graphs, trees and combinatorics. There is an emphasis on how discrete mathematics applies to computer science in general, and algorithm analysis in particular.
Prereq: C SC 2421. Cross-listed with C SC 2511.

MATH 2810
(1 to 3 credit hrs)

Topics.

Topics in mathematics with various subtitles reflecting course content.
Prereq: permission of instructor.

MATH 2830
(3 credit hrs)

Introductory Statistics: GT-MA1.

Basic statistical concepts, summarizing data, probability concepts, distributions, confidence intervals, hypothesis testing.
Colorado state-wide Guaranteed Transfer course. Prereq: intermediate algebra. Approved Core Course.

MATH 2939
(1 to 3 credit hrs)

Internship/Cooperative Education.

Experiences involving application of specific, relevant concepts and skills in supervised employment situations.
Prereq: 15 hours of 2.75 GPA.

MATH 3000
(3 credit hrs)

Introduction to Abstract Mathematics.

Students learn to prove and critique proofs of theorems by studying elementary topics in abstract mathematics, including logic, sets, functions, equivalence relations and elementary combinatorics.
Prereq: MATH 1401.

MATH 3040
(3 credit hrs)

Mathematics for Elementary Teachers.

Topics include intuitive and logical development of geometric ideas relevant to K-6 curriculum; measurement of length, area, volume, mass, angle, temperature, time and the metric system. Further study of the rational number system, probably and statistics, applications and problem solving.
Note: carries credit only for elementary education majors. Prereq: three years of high school mathematics.

MATH 3140
(3 credit hrs)

Introduction to Modern Algebra.

Studies the fundamental algebraic structures used in modern mathematics. Topics include groups, rings, fields and polynomials.
Prereq: MATH 3000.

MATH 3191
(3 credit hrs)

Applied Linear Algebra.

Topics include systems of equations, Gaussian elimination with partial pivoting, LU--decomposition of matrices, matrix algebra, determinants, vector spaces, linear transformations, eigenvalues and applications.
Prereq: MATH 2411.

MATH 3195
(4 credit hrs)

Linear Algebra and Differential Equations.

Presents the essential ideas and methods of linear algebra and differential equations, emphasizing the connections between and the applications of both subjects. The course is designed for students in the sciences and engineering.
Prereq: MATH 2411.

MATH 3200
(3 credit hrs)

Elementary Differential Equations.

First and second order differential equations, Laplace transforms, systems of equations, with an emphasis on modeling and applications.
Prereq: MATH 2411; coreq is MATH 3191.

MATH 3210
(3 credit hrs)

Higher Geometry I.

Studies the foundations of modern geometry by examining axiomatic systems for various geometrics, with an emphasis on non-Euclidean hyperbolic geometry.
Prereq: MATH 3000.

MATH 3250
(3 credit hrs)

Problem Solving Tools.

Fall. Students learn and refine both problem solving techniques and computer programming skills. Examples, exercises, and projects are taken from a wide range of mathematical topics including algebra, calculus, linear algebra and probability.
Note: This course will not count toward a graduate degree in applied mathematics. Prereq: MATH 2421. Cross-listed with MATH 5250.

MATH 3301
(3 credit hrs)

Introduction to Operations Research I - Deterministic Systems.

A mathematical approach to decision making based on optimization. Topics include linear programming, network flows and production models.
Prereq: MATH 3191 or 3195.

MATH 3302
(3 credit hrs)

Operations Research II.

Elementary stochastic processes and standard nondeterministic operations research models: Markov chains, Poisson processes, renewal processes, queuing theory, inventory models, Markov decision processes, simulation.
Prereq: MATH 3800 and 3191.

MATH 3440
(3 credit hrs)

Introduction to Symbolic Logic.

Spring. Covers truth functional and quantificational logic through polyadic first order predicate calculus and theory of identity. Attention is given to such problems in metatheory as proofs of the completeness and consistency of systems of logic.
Prereq: MATH 3000. Cross-listed with PHIL 3440.

MATH 3511
(4 credit hrs)

Mathematics of Chemistry.

Fall. Multivariate functions, probability and statistics for chemistry, matrices and vectors, mathematics of reaction kinetics and symmetry point groups. Course covers mathematics needed for CHEM 4511 and 4521. Can also be an elective for the mathematics minor.
Prereq: MATH 2411, CHEM 2031, CHEM 2061.

MATH 3800
(3 credit hrs)

Probability and Statistics for Engineers

Basic probability theory, discrete and continuous random variables, point and interval estimation, test of hypotheses, one-way analysis of variance, and simple linear regression.
Note: no co-credit with MATH 4810. Prereq: MATH 2411; coreq: MATH 2421.

MATH 3939
(1 to 3 credit hrs)

Internship/Cooperative Education.

Designed experiences involving application of specific, relevant concepts and skills in supervised employment situations.
Prereq: junior standing and 2.75 GPA.

MATH 4010
(3 credit hrs)

History of Mathematics.

Spring. A history of the development of mathematical techniques and ideas from early civilization to the present, including the inter-relationships of mathematics and sciences.
Prereq: MATH 1401. Cross-listed with MATH 5010.

MATH 4027
(3 credit hrs)

Topics in Mathematics.

Special topics in mathematics will be covered; consult ´Schedule Planner´ for current topics and prerequisites.

MATH 4101
(3 credit hrs)

Applied Statistics Using SAS and SPSS I.

Teaches the practical statistical tools social scientists use to analyze real-world problems. Course split into four modules, each taught by a different instructor. The first module introduces SAS and SPSS; modules 2-4 are problem-based and cover topics such as ANOVA, multivariate regression, and cluster analysis.
Prereq: any statistics course.

MATH 4102
(3 credit hrs)

Applied Statistics Using SAS and SPSS II.

Spring. (Continuation of MATH 4101.) Students use the skills they learned in the previous semester to analyze a social issue of their choosing and present their findings. In addition to lectures, weekly one-on-one meetings between faculty and student are are required.
Prereq: MATH 4101.

MATH 4110
(3 credit hrs)

Theory of Numbers.

Every other year. Topics include divisibility, prime numbers, congruencies, number theoretic functions, quadratic reciprocity, and special diophantine equations, with applications in engineering.
Prereq: MATH 3000. Cross-listed with MATH 5110.

MATH 4201
(3 credit hrs)

Topology.

Spring. Metric spaces and topological spaces, compactness, separation properties, and connectedness.
Prereq: MATH 3000.

MATH 4220
(3 credit hrs)

Higher Geometry II.

Studies affine and projective geometrics. Coordinates are introduced in this framework. Planes and higher dimensional spaces are examined.
Prereq: MATH 3191.

MATH 4310
(4 credit hrs)

Introduction to Real Analysis I.

Fall. Calculus of one variable, the real number system, continuity, differentiation, integration theory, sequence and series.
Prereq: MATH 2421 and 3000.

MATH 4320
(4 credit hrs)

Introduction to Real Analysis II.

Spring. Convergence, uniform convergence; Taylor´s theorem; calculus of several variables including continuity, differentiation and integration; Picard´s theorem in ordinary differential equations and Fourier series.
Prereq: MATH 4310.

MATH 4387
(3 credit hrs)

Regression Analysis, Modeling and Time Series

Fall. Topics include linear and multiple regression, basic experimental designs, one-way analysis of variance. Emphasis is on practical aspects and applications of linear models to the analysis of data in business engineering, behavioral, biological and physical sciences.
Prereq: MATH 3191 and 3800/4820. Cross-listed with MATH 5387.

MATH 4390
(3 credit hrs)

Game Theory.

Annual. Begins with an introduction to the mathematical theory of games and the definition of a solution, including extensive and normal forms of representation. The fundamental minimax theorem is presented first as the foundation for two-person matrix games, then extended with fixed point theory to other games. Principles of dominance and solution methods are presented, plus applications to economics, political science, engineering, and other fields. An introduction to n-person game theory is included, with basic terms and concepts.
Prereq: MATH 2421, 3191 and 3800/4810. Cross-listed with MATH 5390.

MATH 4394
(3 credit hrs)

Experimental Designs.

Infrequent. Completely randomized block designs factorial and fractional factorial experiments, balanced incomplete block designs, responses surface methods.
Prereq: MATH 4387. Cross-listed with MATH 5394.

MATH 4408
(3 credit hrs)

Applied Graph Theory.

Introduces discrete structures and applications of graph theory to computer science, engineering, operations research, social science, and biology. Topics include connectivity, coloring, trees, Euler and Hamiltonian paths and circuits, matching and covering problems, shortest route and network flows.
Prereq: C SC/MATH 2511 or MATH 3000. Cross-listed with C SC 4408.

MATH 4409
(3 credit hrs)

Applied Combinatorics.

Every other year. Major emphasis is on applied combinatorics and combinatorial algorithms, with applications in computer science and operations Topics include general counting methods, generating functions, recurrence relations, inclusion-exclusion, and block designs.
Prereq: MATH 4408 and 3140.

MATH 4410
(3 credit hrs)

Mathematics of Coding Theory.

Error correcting codes are used to recapture information that has been distorted in some transmission process. Various coding schemes use block codes obtained from algebraic, geometric, and combinatorial structures. Topics include: fundamentals of coding theory, linear, Reed-Muller, Golay, cyclic and BCH codes.
Prereq: MATH 3191.

MATH 4450
(3 credit hrs)

Complex Variables.

Infrequent. Topics include complex algebra, Cauchy-Riemann equations, Laurent expansions, theory of residues, complex integration, and introduction to conformal mapping.
Prereq: MATH 2421 and MATH 3000.

MATH 4576
(3 credit hrs)

Mathematical Foundations of Artificial Intelligence I.

Infrequent. Fundamentals course that complements other approaches, such as in engineering, psychology, and business administration. Here, the emphasis is on the mathematical foundations. Topics include logical inference, problem solving, heuristic search, neural search, neural nets, analogical reasoning and learning. Models and paradigms also consider different measures of uncertainty.
Prereq: C SC 2511, MATH 2511/3000 and 3191. Cross-listed with MATH 5576.

MATH 4650
(3 credit hrs)

Numerical Analysis I.

(Same as CSMC 4650 at Colorado School of Mines.) Methods and analysis of techniques used to resolve continuous mathematics problems on the computer. Solution of linear and nonlinear equations, interpolation and integration.
Prereq: MATH 2411, 3191 or MATH 3195 and programming experience. Cross-listed with C SC 4650, 5660 and MATH 5660.

MATH 4660
(3 credit hrs)

Numerical Analysis II.

Spring. Numerical differentiation and integration, numerical solution of ordinary differential equations, the Galerkin method for the Poisson equation.
Prereq: MATH 3195 or both 3191 and 3200; MATH or C SC 4650 or 5660; or programming experience. Cross-listed with MATH 5661, C SC 4660 and 5661.

MATH 4674
(3 credit hrs)

Parallel Computing and Architectures.

Infrequent. Examines a range of topics involved in using parallel operations to improve computational performance. Parallel architectures, parallel algorithms, parallel programming languages, interconnection networks, and their relation to specific computer architectures.
Prereq: MATH 4650. Cross-listed with MATH 5674.

MATH 4733
(3 credit hrs)

Partial Differential Equations.

Infrequent. Initial/Boundary value problems for first-order, wave, heat and Laplace Equations; maximum principles; Fourier Series and applications.
Prereq: MATH 2421 and 3200. Cross-listed with MATH 5733.

MATH 4779
(3 credit hrs)

Math Clinic.

The clinic is intended to illustrate the applicability and utility of mathematical concepts. Research problems investigated originate from a variety of sources--industry, government agencies, educational institutions, or nonprofit organizations.
Prereq: consult Schedule Planner or instructor. Cross-listed with MATH 5779.

MATH 4788
(3 credit hrs)

Bioinformatics

Provides a broad exposure to the basic concepts and methodologies of bioinformatics and their application to analyzing genomic and proteomic data. Topics may include dynamic programming algorithms, graph theoretic techniques, hidden Markov models, phylogenetic trees, RNA/protein structure prediction and microarray analysis.
Prereq: C SC 1410 and MATH 3191 or 3195. Cross-listed with C SC 4788, PHYS 4788.

MATH 4791
(3 credit hrs)

Continuous Modeling.

Every other year. Surveys mathematical problems that arise in natural sciences and engineering. Topics may include population models, epidemic models, mechanics, heat transfer and diffusion, tomography, pharmaco-kinetics, traffic flow, fractal models, wave phenomena, and natural resource management. Most models discussed are based on differential and integral equations. Emphasis is formulation and validation of models as well as methods of solution.
Prereq: MATH 3191 and 3200. Cross-listed with MATH 5791.

MATH 4792
(3 credit hrs)

Probabilistic Modeling.

Every other year. Markov chains; Poisson processes, continuous time Markov chains, elementary topics in queuing theory, and some mathematical aspects of Monte Carlo simulation, including random variate generation, variance reduction, and output analysis.
Prereq: MATH 4810, 5310 and some programming experience. Cross-listed with MATH 5792.

MATH 4793
(3 credit hrs)

Discrete Math Modeling.

Every other year. Focuses on the use of graph theory and combinatorics to solve problems in a wide variety of disciplines. Applications are selected from computer science, communication networks, economics, operations research, and the social, biological and environmental sciences.
Prereq: MATH 3191 and 4408. Cross-listed with MATH 5793.

MATH 4794
(3 credit hrs)

Optimization Modeling.

Every other year. Principles of model formulation and analysis are developed by presenting a wide variety of applications, both for natural phenomena and social systems. Examples of optimization models to represent natural phenomena include principles of least time and energy. Examples in social systems include resource allocation, environmental control and land management. Specific applications vary, but are chosen to cover a wide scope that considers dichotomies, such as discrete vs. continuous, static vs. dynamic, and deterministic vs. stochastic. Some computer modeling language (like GAMS) is taught.
Prereq: MATH 2421 and 3191. Cross-listed with MATH 5794.

MATH 4810
(3 credit hrs)

Probability.

Examines elementary theory of probability, including independence, conditional probability, and Bayes´ theorem; random variables, expectations and probability distributions; joint and conditional distributions; functions of random variables; limit theorems, including the central limit theorem.
Prereq: MATH 2421 and 3191. Cross-listed with MATH 5310.

MATH 4820
(3 credit hrs)

Statistics.

Spring. Point and confidence interval estimation, principles of maximum likelihood, sufficiency and completeness, tests of simple and composite hypothesis, linear models and multiple regression, analysis of variance.
Prereq: MATH 3800. MATH 4810 highly recommended, but not required. Cross-listed with MATH 5320.

MATH 4830
(3 credit hrs)

Applied Statistics.

Spring. Review of estimation, confidence intervals and hypothesis testing; ANOVA; categorical data analysis; non-parametric tests; linear and logistic regression.
Prereq: an introductory course in statistics such as MATH 2830 or permission of instructor. Cross-listed with MATH 5830.

MATH 4840
(1 to 3 credit hrs)

Independent Study.

Variable credit depending on the student´s needs. Offered for the advanced student who desires to pursue a specific topic in considerable depth.
Note: Supervision by a full-time faculty member is necessary, and the dean´s office must concur. Students may register for this course more than once with departmental approval.

MATH 5000
(4 credit hrs)

Algebraic Patterns and Functions I RM-MSMSP.

Systematic study of the core elements of algebra: linear, quadratic, exponential, logarithmic functions and their graphs. Includes modeling using graphing calculators and real world applications. Concepts are linked to other scientific, mathematical, and pedagogical domains. This course is not applicable toward any degree in the College of Liberal Arts and Sciences.
Prereq: permission of project director.

MATH 5002
(4 credit hrs)

Algebraic Patterns and Functions II RM-MSMSP.

This course is a continuation of the material covered in Math 5000. Topics that will be covered include logarithmic, exponential and trigonometric functions and applications, parametric equations, systems of equations and inequalities, matrices and linear programming. This course is not applicable toward any degree in the College of Liberal Arts and Sciences.
Prereq: MATH 5000 or permission of instructor.

MATH 5004
(4 credit hrs)

RM-MSMSP: Statistics and Probability.

Studies the collection, presentation, and analysis of data; and elements and applications of counting discrete probability. Includes real world applications and technology. Concepts are linked to other scientific, mathematical, and pedagogical domains. This course is not applicable toward any degree in the College of Liberal Arts and Sciences.
Prereq: permission of project director.

MATH 5005
(4 credit hrs)

RM-MSMSP: Geometry.

Systematic study of advanced geometric concepts: history of geometry and measurement, patterns among shapes, 2- and 3-dimensional shapes, constructions, symmetry or transformational geometry. Includes applications and activity-oriented instruction. Concepts are linked to other scientific, mathematical, and pedagogical domains. This course is not applicable toward any degree in the College of Liberal and Sciences.
Prereq: permission of project director.

MATH 5006
(4 credit hrs)

RM-MSMSP: Mathematics of Change.

Systematic study of the application of calculus to the analysis of changing systems in real world applications. Emphasizes the connections that exist between calculus and aspects of middle school curricula. Concepts are linked to other scientific, mathematical, and pedagogical domains. This course is not applicable toward any degree in the College of Liberal Arts and Sciences.
Prereq: MATH 5000 (or equivalent) or permission of project director.

MATH 5007
(4 credit hrs)

RM-MSMSP: Discrete Math--Counting the Possibilities.

Systematic study of basic techniques in discrete mathematics and their various applications: permutations and combinations, inclusion or exclusion, pigeonhole principle, graph theory, and recursive pattern solving. Applications to topics such as network analysis and voting theory are stressed. Concepts are linked to other scientific, mathematical, pedagogical domains. This course is not applicable toward any degree in the College of Liberal Arts and Sciences.
Prereq: MATH 5000 (or equivalent) or permission of project director.

MATH 5008
(4 credit hrs)

RM-MSMSP: Discovery and Use of the History of Math.

Systematic study of the people, events, ideas and issues from the history of mathematics, focusing on historical topics that are central to the discipline and teaching of mathematics and emphasizing web research of historical topics of interest. Concepts are linked to other scientific, mathematical, and pedagogical domains.
Note: This course is not applicable toward any degree in the College of Liberal Arts and Sciences. Prereq: permission of the project director.

MATH 5009
(4 credit hrs)

RM-MSMSP: Math Modeling--Using and Applying Math.

Systematic study of math modeling using algebra, geometry, discrete mathematics, rates of change, and statistics to solve real-world problems in areas such as finance, biology, economics, and physics. Concepts are linked to other scientific, mathematical, societal, and pedagogical domains. This course is not applicable toward any degree in the College of Liberal Arts and Sciences.
Prereq: MATH 5006 (or equivalent) or permission of instructor.

MATH 5010
(3 credit hrs)

History of Mathematics.

Spring. A history of the development of mathematical techniques and ideas from early civilization to the present, including the inter-relationships of mathematics and sciences.
Prereq: MATH 1401. Not open to students who have had MATH 4010. No credit for applied math graduate students. Cross-listed with MATH 4010.

MATH 5017
(1 to 3 credit hrs)

Topics in Mathematics for Teachers.

Topics vary from semester to semester. Designed for professional mathematics teachers.
Note: This course will not count toward a degree in applied mathematics. Prereq: permission of instructor.

MATH 5027
(3 credit hrs)

Topics in Applied Mathematics.

Selected topics in mathematical problems arising from various applied fields such as mechanics, electromagnetic theory, economics and biological sciences.
Prereq: permission of instructor.

MATH 5070
(3 credit hrs)

Applied Analysis.

Spring. Designed to serve as an introduction to real analysis. Topics include: fundamentals of logic and theorem proving, infimum and supremum, real numbers, point-set topology in metric spaces, properties of functions, sequences and series of functions, fixed point theorems, Riemann integral, power series.
Prereq: MATH 4320.

MATH 5110
(3 credit hrs)

Theory of Numbers.

Every other year. Topics include divisibility, prime numbers, congruences, number theoretic functions, quadratic reciprocity, and special diophantine equations, with applications in engineering.
Prereq: MATH 3000. Cross-listed with MATH 4110.

MATH 5135
(3 credit hrs)

Functions of a Complex Variable.

Infrequent. The complex plane, infinite series and products, elementary special functions, Cauchy-Riemann equations, conformal mapping, complex integration, Cauchy integral theory, and residue theory.
Prereq: MATH 4320; MATH 5070 recommended.

MATH 5198
(3 credit hrs)

Mathematics for Bioscientists.

Fall. Develops mathematical reasoning; introduces linear algebra, discrete structures, graph theory, probability, and differential equations, using applications to molecular biology.
Note: No credit for mathematics or engineering students. Prereq: MATH 2411.

MATH 5250
(3 credit hrs)

Problem Solving Tools.

Fall. Students learn and refine both problem solving techniques and computer programming skills. Examples, exercises, and projects are taken from a wide range of mathematical topics including algebra, calculus, linear algebra and probability.
Note: This course will not count toward a graduate degree in applied mathematics. Coreq: MATH 2421. Cross-listed with MATH 3250.

MATH 5310
(3 credit hrs)

Probability.

Examines elementary theory of probability, including independence, conditional probability, and Bayes´ theorem; random variables, expectations and probability distributions; joint and conditional distributions; functions of random variables; limit theorems, including the central limit theorem.
Prereq: Math 2421 and 3191. Cross-listed with MATH 4810.

MATH 5320
(3 credit hrs)

Statistics.

Spring. Point and confidence interval estimation, principles of maximum likelihood, sufficiency and completeness, tests of simple and composite hypothesis, linear models and multiple regression, analysis of variance.
Prereq: MATH 3800. MATH 4810 highly recommended, but not required. Cross-listed with MATH 4820.

MATH 5387
(3 credit hrs)

Regression Analysis, Modeling and Time Series

Fall. Topics include linear and multiple regression, basic experimental designs, one-way analysis of variance. Emphasis is on practical aspects and applications of linear models to the analysis of data in business engineering and behavioral, biological and physical sciences.
Prereq: MATH 3191 and 3800/4820. Cross-listed with MATH 4387.

MATH 5390
(3 credit hrs)

Game Theory.

Infrequent. Begins with an introduction to the mathematical theory of games and the definition of a solution, including extensive and normal forms of representation. The fundamental minimax theorem is presented first, as the foundation for two-person matrix games, then extended with fixed point theory to other games. Principles of dominance and solution methods are presented, plus applications to economics, political science, engineering, and other fields. An introduction to n-person game theory is included, with basic terms and concepts.
Prereq: MATH 2421, 3191 and 3800/4810. Cross-listed with MATH 4390.

MATH 5394
(3 credit hrs)

Experimental Designs.

Infrequent. Completely randomized block designs, factorial and fractional factorial experiments, balanced incomplete block designs, response surface methods.
Prereq: MATH 4387 and 5387. Cross-listed with MATH 4394.

MATH 5410
(3 credit hrs)

Modern Cryptology.

Every other year. Deals with the mathematics that underlies modern cryptology. Topics include: classical cryptology, public and private key cryptosystems, secret sharing schemes, authentication schemes, linear feedback shift registers, discrete logarithm and elliptic curve-based schemes.
Prereq: MATH 3191.

MATH 5432
(3 credit hrs)

Computational Graph Theory.

Infrequent. Algorithmic techniques in graph theory and other discrete mathematics areas. Typical topics include: branch-bound algorithms, matching, colorings, domination, min-plus algebra, simulated annealing and related heuristics, NP-completeness theory.
Prereq: a course in graph theory and some programming experience.

MATH 5446
(3 credit hrs)

Theory of Automata.

Infrequent. Studies the relationships between classes of formal languages (regular, context-free, context-sensitive, phrase-structure) and classes of automata (finite-state, pushdown, Turing machines). Additional topics include decidability and computability issues.
Prereq: MATH 3000 and 3140. Cross-listed with C SC 5446.

MATH 5490
(3 credit hrs)

Network Flows.

Infrequent. Begins with the classical min-cost flow problem, defined on an ordinary network. Other problems, such as shortest path, are also shown in this class. Both theory and algorithms are presented. Extensions include generalized networks, nonlinear costs, fixed charges, multi-commodity flows, and and additional applications, such as in communications networks.
Prereq: graduate standing in math or computer science.

MATH 5576
(3 credit hrs)

Mathematical Foundations of Artificial Intelligence I.

Infrequent. A fundamentals course that complements other approaches, such as in engineering, psychology, and business administration. Here the emphasis is on the mathematical foundations. Topics include logical inference, problem solving, heuristic search, neural nets, analogical reasoning and learning. Models and paradigms also consider different measures of uncertainty.
Prereq: C SC 2511, MATH 2511/3000 and 3191. Cross-listed with MATH 4576.

MATH 5593
(3 credit hrs)

Linear Programming.

Fall. A linear program is an optimization problem that seeks to minimize or maximize a linear function subject to a system of linear inequalities and equations. This course begins with examples of linear programs and variations in their representations. Basic theoretical foundations covered include polyhedra, convexity, linear inequalities and duality. Two classes of solution algorithms are given: simplex methods and interior point methods. The primary emphasis of this course is on mathematical foundations, and applications are used to illustrate the main results.
Prereq: MATH 3191.

MATH 5610
(3 credit hrs)

Computational Biology.

Spring. Basic introduction and mathematical foundations. Topics include comparative genomics; proteomics; phylogeny; dynamic programming and sequence alignment; gene expression arrays and clustering; Bayesian networks; structure prediction and hidden Markov models.
Prereq: C SC 1410 or equivalent programming experience, and MATH 3191 or 3195.

MATH 5660
(3 credit hrs)

Numerical Analysis I.

(Same as CSMC 4650 at Colorado School of Mines.) Methods and analysis of techniques used to resolve continuous mathematics problems on the computer. Solution of linear and nonlinear equations, interpolation and integration.
Prereq: MATH 2411, 3191 or MATH 3195 and programming experience. Cross-listed with C SC 4650, 5660 and MATH 4650.

MATH 5661
(3 credit hrs)

Numerical Analysis II.

Spring. Numerical differentiation and integration, numerical solution of ordinary differential equations, the Galerkin method for the Poisson equation.
Prereq: MATH 3195 or both 3191 and 3200; MATH or C SC 4650 or 5660; or programming experience. Cross-listed with MATH 4660, C SC 4660 and 5661.

MATH 5674
(3 credit hrs)

Parallel Computing and Architectures.

Infrequent. Examines a range of topics involved in using parallel operations to improve computational performance. Parallel architectures, parallel algorithms, parallel programming languages, interconnection networks, and their relation to specific computer architectures.
Prereq: MATH 4650. Cross-listed with MATH 4674.

MATH 5718
(3 credit hrs)

Applied Linear Algebra.

Fall. Topics include: vector spaces, practical solution of systems of equations, projections, eigenvalues and eigenvectors, unitary transformations, Schur QR, singular value decompositions, similarity transformations, Jordan forms, and positive definite matrices.
Prereq: MATH 3191.

MATH 5733
(3 credit hrs)

Partial Differential Equations.

Infrequent. Initial/Boundary value problems for first-order, wave, heat and Laplace Equations; maximum principles; Fourier Series and applications.
Prereq: MATH 2421 and 3200; graduate standing. Cross-listed with MATH 4733.

MATH 5779
(3 credit hrs)

Math Clinic.

The clinic is intended to illustrate the applicability and utility of mathematical concepts. Research problems investigated originate from a variety of sources--industry, government agencies, educational institutions, or nonprofit organizations.
Prereq: consult Schedule Planner or instructor. Cross-listed with MATH 4779.

MATH 5791
(3 credit hrs)

Continuous Modeling.

Every other year. Surveys mathematical problems that arise in natural sciences and engineering. Topics may include population models, epidemic models, mechanics, heat transfer and diffusion, tomography, pharmaco-kinetics, traffic flow, fractal models, wave phenomena, and natural resource management. Most models discussed are based on differential and integral equations. Emphasis is formulation and validation of models as well as methods of solution.
Prereq: MATH 3191 and 3200. Cross-listed with MATH 4791.

MATH 5792
(3 credit hrs)

Probabilistic Modeling.

Every other year. Markov chains; Poisson processes, continuous time Markov chains, elementary topics in queuing theory, and some mathematical aspects of Monte Carlo simulation, including random variate generation, variance reduction, and output analysis.
Prereq: MATH 4810, 5310 and some programming experience. Cross-listed with MATH 4792.

MATH 5793
(3 credit hrs)

Discrete Math Modeling.

Every other year. Focuses on the use of graph theory and combinatorics to solve problems in a wide variety of disciplines. Applications are selected from computer science, communication networks, economics, operations research, and the social, biological and environmental sciences.
Prereq: MATH 3191 and 4408. Cross-listed with MATH 4793.

MATH 5794
(3 credit hrs)

Optimization Modeling.

Every other year. Principles of model formulation and analysis are developed by presenting a wide variety of applications, both for natural phenomena and social systems. Examples of optimization models to represent natural phenomena include principles of least time and energy. Examples in social systems include resource allocation, environmental control and land management. Specific applications vary, but are chosen to cover a wide scope that considers dichotomies, such as discrete vs. continuous, static vs. dynamic, and deterministic vs. stochastic. Some computer modeling language (like GAMS) is taught.
Prereq: MATH 2421 and 3191. Cross-listed with MATH 4794.

MATH 5830
(3 credit hrs)

Applied Statistics.

Spring. Review of estimation, confidence intervals and hypothesis testing; ANOVA; categorical data analysis; non-parametric tests; linear and logistic regression.
Prereq: an introductory course in statistics such as MATH 2830 or permission of instructor. Cross-listed with MATH 4830.

MATH 5840
(1 to 3 credit hrs)

Independent Study.

Available only with approval of graduate advisor.
Subjects arranged.

MATH 5939
(1 to 6 credit hrs)

Internship/Cooperative Education.

MATH 5950
(1 to 8 credit hrs)

Master´s Thesis.

MATH 5960
(1 to 8 credit hrs)

Master´s Project.

Note: This course will not count toward a graduate degree in Applied Mathematics.

MATH 6023
(3 credit hrs)

Topics in Discrete Math.

Topics may include graph theory, combinatorics, matroid theory, combinatorial matrix theory, finite geometry, design theory, and discrete algorithms.
Note: Since topic varies by semester, students may register for this course more than once. Prereq: permission of the instructor.

MATH 6131
(3 credit hrs)

Real Analysis.

Every other year. Lebesque measure and integration, general measure and integration theory, Radon-Nikodyn Theorem, Fubini Theorem.
Prereq: MATH 4320 or 5070.

MATH 6330
(3 credit hrs)

Workshop in Statistical Consulting

Students participate as consultants in a drop-in consulting service operated by the department. Seminars provide students with supervised experience in short term statistical consulting.
Note: Since problems vary each semester, students may register for this course more than once. Prereq: MATH 5387.

MATH 6360
(3 credit hrs)

Exploratory Data Analysis.

Every other year. Philosophy and techniques associated with exploratory (vs. confirmatory) data analysis, both as originally presented (John Tukey) and current computer-based implementations. Graphical displays, robust-resistant methods (lines, two-way fits), diagnostic plots, standardization.
Prereq: previous statistics course or permission of instructor.

MATH 6376
(3 credit hrs)

Statistical Computing.

Computationally-intensive methods in statistics, including random number generation and Monte Carlo methods, data partitioning and re-sampling, numerical and graphical methods, nonparametric function estimation, statistical models and data mining methodology, analysis of large data sets.
Prereq: MATH 4820/4830 and 4387. Cross-listed with MATH 7376.

MATH 6380
(3 credit hrs)

Stochastic Processes.

Every other year. Markov processes in discrete and continuous time, renewal theory, martingales, Brownian motion, branching processes, and stationary processes. Applications include queuing theory, performance evaluation of computer and communication systems and finance.
Prereq: MATH 3191, 3200, and 4810/5310.

MATH 6384
( credit hrs)

Analysis of Dependent Data.

Infrequent. Statistical methods for the analysis of data with temporal and/or spatial dependence. Longitudinal data, stationary and non-stationary time series models, geostatistical and lattice spatial models, point processes, hierarchical models.
Prereq: MATH 4820 or 4830 and MATH 4387.

MATH 6388
(3 credit hrs)

Advanced Statistical Methods for Research.

Infrequent. The second in a two-semester course in applied statistics. Topics include multifactor analysis of variance and covariance, categorical data, general linear models, bootstrapping, and other computationally intensive statistical methods.
Prereq: MATH 5387.

MATH 6391
(3 credit hrs)

Stochastic Differential Equations.

Infrequent. Ito integral, Ito formula, weak and strong solutions, martingale representation formula, filtering, stochastic optimal control, diffusions, boundary value problems.
Prereq: MATH 6383.

MATH 6393
(3 credit hrs)

Introduction to Bayesian Statistics.

Prior and posterior distributions, conjugate models, single and multiparameter models, hierarchical models, mixture models, numerical methods for evaluating posteriors, Monte Carlo methods, and Markov chain Monte Carlo.
Prereq: MATH 3800 or both MATH 4810 and 4820 (or equivalent). Some computer programming experience.

MATH 6395
(3 credit hrs)

Multivariate Methods.

Every other year. Multivariate distributions, hypothesis testing and estimation. Multivariate analysis of variance, discriminant analysis, multidimensional scaling, factor analysis, principal components.
Prereq: MATH 5387.

MATH 6398
(3 credit hrs)

Calculus of Variations and Optimal Control.

Infrequent. Standard variational problems (geodesic, time-of-transit, isoperimetric, surface, area), Euler-Lagrange equations, variational principles in mechanics, optimal control problems, necessary conditions for optimality, Pontryagin principle.
Prereq: MATH 4320.

MATH 6404
(3 credit hrs)

Applied Graph Theory.

Every other year. Emphasis on graph theory. Topics will include trees, digraphs and networks, intersection graphs, coloring, clique coverings, distance, paths and cycles. Topics are motivated by applications.
Prereq: graduate standing.

MATH 6595
(3 credit hrs)

Computational Methods in Nonlinear Programming.

Every other year. Introduces fundamental algorithms and theory for nonlinear optimization problems. Topics include Newton, quasi-Newtown and conjugate direction methods; linesearch and trust-region methods; active set, penalty and barrier methods for constrained optimization; convergence analysis and duality theory.
Prereq: MATH 4320 and 5718.

MATH 6653
(3 credit hrs)

Introduction to Finite Element Methods.

Every other year. The Finite Element Method (FEM) is introduced as a generic tool for the approximation of partial differential equations that model engineering and physics problems of interest. Elliptic, hyperbolic, and parabolic equations are solved with FEM.
Prereq: MATH 5660.

MATH 6735
(3 credit hrs)

Continuum Mechanics.

Every other year. Indicial notation. Eulerian and Lagrangian coordinates. Deformation, strain, strain rate, stress. Conservation of mass, momentum, and energy. Exploitation of entropy production inequality to obtain constitutive equations for elastic, viscous, visco elastic, plastic, or porous materials.
Prereq: MATH 3191 and 3200 or graduate standing.

MATH 6840
(1 to 3 credit hrs)

Independent Study.

MATH 7101
(3 credit hrs)

Topology.

Every other year. Topological spaces, compactness, separation properties and connectedness.
Prereq: MATH 4320.

MATH 7132
(3 credit hrs)

Functional Analysis.

Every other year. Linear metric and topological spaces, duality, weak topology, spaces of functions, linear operators, compact operators, elements of spectral theory, and operator calculus.
Prereq: MATH 6131.

MATH 7376
(3 credit hrs)

Statistical Computing.

Computationally-intensive methods in statistics, including random number generation and Monte Carlo methods, data partitioning and re-sampling, numerical and graphical methods, nonparametric function estimation, statistical models and data mining methodology, analysis of large data sets.
Prereq: MATH 4820/4830 and 4387. Cross-listed with MATH 6376.

MATH 7381
(3 credit hrs)

Mathematical Statistics I.

Every other year. Mathematical theory of statistics. Parametric inference: discrete and continuous distributions, methods of parameter estimation, confidence intervals.
Prereq: MATH 3191 and 4820/5320.

MATH 7382
(3 credit hrs)

Mathematical Statistics II.

Every other year. (Continuation of MATH 7381.) Hypothesis testing, robust estimation, tolerance intervals, nonparametric inference, sequential methods.
Prereq: MATH 7381.

MATH 7384
(3 credit hrs)

Mathematical Probability.

Every other year. Measurable spaces, probability measures, random variables, conditional expectations and martingales. Convergence in probability, almost sure convergence, convergence in distribution, limit theorems (law of large numbers, central limit theorem, law of iterated logarithm).
Prereq: MATH 4810/5310 and MATH 5070 or MATH 6131.

MATH 7385
(3 credit hrs)

Stochastic Differential Equations.

Brownian motion, Ito integral, Ito formula, Dynkin´s formula, stochastic optimal control, boundary value problems, Girsanov theorem, mathematical finance, optimal stopping.
Prereq: MATH 6383.

MATH 7397
(3 credit hrs)

Nonparametric Statistics.

Every three years. Statistical inference without strong model assumptions. Hypothesis testing and estimation using permutations and ranks, analysis of variance, and nonparametric model fitting.
Prereq: applied mathematics - statistics (PhD.)

MATH 7405
(3 credit hrs)

Advanced Graph Theory

Continuation of MATH 6404. Topics to be covered include: trees and optimization, encoding and embedding of graphs, generalized colorings and applications, perfect graphs, extremal problems, substructures, connectedness and cycles.
Prereq: MATH 6404 or permission of instructor.

MATH 7409
(3 credit hrs)

Applied Combinatorics.

Every other year. Emphasis is on enumerative combinatorics. Topics include multinomial coefficients, generating functions, SDRs, Polya's enumeration theory, pigeon-hole principle, inclusion/exclusion and Moebius inversion of finite posets. Topics may also include introduction to designs and finite geometry.

MATH 7410
(3 credit hrs)

Combinatorial Structures.

Every other year. Finite combinatorial structures; existence, construction and applications. Topics include Latin squares, Hadamard matrices, block designs, finite geometries, and extremal and non-constructive combinatorics.
Prereq: MATH 5718 and MATH 7409 or permission of instructor.

MATH 7413
(3 credit hrs)

Modern Algebra I.

Every other year. Groups, rings and ideals, integral domains.
Prereq: MATH 3140. Coreq: MATH 5718.

MATH 7414
(3 credit hrs)

Modern Algebra II.

Every other year. Field theory, Galois theory, modules over rings, especially over integral domains.
Prereq: MATH 5718 and MATH 7413.

MATH 7421
(3 credit hrs)

Projective Geometry.

Every other year. Synthetic and algebraic development of projective spaces. Collineation groups, representation theorems, quadratic sets and applications. Emphasis is on finite projective spaces.
Prereq: MATH 5718 and 7409.

MATH 7593
(3 credit hrs)

Advanced Linear Programming.

Every three years. A PhD level course that goes deeper into linear programming, starting from where a graduate-level course (5593) ends. Topics include advanced sensitivity analysis, sparse matrix techniques, and special structures. Additional topics, which vary, include deeper analysis of algorithms, principles of model formulation and solution analysis.
Prereq: MATH 5593.

MATH 7594
(3 credit hrs)

Integer Programming.

Every three years. A PhD level course that uses linear programming (5593), especially polyhedral theory, to introduce concepts of valid inequalities and superadditivity. Early group-theoretic methods by Gomory and Chvatal´s rounding function are put into modern context, including their role in algorithm design and analysis. Duality theory and relaxation methods are presented for general foundation and analyzed for particular problem classes. Among the special problems considered are knapsack, covering, partitioning, packing, fix-charge, traveling salesman, generalized assignment matchings. Matroids are introduced and some greedy algorithms are analyzed. Additional topics, which vary, include representability theory, heuristic search and complexity analysis.
Prereq: MATH 5593.

MATH 7595
(3 credit hrs)

Advanced Nonlinear Programming.

Every three years. Focuses primarily on the fundamental theory of nonlinear programming. Topics include convex analysis, optimality criteria, Lagrangian and conjugate duality, stability and sensitivity analysis. Other topics vary depending on the research interests of the instructor.
Prereq: MATH 5595.

MATH 7663
(3 credit hrs)

Finite Difference Methods For Partial Differential Equations.

Every other year. Consistency, stability, and convergence for difference schemes. Derivations based on Taylor series and finite series. Methods for parabolic and hyperbolic initial value problems and initial-boundary-value problems, elliptic boundary-value problems, some nonlinear problems.
Prereq: MATH 5070 and 5733.

MATH 7665
(3 credit hrs)

Numerical Linear Algebra.

Every other year. Solution of linear equations, eigenvector and eigenvalue calculation, matrix error analysis, orthogonal transformation, iterative methods.
Prereq: MATH 5660 and 5718.

MATH 7667
(3 credit hrs)

Introduction to Approximation Theory.

Every other year. Linear normed and Banach spaces, convexity, existence and uniqueness of best approximations, least square approximation and orthogonal polynomials, Chebychev approximation by polynomials and other related families, splines.
Prereq: MATH 5070 and 5718.

MATH 7821
(3 credit hrs)

Topics in Projective Geometry.

Infrequent. Advanced topics in projective geometry. Topics may include finite projective planes, free projective planes, derivation, collineation groups, higher dimensional projective spaces, ovals and ovoids.
Prereq: MATH 7421.

MATH 7822
(3 credit hrs)

Topics in Linear Algebra.

Infrequent. Topics may include canonical forms, bilinear and quadratic forms, and combinatorial matrix theory.
Prereq: MATH 5718.

MATH 7823
(3 credit hrs)

Topics in Discrete Math.

Advanced topics in discrete mathematics; will change from semester to semester.
Prereq: MATH 7113, 6404 and 7409, or permission of instructor.

MATH 7824
(3 credit hrs)

Topics in Computational Mathematics.

Infrequent. Topics include methods for differential equations, numerical optimization, approximation theory, inverse problems, and Fourier analysis.
Prereq: permission of instructor.

MATH 7825
(3 credit hrs)

Topics in Optimization.

Infrequent. Some topics are extensions of those introduced in MATH 6595, while other topics are new. Examples of topics are: duality, stability, sensitivity, consistency, redundancy, principles of optimality, control theory, calculus of variations, global (non-convex) optimization and model reformulation.
Prereq: permission of the instructor.

MATH 7826
(3 credit hrs)

Topics in Probability and Statistics.

Infrequent. Topics may include generalized linear models, information theory, robust methods, spatial statistics, sequential analysis, Monte Carlo methods, queuing theory.
Note: Since topics vary each semester, students may register for this course more than once. Prereq: permission of instructor.

MATH 7827
(3 credit hrs)

Topics in Applied Mathematics.

Infrequent. Topics include problems in differential equations, optimization, mathematical modeling, Fourier analysis and approximation theory.
Note: Since topics vary each semester, students may register for this course more than once.

MATH 7840
(1 to 3 credit hrs)

Independent Study.

Available only to PhD students.

MATH 7921 to 7927
(1 credit hrs)

Readings in Mathematics.

Annual. Seven readings courses are offered regularly primarily for PhD students at the research level in the designated fields. The seminar format requires significant student participation.
Prereq: permission of instructor.

MATH 8660
(3 credit hrs)

Mathematical Foundations of Finite Element Methods.

Every other year. Theoretical foundations of finite element methods for elliptic boundary value problems, Sobolev spaces, interpolation of Sobolev spaces, variational formulation of elliptic boundary-value problems, basic error estimates, applications to elasticity, practical aspects of the finite element method.
Prereq: MATH 6653 (or equivalent programming experience), and MATH 6131/7132.

MATH 8664
(3 credit hrs)

Iterative Methods in Numerical Linear Algebra.

Every other year. Preconditioned iterative methods for linear systems and eigen problems, conjugate gradients, multigrid and domain decomposition.
Prereq: MATH 5660 and 7665.

MATH 8990
(1 to 10 credit hrs)

Doctoral Dissertation.

Only for students working on their PhD research.

Course No.	Course Title	Description
MATH 1009 (3 credit hrs)	Computer-Based Algebraic Problem Solving.	A laboratory-based problem solving course focused on personal computing applications. Topics include general problem solving techniques, deductive reasoning, elementary probability, computer algebraic software, optimization, graphical analysis, systems of equations, spreadsheets, functions, descriptive statistics, linear programming and elementary programming logic. Prereq: basic high school algebra and some familiarity with Microsoft Windows.
MATH 1010 (3 credit hrs)	Mathematics for the Liberal Arts: GT-MA1.	Designed to give liberal arts students the skills required to understand and interpret quantitative information that they encounter in the news and in their studies, and to make quantitatively-based decisions in their lives. Topics include a survey of logic and analysis of arguments, identifying fallacies in reasoning, working with numbers and units, linear and exponential relations and essentials of probability and statistics. The emphasis is on applications with case studies in economics, finance, environmental sciences, health, music and science. Colorado state-wide Guaranteed Transfer course. Prereq: three years of high school mathematics. Approved Core Course.
MATH 1070 (3 credit hrs)	Algebra for Social Sciences and Business: GT-MA1.	Topics in algebra designed for students who intend to take business calculus. Functions, graphs, scatter plots, curve-fitting, solving systems of equations, polynomial and rational functions, and selected other topics. Colorado state-wide Guaranteed Transfer course. NOTE: Graphics calculator required. No co-credit with MATH 1110 or MATH 1130. Prereq: intermediate algebra and satisfactory score on the placement exam. Approved Core Course.
MATH 1075 (1 credit hrs)	Linear Programming and Probability.	Designed to fulfill College of Business requirements for students who have had college algebra but not MATH 1070. Emphasis is on applications of linear programming and probability. Prereq: MATH 1110. No co-credit with MATH 1070.
MATH 1080 (3 credit hrs)	Polynomial Calculus: GT-MA1.	A one-semester course in single-variable calculus. Topics include limits, derivatives, differentiation rules, integration and integration rules. Emphasis is on applications to business and social sciences. Colorado state-wide Guaranteed Transfer course. Note: No knowledge of trigonometry is required. Those planning to take more than one semester of calculus should take MATH 1401 instead of MATH 1080. Prereq: MATH 1070 or 1110. No co-credit with MATH 1401. Approved Core Course.
MATH 1110 (3 credit hrs)	College Algebra: GT-MA1.	Topics in algebra designed for students who intend to take the calculus sequence. Functions, domains, ranges, graphs, data scatter plots and curve fitting, solving equations and systems of equations, polynomial functions, rational functions, and selected other topics. Graphic calculators and/or computer algebra systems are used extensively. Applications are emphasized. Colorado state-wide Guaranteed Transfer course. Note: No co-credit with either MATH 1070 or 1130. Prereq: intermediate algebra and satisfactory score on the placement exam. Approved Core Course.
MATH 1111 (1 to 3 credit hrs)	Freshman Seminar.
MATH 1120 (3 credit hrs)	College Trigonometry: GT-MA1.	Topics in trigonometry, analytic geometry, and elementary functions designed for students who intend to take the calculus sequence. Angles and trigonometry functions of acute angles, analytic trigonometry, fundamental trigonometric functions and identities including hyperbolic trigonometry, parametric equations, and polar coordinate system. Graphic calculators and/or computer algebra systems are used extensively. Applications are emphasized. Colorado state-wide Guaranteed Transfer course. Prereq: MATH 1110 and placement test. No joint credit with MATH 1130. Approved Core Course.
MATH 1130 (4 credit hrs)	Precalculus Mathematics: GT-MA1.	Condensed treatment of the topics in MATH 1110 and 1120. Colorado state-wide Guaranteed Transfer course. Prereq: satisfactory score on the placement exam. No co-credit with MATH 1070, 1110 or 1120. Approved Core Course.
MATH 1401 (4 credit hrs)	Calculus I: GT-MA1.	First course of a three-semester sequence (MATH 1401, 2411, 2421) in calculus. Topics covered include limits, derivatives, applications of derivatives, and the definite integral. Colorado state-wide Guaranteed Transfer course. Note: Students cannot receive credit for both MATH 1080 and 1401. Prereq: MATH 1120 or 1130 and satisfactory score on the placement exam. Approved Core Course.
MATH 2411 (4 credit hrs)	Calculus II: GT-MA1.	The second of a three-semester sequence (MATH 1401, 2411, 2421) in calculus. Topics covered include exponential, logarithmic, and trigonometric functions, techniques of integration, indeterminate forms, improper integrals and infinite series. Colorado state-wide Guaranteed Transfer course. Prereq: MATH 1401. Approved Core Course.
MATH 2421 (4 credit hrs)	Calculus III: GT-MA1.	The third of a three-semester sequence in calculus (MATH 1401, 2411 and 2421). Topics include vectors, vector-valued functions, partial differentiation, differentiation, multiple integration, and vector calculus. Colorado state-wide Guaranteed Transfer course. Prereq: MATH 2411. Approved Core Course.
MATH 2511 (3 credit hrs)	Discrete Structures.	Covers the fundamentals of discrete mathematics, including: logic, sets, functions, growth of functions, algorithms, matrices, mathematical reasoning, proofs, induction, relations, graphs, trees and combinatorics. There is an emphasis on how discrete mathematics applies to computer science in general, and algorithm analysis in particular. Prereq: C SC 2421. Cross-listed with C SC 2511.
MATH 2810 (1 to 3 credit hrs)	Topics.	Topics in mathematics with various subtitles reflecting course content. Prereq: permission of instructor.
MATH 2830 (3 credit hrs)	Introductory Statistics: GT-MA1.	Basic statistical concepts, summarizing data, probability concepts, distributions, confidence intervals, hypothesis testing. Colorado state-wide Guaranteed Transfer course. Prereq: intermediate algebra. Approved Core Course.
MATH 2939 (1 to 3 credit hrs)	Internship/Cooperative Education.	Experiences involving application of specific, relevant concepts and skills in supervised employment situations. Prereq: 15 hours of 2.75 GPA.
MATH 3000 (3 credit hrs)	Introduction to Abstract Mathematics.	Students learn to prove and critique proofs of theorems by studying elementary topics in abstract mathematics, including logic, sets, functions, equivalence relations and elementary combinatorics. Prereq: MATH 1401.
MATH 3040 (3 credit hrs)	Mathematics for Elementary Teachers.	Topics include intuitive and logical development of geometric ideas relevant to K-6 curriculum; measurement of length, area, volume, mass, angle, temperature, time and the metric system. Further study of the rational number system, probably and statistics, applications and problem solving. Note: carries credit only for elementary education majors. Prereq: three years of high school mathematics.
MATH 3140 (3 credit hrs)	Introduction to Modern Algebra.	Studies the fundamental algebraic structures used in modern mathematics. Topics include groups, rings, fields and polynomials. Prereq: MATH 3000.
MATH 3191 (3 credit hrs)	Applied Linear Algebra.	Topics include systems of equations, Gaussian elimination with partial pivoting, LU--decomposition of matrices, matrix algebra, determinants, vector spaces, linear transformations, eigenvalues and applications. Prereq: MATH 2411.
MATH 3195 (4 credit hrs)	Linear Algebra and Differential Equations.	Presents the essential ideas and methods of linear algebra and differential equations, emphasizing the connections between and the applications of both subjects. The course is designed for students in the sciences and engineering. Prereq: MATH 2411.
MATH 3200 (3 credit hrs)	Elementary Differential Equations.	First and second order differential equations, Laplace transforms, systems of equations, with an emphasis on modeling and applications. Prereq: MATH 2411; coreq is MATH 3191.
MATH 3210 (3 credit hrs)	Higher Geometry I.	Studies the foundations of modern geometry by examining axiomatic systems for various geometrics, with an emphasis on non-Euclidean hyperbolic geometry. Prereq: MATH 3000.
MATH 3250 (3 credit hrs)	Problem Solving Tools.	Fall. Students learn and refine both problem solving techniques and computer programming skills. Examples, exercises, and projects are taken from a wide range of mathematical topics including algebra, calculus, linear algebra and probability. Note: This course will not count toward a graduate degree in applied mathematics. Prereq: MATH 2421. Cross-listed with MATH 5250.
MATH 3301 (3 credit hrs)	Introduction to Operations Research I - Deterministic Systems.	A mathematical approach to decision making based on optimization. Topics include linear programming, network flows and production models. Prereq: MATH 3191 or 3195.
MATH 3302 (3 credit hrs)	Operations Research II.	Elementary stochastic processes and standard nondeterministic operations research models: Markov chains, Poisson processes, renewal processes, queuing theory, inventory models, Markov decision processes, simulation. Prereq: MATH 3800 and 3191.
MATH 3440 (3 credit hrs)	Introduction to Symbolic Logic.	Spring. Covers truth functional and quantificational logic through polyadic first order predicate calculus and theory of identity. Attention is given to such problems in metatheory as proofs of the completeness and consistency of systems of logic. Prereq: MATH 3000. Cross-listed with PHIL 3440.
MATH 3511 (4 credit hrs)	Mathematics of Chemistry.	Fall. Multivariate functions, probability and statistics for chemistry, matrices and vectors, mathematics of reaction kinetics and symmetry point groups. Course covers mathematics needed for CHEM 4511 and 4521. Can also be an elective for the mathematics minor. Prereq: MATH 2411, CHEM 2031, CHEM 2061.
MATH 3800 (3 credit hrs)	Probability and Statistics for Engineers	Basic probability theory, discrete and continuous random variables, point and interval estimation, test of hypotheses, one-way analysis of variance, and simple linear regression. Note: no co-credit with MATH 4810. Prereq: MATH 2411; coreq: MATH 2421.
MATH 3939 (1 to 3 credit hrs)	Internship/Cooperative Education.	Designed experiences involving application of specific, relevant concepts and skills in supervised employment situations. Prereq: junior standing and 2.75 GPA.
MATH 4010 (3 credit hrs)	History of Mathematics.	Spring. A history of the development of mathematical techniques and ideas from early civilization to the present, including the inter-relationships of mathematics and sciences. Prereq: MATH 1401. Cross-listed with MATH 5010.
MATH 4027 (3 credit hrs)	Topics in Mathematics.	Special topics in mathematics will be covered; consult ´Schedule Planner´ for current topics and prerequisites.
MATH 4101 (3 credit hrs)	Applied Statistics Using SAS and SPSS I.	Teaches the practical statistical tools social scientists use to analyze real-world problems. Course split into four modules, each taught by a different instructor. The first module introduces SAS and SPSS; modules 2-4 are problem-based and cover topics such as ANOVA, multivariate regression, and cluster analysis. Prereq: any statistics course.
MATH 4102 (3 credit hrs)	Applied Statistics Using SAS and SPSS II.	Spring. (Continuation of MATH 4101.) Students use the skills they learned in the previous semester to analyze a social issue of their choosing and present their findings. In addition to lectures, weekly one-on-one meetings between faculty and student are are required. Prereq: MATH 4101.
MATH 4110 (3 credit hrs)	Theory of Numbers.	Every other year. Topics include divisibility, prime numbers, congruencies, number theoretic functions, quadratic reciprocity, and special diophantine equations, with applications in engineering. Prereq: MATH 3000. Cross-listed with MATH 5110.
MATH 4201 (3 credit hrs)	Topology.	Spring. Metric spaces and topological spaces, compactness, separation properties, and connectedness. Prereq: MATH 3000.
MATH 4220 (3 credit hrs)	Higher Geometry II.	Studies affine and projective geometrics. Coordinates are introduced in this framework. Planes and higher dimensional spaces are examined. Prereq: MATH 3191.
MATH 4310 (4 credit hrs)	Introduction to Real Analysis I.	Fall. Calculus of one variable, the real number system, continuity, differentiation, integration theory, sequence and series. Prereq: MATH 2421 and 3000.
MATH 4320 (4 credit hrs)	Introduction to Real Analysis II.	Spring. Convergence, uniform convergence; Taylor´s theorem; calculus of several variables including continuity, differentiation and integration; Picard´s theorem in ordinary differential equations and Fourier series. Prereq: MATH 4310.
MATH 4387 (3 credit hrs)	Regression Analysis, Modeling and Time Series	Fall. Topics include linear and multiple regression, basic experimental designs, one-way analysis of variance. Emphasis is on practical aspects and applications of linear models to the analysis of data in business engineering, behavioral, biological and physical sciences. Prereq: MATH 3191 and 3800/4820. Cross-listed with MATH 5387.
MATH 4390 (3 credit hrs)	Game Theory.	Annual. Begins with an introduction to the mathematical theory of games and the definition of a solution, including extensive and normal forms of representation. The fundamental minimax theorem is presented first as the foundation for two-person matrix games, then extended with fixed point theory to other games. Principles of dominance and solution methods are presented, plus applications to economics, political science, engineering, and other fields. An introduction to n-person game theory is included, with basic terms and concepts. Prereq: MATH 2421, 3191 and 3800/4810. Cross-listed with MATH 5390.
MATH 4394 (3 credit hrs)	Experimental Designs.	Infrequent. Completely randomized block designs factorial and fractional factorial experiments, balanced incomplete block designs, responses surface methods. Prereq: MATH 4387. Cross-listed with MATH 5394.
MATH 4408 (3 credit hrs)	Applied Graph Theory.	Introduces discrete structures and applications of graph theory to computer science, engineering, operations research, social science, and biology. Topics include connectivity, coloring, trees, Euler and Hamiltonian paths and circuits, matching and covering problems, shortest route and network flows. Prereq: C SC/MATH 2511 or MATH 3000. Cross-listed with C SC 4408.
MATH 4409 (3 credit hrs)	Applied Combinatorics.	Every other year. Major emphasis is on applied combinatorics and combinatorial algorithms, with applications in computer science and operations Topics include general counting methods, generating functions, recurrence relations, inclusion-exclusion, and block designs. Prereq: MATH 4408 and 3140.
MATH 4410 (3 credit hrs)	Mathematics of Coding Theory.	Error correcting codes are used to recapture information that has been distorted in some transmission process. Various coding schemes use block codes obtained from algebraic, geometric, and combinatorial structures. Topics include: fundamentals of coding theory, linear, Reed-Muller, Golay, cyclic and BCH codes. Prereq: MATH 3191.
MATH 4450 (3 credit hrs)	Complex Variables.	Infrequent. Topics include complex algebra, Cauchy-Riemann equations, Laurent expansions, theory of residues, complex integration, and introduction to conformal mapping. Prereq: MATH 2421 and MATH 3000.
MATH 4576 (3 credit hrs)	Mathematical Foundations of Artificial Intelligence I.	Infrequent. Fundamentals course that complements other approaches, such as in engineering, psychology, and business administration. Here, the emphasis is on the mathematical foundations. Topics include logical inference, problem solving, heuristic search, neural search, neural nets, analogical reasoning and learning. Models and paradigms also consider different measures of uncertainty. Prereq: C SC 2511, MATH 2511/3000 and 3191. Cross-listed with MATH 5576.
MATH 4650 (3 credit hrs)	Numerical Analysis I.	(Same as CSMC 4650 at Colorado School of Mines.) Methods and analysis of techniques used to resolve continuous mathematics problems on the computer. Solution of linear and nonlinear equations, interpolation and integration. Prereq: MATH 2411, 3191 or MATH 3195 and programming experience. Cross-listed with C SC 4650, 5660 and MATH 5660.
MATH 4660 (3 credit hrs)	Numerical Analysis II.	Spring. Numerical differentiation and integration, numerical solution of ordinary differential equations, the Galerkin method for the Poisson equation. Prereq: MATH 3195 or both 3191 and 3200; MATH or C SC 4650 or 5660; or programming experience. Cross-listed with MATH 5661, C SC 4660 and 5661.
MATH 4674 (3 credit hrs)	Parallel Computing and Architectures.	Infrequent. Examines a range of topics involved in using parallel operations to improve computational performance. Parallel architectures, parallel algorithms, parallel programming languages, interconnection networks, and their relation to specific computer architectures. Prereq: MATH 4650. Cross-listed with MATH 5674.
MATH 4733 (3 credit hrs)	Partial Differential Equations.	Infrequent. Initial/Boundary value problems for first-order, wave, heat and Laplace Equations; maximum principles; Fourier Series and applications. Prereq: MATH 2421 and 3200. Cross-listed with MATH 5733.
MATH 4779 (3 credit hrs)	Math Clinic.	The clinic is intended to illustrate the applicability and utility of mathematical concepts. Research problems investigated originate from a variety of sources--industry, government agencies, educational institutions, or nonprofit organizations. Prereq: consult Schedule Planner or instructor. Cross-listed with MATH 5779.
MATH 4788 (3 credit hrs)	Bioinformatics	Provides a broad exposure to the basic concepts and methodologies of bioinformatics and their application to analyzing genomic and proteomic data. Topics may include dynamic programming algorithms, graph theoretic techniques, hidden Markov models, phylogenetic trees, RNA/protein structure prediction and microarray analysis. Prereq: C SC 1410 and MATH 3191 or 3195. Cross-listed with C SC 4788, PHYS 4788.
MATH 4791 (3 credit hrs)	Continuous Modeling.	Every other year. Surveys mathematical problems that arise in natural sciences and engineering. Topics may include population models, epidemic models, mechanics, heat transfer and diffusion, tomography, pharmaco-kinetics, traffic flow, fractal models, wave phenomena, and natural resource management. Most models discussed are based on differential and integral equations. Emphasis is formulation and validation of models as well as methods of solution. Prereq: MATH 3191 and 3200. Cross-listed with MATH 5791.
MATH 4792 (3 credit hrs)	Probabilistic Modeling.	Every other year. Markov chains; Poisson processes, continuous time Markov chains, elementary topics in queuing theory, and some mathematical aspects of Monte Carlo simulation, including random variate generation, variance reduction, and output analysis. Prereq: MATH 4810, 5310 and some programming experience. Cross-listed with MATH 5792.
MATH 4793 (3 credit hrs)	Discrete Math Modeling.	Every other year. Focuses on the use of graph theory and combinatorics to solve problems in a wide variety of disciplines. Applications are selected from computer science, communication networks, economics, operations research, and the social, biological and environmental sciences. Prereq: MATH 3191 and 4408. Cross-listed with MATH 5793.
MATH 4794 (3 credit hrs)	Optimization Modeling.	Every other year. Principles of model formulation and analysis are developed by presenting a wide variety of applications, both for natural phenomena and social systems. Examples of optimization models to represent natural phenomena include principles of least time and energy. Examples in social systems include resource allocation, environmental control and land management. Specific applications vary, but are chosen to cover a wide scope that considers dichotomies, such as discrete vs. continuous, static vs. dynamic, and deterministic vs. stochastic. Some computer modeling language (like GAMS) is taught. Prereq: MATH 2421 and 3191. Cross-listed with MATH 5794.
MATH 4810 (3 credit hrs)	Probability.	Examines elementary theory of probability, including independence, conditional probability, and Bayes´ theorem; random variables, expectations and probability distributions; joint and conditional distributions; functions of random variables; limit theorems, including the central limit theorem. Prereq: MATH 2421 and 3191. Cross-listed with MATH 5310.
MATH 4820 (3 credit hrs)	Statistics.	Spring. Point and confidence interval estimation, principles of maximum likelihood, sufficiency and completeness, tests of simple and composite hypothesis, linear models and multiple regression, analysis of variance. Prereq: MATH 3800. MATH 4810 highly recommended, but not required. Cross-listed with MATH 5320.
MATH 4830 (3 credit hrs)	Applied Statistics.	Spring. Review of estimation, confidence intervals and hypothesis testing; ANOVA; categorical data analysis; non-parametric tests; linear and logistic regression. Prereq: an introductory course in statistics such as MATH 2830 or permission of instructor. Cross-listed with MATH 5830.
MATH 4840 (1 to 3 credit hrs)	Independent Study.	Variable credit depending on the student´s needs. Offered for the advanced student who desires to pursue a specific topic in considerable depth. Note: Supervision by a full-time faculty member is necessary, and the dean´s office must concur. Students may register for this course more than once with departmental approval.
MATH 5000 (4 credit hrs)	Algebraic Patterns and Functions I RM-MSMSP.	Systematic study of the core elements of algebra: linear, quadratic, exponential, logarithmic functions and their graphs. Includes modeling using graphing calculators and real world applications. Concepts are linked to other scientific, mathematical, and pedagogical domains. This course is not applicable toward any degree in the College of Liberal Arts and Sciences. Prereq: permission of project director.
MATH 5002 (4 credit hrs)	Algebraic Patterns and Functions II RM-MSMSP.	This course is a continuation of the material covered in Math 5000. Topics that will be covered include logarithmic, exponential and trigonometric functions and applications, parametric equations, systems of equations and inequalities, matrices and linear programming. This course is not applicable toward any degree in the College of Liberal Arts and Sciences. Prereq: MATH 5000 or permission of instructor.
MATH 5004 (4 credit hrs)	RM-MSMSP: Statistics and Probability.	Studies the collection, presentation, and analysis of data; and elements and applications of counting discrete probability. Includes real world applications and technology. Concepts are linked to other scientific, mathematical, and pedagogical domains. This course is not applicable toward any degree in the College of Liberal Arts and Sciences. Prereq: permission of project director.
MATH 5005 (4 credit hrs)	RM-MSMSP: Geometry.	Systematic study of advanced geometric concepts: history of geometry and measurement, patterns among shapes, 2- and 3-dimensional shapes, constructions, symmetry or transformational geometry. Includes applications and activity-oriented instruction. Concepts are linked to other scientific, mathematical, and pedagogical domains. This course is not applicable toward any degree in the College of Liberal and Sciences. Prereq: permission of project director.
MATH 5006 (4 credit hrs)	RM-MSMSP: Mathematics of Change.	Systematic study of the application of calculus to the analysis of changing systems in real world applications. Emphasizes the connections that exist between calculus and aspects of middle school curricula. Concepts are linked to other scientific, mathematical, and pedagogical domains. This course is not applicable toward any degree in the College of Liberal Arts and Sciences. Prereq: MATH 5000 (or equivalent) or permission of project director.
MATH 5007 (4 credit hrs)	RM-MSMSP: Discrete Math--Counting the Possibilities.	Systematic study of basic techniques in discrete mathematics and their various applications: permutations and combinations, inclusion or exclusion, pigeonhole principle, graph theory, and recursive pattern solving. Applications to topics such as network analysis and voting theory are stressed. Concepts are linked to other scientific, mathematical, pedagogical domains. This course is not applicable toward any degree in the College of Liberal Arts and Sciences. Prereq: MATH 5000 (or equivalent) or permission of project director.
MATH 5008 (4 credit hrs)	RM-MSMSP: Discovery and Use of the History of Math.	Systematic study of the people, events, ideas and issues from the history of mathematics, focusing on historical topics that are central to the discipline and teaching of mathematics and emphasizing web research of historical topics of interest. Concepts are linked to other scientific, mathematical, and pedagogical domains. Note: This course is not applicable toward any degree in the College of Liberal Arts and Sciences. Prereq: permission of the project director.
MATH 5009 (4 credit hrs)	RM-MSMSP: Math Modeling--Using and Applying Math.	Systematic study of math modeling using algebra, geometry, discrete mathematics, rates of change, and statistics to solve real-world problems in areas such as finance, biology, economics, and physics. Concepts are linked to other scientific, mathematical, societal, and pedagogical domains. This course is not applicable toward any degree in the College of Liberal Arts and Sciences. Prereq: MATH 5006 (or equivalent) or permission of instructor.
MATH 5010 (3 credit hrs)	History of Mathematics.	Spring. A history of the development of mathematical techniques and ideas from early civilization to the present, including the inter-relationships of mathematics and sciences. Prereq: MATH 1401. Not open to students who have had MATH 4010. No credit for applied math graduate students. Cross-listed with MATH 4010.
MATH 5017 (1 to 3 credit hrs)	Topics in Mathematics for Teachers.	Topics vary from semester to semester. Designed for professional mathematics teachers. Note: This course will not count toward a degree in applied mathematics. Prereq: permission of instructor.
MATH 5027 (3 credit hrs)	Topics in Applied Mathematics.	Selected topics in mathematical problems arising from various applied fields such as mechanics, electromagnetic theory, economics and biological sciences. Prereq: permission of instructor.
MATH 5070 (3 credit hrs)	Applied Analysis.	Spring. Designed to serve as an introduction to real analysis. Topics include: fundamentals of logic and theorem proving, infimum and supremum, real numbers, point-set topology in metric spaces, properties of functions, sequences and series of functions, fixed point theorems, Riemann integral, power series. Prereq: MATH 4320.
MATH 5110 (3 credit hrs)	Theory of Numbers.	Every other year. Topics include divisibility, prime numbers, congruences, number theoretic functions, quadratic reciprocity, and special diophantine equations, with applications in engineering. Prereq: MATH 3000. Cross-listed with MATH 4110.
MATH 5135 (3 credit hrs)	Functions of a Complex Variable.	Infrequent. The complex plane, infinite series and products, elementary special functions, Cauchy-Riemann equations, conformal mapping, complex integration, Cauchy integral theory, and residue theory. Prereq: MATH 4320; MATH 5070 recommended.
MATH 5198 (3 credit hrs)	Mathematics for Bioscientists.	Fall. Develops mathematical reasoning; introduces linear algebra, discrete structures, graph theory, probability, and differential equations, using applications to molecular biology. Note: No credit for mathematics or engineering students. Prereq: MATH 2411.
MATH 5250 (3 credit hrs)	Problem Solving Tools.	Fall. Students learn and refine both problem solving techniques and computer programming skills. Examples, exercises, and projects are taken from a wide range of mathematical topics including algebra, calculus, linear algebra and probability. Note: This course will not count toward a graduate degree in applied mathematics. Coreq: MATH 2421. Cross-listed with MATH 3250.
MATH 5310 (3 credit hrs)	Probability.	Examines elementary theory of probability, including independence, conditional probability, and Bayes´ theorem; random variables, expectations and probability distributions; joint and conditional distributions; functions of random variables; limit theorems, including the central limit theorem. Prereq: Math 2421 and 3191. Cross-listed with MATH 4810.
MATH 5320 (3 credit hrs)	Statistics.	Spring. Point and confidence interval estimation, principles of maximum likelihood, sufficiency and completeness, tests of simple and composite hypothesis, linear models and multiple regression, analysis of variance. Prereq: MATH 3800. MATH 4810 highly recommended, but not required. Cross-listed with MATH 4820.
MATH 5387 (3 credit hrs)	Regression Analysis, Modeling and Time Series	Fall. Topics include linear and multiple regression, basic experimental designs, one-way analysis of variance. Emphasis is on practical aspects and applications of linear models to the analysis of data in business engineering and behavioral, biological and physical sciences. Prereq: MATH 3191 and 3800/4820. Cross-listed with MATH 4387.
MATH 5390 (3 credit hrs)	Game Theory.	Infrequent. Begins with an introduction to the mathematical theory of games and the definition of a solution, including extensive and normal forms of representation. The fundamental minimax theorem is presented first, as the foundation for two-person matrix games, then extended with fixed point theory to other games. Principles of dominance and solution methods are presented, plus applications to economics, political science, engineering, and other fields. An introduction to n-person game theory is included, with basic terms and concepts. Prereq: MATH 2421, 3191 and 3800/4810. Cross-listed with MATH 4390.
MATH 5394 (3 credit hrs)	Experimental Designs.	Infrequent. Completely randomized block designs, factorial and fractional factorial experiments, balanced incomplete block designs, response surface methods. Prereq: MATH 4387 and 5387. Cross-listed with MATH 4394.
MATH 5410 (3 credit hrs)	Modern Cryptology.	Every other year. Deals with the mathematics that underlies modern cryptology. Topics include: classical cryptology, public and private key cryptosystems, secret sharing schemes, authentication schemes, linear feedback shift registers, discrete logarithm and elliptic curve-based schemes. Prereq: MATH 3191.
MATH 5432 (3 credit hrs)	Computational Graph Theory.	Infrequent. Algorithmic techniques in graph theory and other discrete mathematics areas. Typical topics include: branch-bound algorithms, matching, colorings, domination, min-plus algebra, simulated annealing and related heuristics, NP-completeness theory. Prereq: a course in graph theory and some programming experience.
MATH 5446 (3 credit hrs)	Theory of Automata.	Infrequent. Studies the relationships between classes of formal languages (regular, context-free, context-sensitive, phrase-structure) and classes of automata (finite-state, pushdown, Turing machines). Additional topics include decidability and computability issues. Prereq: MATH 3000 and 3140. Cross-listed with C SC 5446.
MATH 5490 (3 credit hrs)	Network Flows.	Infrequent. Begins with the classical min-cost flow problem, defined on an ordinary network. Other problems, such as shortest path, are also shown in this class. Both theory and algorithms are presented. Extensions include generalized networks, nonlinear costs, fixed charges, multi-commodity flows, and and additional applications, such as in communications networks. Prereq: graduate standing in math or computer science.
MATH 5576 (3 credit hrs)	Mathematical Foundations of Artificial Intelligence I.	Infrequent. A fundamentals course that complements other approaches, such as in engineering, psychology, and business administration. Here the emphasis is on the mathematical foundations. Topics include logical inference, problem solving, heuristic search, neural nets, analogical reasoning and learning. Models and paradigms also consider different measures of uncertainty. Prereq: C SC 2511, MATH 2511/3000 and 3191. Cross-listed with MATH 4576.
MATH 5593 (3 credit hrs)	Linear Programming.	Fall. A linear program is an optimization problem that seeks to minimize or maximize a linear function subject to a system of linear inequalities and equations. This course begins with examples of linear programs and variations in their representations. Basic theoretical foundations covered include polyhedra, convexity, linear inequalities and duality. Two classes of solution algorithms are given: simplex methods and interior point methods. The primary emphasis of this course is on mathematical foundations, and applications are used to illustrate the main results. Prereq: MATH 3191.
MATH 5610 (3 credit hrs)	Computational Biology.	Spring. Basic introduction and mathematical foundations. Topics include comparative genomics; proteomics; phylogeny; dynamic programming and sequence alignment; gene expression arrays and clustering; Bayesian networks; structure prediction and hidden Markov models. Prereq: C SC 1410 or equivalent programming experience, and MATH 3191 or 3195.
MATH 5660 (3 credit hrs)	Numerical Analysis I.	(Same as CSMC 4650 at Colorado School of Mines.) Methods and analysis of techniques used to resolve continuous mathematics problems on the computer. Solution of linear and nonlinear equations, interpolation and integration. Prereq: MATH 2411, 3191 or MATH 3195 and programming experience. Cross-listed with C SC 4650, 5660 and MATH 4650.
MATH 5661 (3 credit hrs)	Numerical Analysis II.	Spring. Numerical differentiation and integration, numerical solution of ordinary differential equations, the Galerkin method for the Poisson equation. Prereq: MATH 3195 or both 3191 and 3200; MATH or C SC 4650 or 5660; or programming experience. Cross-listed with MATH 4660, C SC 4660 and 5661.
MATH 5674 (3 credit hrs)	Parallel Computing and Architectures.	Infrequent. Examines a range of topics involved in using parallel operations to improve computational performance. Parallel architectures, parallel algorithms, parallel programming languages, interconnection networks, and their relation to specific computer architectures. Prereq: MATH 4650. Cross-listed with MATH 4674.
MATH 5718 (3 credit hrs)	Applied Linear Algebra.	Fall. Topics include: vector spaces, practical solution of systems of equations, projections, eigenvalues and eigenvectors, unitary transformations, Schur QR, singular value decompositions, similarity transformations, Jordan forms, and positive definite matrices. Prereq: MATH 3191.
MATH 5733 (3 credit hrs)	Partial Differential Equations.	Infrequent. Initial/Boundary value problems for first-order, wave, heat and Laplace Equations; maximum principles; Fourier Series and applications. Prereq: MATH 2421 and 3200; graduate standing. Cross-listed with MATH 4733.
MATH 5779 (3 credit hrs)	Math Clinic.	The clinic is intended to illustrate the applicability and utility of mathematical concepts. Research problems investigated originate from a variety of sources--industry, government agencies, educational institutions, or nonprofit organizations. Prereq: consult Schedule Planner or instructor. Cross-listed with MATH 4779.
MATH 5791 (3 credit hrs)	Continuous Modeling.	Every other year. Surveys mathematical problems that arise in natural sciences and engineering. Topics may include population models, epidemic models, mechanics, heat transfer and diffusion, tomography, pharmaco-kinetics, traffic flow, fractal models, wave phenomena, and natural resource management. Most models discussed are based on differential and integral equations. Emphasis is formulation and validation of models as well as methods of solution. Prereq: MATH 3191 and 3200. Cross-listed with MATH 4791.
MATH 5792 (3 credit hrs)	Probabilistic Modeling.	Every other year. Markov chains; Poisson processes, continuous time Markov chains, elementary topics in queuing theory, and some mathematical aspects of Monte Carlo simulation, including random variate generation, variance reduction, and output analysis. Prereq: MATH 4810, 5310 and some programming experience. Cross-listed with MATH 4792.
MATH 5793 (3 credit hrs)	Discrete Math Modeling.	Every other year. Focuses on the use of graph theory and combinatorics to solve problems in a wide variety of disciplines. Applications are selected from computer science, communication networks, economics, operations research, and the social, biological and environmental sciences. Prereq: MATH 3191 and 4408. Cross-listed with MATH 4793.
MATH 5794 (3 credit hrs)	Optimization Modeling.	Every other year. Principles of model formulation and analysis are developed by presenting a wide variety of applications, both for natural phenomena and social systems. Examples of optimization models to represent natural phenomena include principles of least time and energy. Examples in social systems include resource allocation, environmental control and land management. Specific applications vary, but are chosen to cover a wide scope that considers dichotomies, such as discrete vs. continuous, static vs. dynamic, and deterministic vs. stochastic. Some computer modeling language (like GAMS) is taught. Prereq: MATH 2421 and 3191. Cross-listed with MATH 4794.
MATH 5830 (3 credit hrs)	Applied Statistics.	Spring. Review of estimation, confidence intervals and hypothesis testing; ANOVA; categorical data analysis; non-parametric tests; linear and logistic regression. Prereq: an introductory course in statistics such as MATH 2830 or permission of instructor. Cross-listed with MATH 4830.
MATH 5840 (1 to 3 credit hrs)	Independent Study.	Available only with approval of graduate advisor. Subjects arranged.
MATH 5939 (1 to 6 credit hrs)	Internship/Cooperative Education.
MATH 5950 (1 to 8 credit hrs)	Master´s Thesis.
MATH 5960 (1 to 8 credit hrs)	Master´s Project.	Note: This course will not count toward a graduate degree in Applied Mathematics.
MATH 6023 (3 credit hrs)	Topics in Discrete Math.	Topics may include graph theory, combinatorics, matroid theory, combinatorial matrix theory, finite geometry, design theory, and discrete algorithms. Note: Since topic varies by semester, students may register for this course more than once. Prereq: permission of the instructor.
MATH 6131 (3 credit hrs)	Real Analysis.	Every other year. Lebesque measure and integration, general measure and integration theory, Radon-Nikodyn Theorem, Fubini Theorem. Prereq: MATH 4320 or 5070.
MATH 6330 (3 credit hrs)	Workshop in Statistical Consulting	Students participate as consultants in a drop-in consulting service operated by the department. Seminars provide students with supervised experience in short term statistical consulting. Note: Since problems vary each semester, students may register for this course more than once. Prereq: MATH 5387.
MATH 6360 (3 credit hrs)	Exploratory Data Analysis.	Every other year. Philosophy and techniques associated with exploratory (vs. confirmatory) data analysis, both as originally presented (John Tukey) and current computer-based implementations. Graphical displays, robust-resistant methods (lines, two-way fits), diagnostic plots, standardization. Prereq: previous statistics course or permission of instructor.
MATH 6376 (3 credit hrs)	Statistical Computing.	Computationally-intensive methods in statistics, including random number generation and Monte Carlo methods, data partitioning and re-sampling, numerical and graphical methods, nonparametric function estimation, statistical models and data mining methodology, analysis of large data sets. Prereq: MATH 4820/4830 and 4387. Cross-listed with MATH 7376.
MATH 6380 (3 credit hrs)	Stochastic Processes.	Every other year. Markov processes in discrete and continuous time, renewal theory, martingales, Brownian motion, branching processes, and stationary processes. Applications include queuing theory, performance evaluation of computer and communication systems and finance. Prereq: MATH 3191, 3200, and 4810/5310.
MATH 6384 ( credit hrs)	Analysis of Dependent Data.	Infrequent. Statistical methods for the analysis of data with temporal and/or spatial dependence. Longitudinal data, stationary and non-stationary time series models, geostatistical and lattice spatial models, point processes, hierarchical models. Prereq: MATH 4820 or 4830 and MATH 4387.
MATH 6388 (3 credit hrs)	Advanced Statistical Methods for Research.	Infrequent. The second in a two-semester course in applied statistics. Topics include multifactor analysis of variance and covariance, categorical data, general linear models, bootstrapping, and other computationally intensive statistical methods. Prereq: MATH 5387.
MATH 6391 (3 credit hrs)	Stochastic Differential Equations.	Infrequent. Ito integral, Ito formula, weak and strong solutions, martingale representation formula, filtering, stochastic optimal control, diffusions, boundary value problems. Prereq: MATH 6383.
MATH 6393 (3 credit hrs)	Introduction to Bayesian Statistics.	Prior and posterior distributions, conjugate models, single and multiparameter models, hierarchical models, mixture models, numerical methods for evaluating posteriors, Monte Carlo methods, and Markov chain Monte Carlo. Prereq: MATH 3800 or both MATH 4810 and 4820 (or equivalent). Some computer programming experience.
MATH 6395 (3 credit hrs)	Multivariate Methods.	Every other year. Multivariate distributions, hypothesis testing and estimation. Multivariate analysis of variance, discriminant analysis, multidimensional scaling, factor analysis, principal components. Prereq: MATH 5387.
MATH 6398 (3 credit hrs)	Calculus of Variations and Optimal Control.	Infrequent. Standard variational problems (geodesic, time-of-transit, isoperimetric, surface, area), Euler-Lagrange equations, variational principles in mechanics, optimal control problems, necessary conditions for optimality, Pontryagin principle. Prereq: MATH 4320.
MATH 6404 (3 credit hrs)	Applied Graph Theory.	Every other year. Emphasis on graph theory. Topics will include trees, digraphs and networks, intersection graphs, coloring, clique coverings, distance, paths and cycles. Topics are motivated by applications. Prereq: graduate standing.
MATH 6595 (3 credit hrs)	Computational Methods in Nonlinear Programming.	Every other year. Introduces fundamental algorithms and theory for nonlinear optimization problems. Topics include Newton, quasi-Newtown and conjugate direction methods; linesearch and trust-region methods; active set, penalty and barrier methods for constrained optimization; convergence analysis and duality theory. Prereq: MATH 4320 and 5718.
MATH 6653 (3 credit hrs)	Introduction to Finite Element Methods.	Every other year. The Finite Element Method (FEM) is introduced as a generic tool for the approximation of partial differential equations that model engineering and physics problems of interest. Elliptic, hyperbolic, and parabolic equations are solved with FEM. Prereq: MATH 5660.
MATH 6735 (3 credit hrs)	Continuum Mechanics.	Every other year. Indicial notation. Eulerian and Lagrangian coordinates. Deformation, strain, strain rate, stress. Conservation of mass, momentum, and energy. Exploitation of entropy production inequality to obtain constitutive equations for elastic, viscous, visco elastic, plastic, or porous materials. Prereq: MATH 3191 and 3200 or graduate standing.
MATH 6840 (1 to 3 credit hrs)	Independent Study.
MATH 7101 (3 credit hrs)	Topology.	Every other year. Topological spaces, compactness, separation properties and connectedness. Prereq: MATH 4320.
MATH 7132 (3 credit hrs)	Functional Analysis.	Every other year. Linear metric and topological spaces, duality, weak topology, spaces of functions, linear operators, compact operators, elements of spectral theory, and operator calculus. Prereq: MATH 6131.
MATH 7376 (3 credit hrs)	Statistical Computing.	Computationally-intensive methods in statistics, including random number generation and Monte Carlo methods, data partitioning and re-sampling, numerical and graphical methods, nonparametric function estimation, statistical models and data mining methodology, analysis of large data sets. Prereq: MATH 4820/4830 and 4387. Cross-listed with MATH 6376.
MATH 7381 (3 credit hrs)	Mathematical Statistics I.	Every other year. Mathematical theory of statistics. Parametric inference: discrete and continuous distributions, methods of parameter estimation, confidence intervals. Prereq: MATH 3191 and 4820/5320.
MATH 7382 (3 credit hrs)	Mathematical Statistics II.	Every other year. (Continuation of MATH 7381.) Hypothesis testing, robust estimation, tolerance intervals, nonparametric inference, sequential methods. Prereq: MATH 7381.
MATH 7384 (3 credit hrs)	Mathematical Probability.	Every other year. Measurable spaces, probability measures, random variables, conditional expectations and martingales. Convergence in probability, almost sure convergence, convergence in distribution, limit theorems (law of large numbers, central limit theorem, law of iterated logarithm). Prereq: MATH 4810/5310 and MATH 5070 or MATH 6131.
MATH 7385 (3 credit hrs)	Stochastic Differential Equations.	Brownian motion, Ito integral, Ito formula, Dynkin´s formula, stochastic optimal control, boundary value problems, Girsanov theorem, mathematical finance, optimal stopping. Prereq: MATH 6383.
MATH 7397 (3 credit hrs)	Nonparametric Statistics.	Every three years. Statistical inference without strong model assumptions. Hypothesis testing and estimation using permutations and ranks, analysis of variance, and nonparametric model fitting. Prereq: applied mathematics - statistics (PhD.)
MATH 7405 (3 credit hrs)	Advanced Graph Theory	Continuation of MATH 6404. Topics to be covered include: trees and optimization, encoding and embedding of graphs, generalized colorings and applications, perfect graphs, extremal problems, substructures, connectedness and cycles. Prereq: MATH 6404 or permission of instructor.
MATH 7409 (3 credit hrs)	Applied Combinatorics.	Every other year. Emphasis is on enumerative combinatorics. Topics include multinomial coefficients, generating functions, SDRs, Polya's enumeration theory, pigeon-hole principle, inclusion/exclusion and Moebius inversion of finite posets. Topics may also include introduction to designs and finite geometry.
MATH 7410 (3 credit hrs)	Combinatorial Structures.	Every other year. Finite combinatorial structures; existence, construction and applications. Topics include Latin squares, Hadamard matrices, block designs, finite geometries, and extremal and non-constructive combinatorics. Prereq: MATH 5718 and MATH 7409 or permission of instructor.
MATH 7413 (3 credit hrs)	Modern Algebra I.	Every other year. Groups, rings and ideals, integral domains. Prereq: MATH 3140. Coreq: MATH 5718.
MATH 7414 (3 credit hrs)	Modern Algebra II.	Every other year. Field theory, Galois theory, modules over rings, especially over integral domains. Prereq: MATH 5718 and MATH 7413.
MATH 7421 (3 credit hrs)	Projective Geometry.	Every other year. Synthetic and algebraic development of projective spaces. Collineation groups, representation theorems, quadratic sets and applications. Emphasis is on finite projective spaces. Prereq: MATH 5718 and 7409.
MATH 7593 (3 credit hrs)	Advanced Linear Programming.	Every three years. A PhD level course that goes deeper into linear programming, starting from where a graduate-level course (5593) ends. Topics include advanced sensitivity analysis, sparse matrix techniques, and special structures. Additional topics, which vary, include deeper analysis of algorithms, principles of model formulation and solution analysis. Prereq: MATH 5593.
MATH 7594 (3 credit hrs)	Integer Programming.	Every three years. A PhD level course that uses linear programming (5593), especially polyhedral theory, to introduce concepts of valid inequalities and superadditivity. Early group-theoretic methods by Gomory and Chvatal´s rounding function are put into modern context, including their role in algorithm design and analysis. Duality theory and relaxation methods are presented for general foundation and analyzed for particular problem classes. Among the special problems considered are knapsack, covering, partitioning, packing, fix-charge, traveling salesman, generalized assignment matchings. Matroids are introduced and some greedy algorithms are analyzed. Additional topics, which vary, include representability theory, heuristic search and complexity analysis. Prereq: MATH 5593.
MATH 7595 (3 credit hrs)	Advanced Nonlinear Programming.	Every three years. Focuses primarily on the fundamental theory of nonlinear programming. Topics include convex analysis, optimality criteria, Lagrangian and conjugate duality, stability and sensitivity analysis. Other topics vary depending on the research interests of the instructor. Prereq: MATH 5595.
MATH 7663 (3 credit hrs)	Finite Difference Methods For Partial Differential Equations.	Every other year. Consistency, stability, and convergence for difference schemes. Derivations based on Taylor series and finite series. Methods for parabolic and hyperbolic initial value problems and initial-boundary-value problems, elliptic boundary-value problems, some nonlinear problems. Prereq: MATH 5070 and 5733.
MATH 7665 (3 credit hrs)	Numerical Linear Algebra.	Every other year. Solution of linear equations, eigenvector and eigenvalue calculation, matrix error analysis, orthogonal transformation, iterative methods. Prereq: MATH 5660 and 5718.
MATH 7667 (3 credit hrs)	Introduction to Approximation Theory.	Every other year. Linear normed and Banach spaces, convexity, existence and uniqueness of best approximations, least square approximation and orthogonal polynomials, Chebychev approximation by polynomials and other related families, splines. Prereq: MATH 5070 and 5718.
MATH 7821 (3 credit hrs)	Topics in Projective Geometry.	Infrequent. Advanced topics in projective geometry. Topics may include finite projective planes, free projective planes, derivation, collineation groups, higher dimensional projective spaces, ovals and ovoids. Prereq: MATH 7421.
MATH 7822 (3 credit hrs)	Topics in Linear Algebra.	Infrequent. Topics may include canonical forms, bilinear and quadratic forms, and combinatorial matrix theory. Prereq: MATH 5718.
MATH 7823 (3 credit hrs)	Topics in Discrete Math.	Advanced topics in discrete mathematics; will change from semester to semester. Prereq: MATH 7113, 6404 and 7409, or permission of instructor.
MATH 7824 (3 credit hrs)	Topics in Computational Mathematics.	Infrequent. Topics include methods for differential equations, numerical optimization, approximation theory, inverse problems, and Fourier analysis. Prereq: permission of instructor.
MATH 7825 (3 credit hrs)	Topics in Optimization.	Infrequent. Some topics are extensions of those introduced in MATH 6595, while other topics are new. Examples of topics are: duality, stability, sensitivity, consistency, redundancy, principles of optimality, control theory, calculus of variations, global (non-convex) optimization and model reformulation. Prereq: permission of the instructor.
MATH 7826 (3 credit hrs)	Topics in Probability and Statistics.	Infrequent. Topics may include generalized linear models, information theory, robust methods, spatial statistics, sequential analysis, Monte Carlo methods, queuing theory. Note: Since topics vary each semester, students may register for this course more than once. Prereq: permission of instructor.
MATH 7827 (3 credit hrs)	Topics in Applied Mathematics.	Infrequent. Topics include problems in differential equations, optimization, mathematical modeling, Fourier analysis and approximation theory. Note: Since topics vary each semester, students may register for this course more than once.
MATH 7840 (1 to 3 credit hrs)	Independent Study.	Available only to PhD students.
MATH 7921 to 7927 (1 credit hrs)	Readings in Mathematics.	Annual. Seven readings courses are offered regularly primarily for PhD students at the research level in the designated fields. The seminar format requires significant student participation. Prereq: permission of instructor.
MATH 8660 (3 credit hrs)	Mathematical Foundations of Finite Element Methods.	Every other year. Theoretical foundations of finite element methods for elliptic boundary value problems, Sobolev spaces, interpolation of Sobolev spaces, variational formulation of elliptic boundary-value problems, basic error estimates, applications to elasticity, practical aspects of the finite element method. Prereq: MATH 6653 (or equivalent programming experience), and MATH 6131/7132.
MATH 8664 (3 credit hrs)	Iterative Methods in Numerical Linear Algebra.	Every other year. Preconditioned iterative methods for linear systems and eigen problems, conjugate gradients, multigrid and domain decomposition. Prereq: MATH 5660 and 7665.
MATH 8990 (1 to 10 credit hrs)	Doctoral Dissertation.	Only for students working on their PhD research.

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