The topics covered will vary from one offering to the next. Topics will be chosen to meet the needs of secondary mathematics teachers for additional training to teach to the Colorado Model Content Standards. Prer., One semester of calculus, or instructor approval. Meets with MATH 4050.

- 0.5 Credits (Minimum); 3 Credits (Maximum)

Methodology for using technology as a teaching/learning tool for high school and college math courses. Use of graphing calculators, computer algebra systems, computer geometry systems and the internet will be emphasized. Students are required to develop and present a portfolio of in-depth projects. Prer., MATH 1360. Meets with MATH 4100.

- 3 Credits

A follow-up to MATH 4100/5100. Students will present demonstrations, projects and/or laboratories they have developed for use in their math courses. Extended in-depth coverage of computer algebra or geometry systems and/or graphing calculators and internet. Basic familiarity with computer algebra or geometry systems and/or graphing calculators is required. Prer., MATH 5100 or consent of instructor.

- 1 Credits (Minimum); 3 Credits (Maximum)

Proof-based treatment of vector spaces, linear transformations and matrices, determinants, eigenvalues, similarity transformations, orthogonal and unitary transformations, normal matrices and quadratic forms. Prer., MATH 3130. Meets with MATH 4130.

- 3 Credits

Continuation of MATH 4140. The relationship between groups and fields is explored via a thorough investigation of Galois theory. Prer., MATH 4140. Meets with MATH 4150.

- 3 Credits

Fundamentals of ring and module theory, including simple and semisimple rings and modules, projective and injective modules, chain conditions on ideals, Jacobson radical, von Neumann regular rings, group rings. Meets with MATH 6170. Prer., MATH 4140.

- 3 Credits

Presents topics in geometry suitable for advanced undergraduates, such as differential geometry of curves and surfaces, geometry of manifolds, or more in-depth exploration of non-Euclidean geometries. Prer., MATH 3130, MATH 3410. Meets with MATH 4210.

- 3 Credits

Introduction to iterated function systems and mathematical aspects of fractal sets. Includes metric spaces and the space fractals live in, transformations, contraction mapping and collage theorem, chaotic dynamics, shadowing theorem, fractal dimension, fractal interpolation, and measures on fractals. Prer., MATH 2350 and MATH 3130. Meets with MATH 4230.

- 3 Credits

Introduction to dynamical systems or processes in motion, defined in discrete time by iteration of simple functions, or in continuous time by differential equations. Emphasis on chaotic behavior of an iterated simple nonlinear function. Orbits, graphical analysis, fixed and periodic points, bifurcations, symbolic dynamics, chaos, fractals, and Julia sets. Prer., MATH 2350. Meets with MATH 4250.

- 3 Credits

The basic ideas, examples, and applications of the theory of error-correcting codes are presented, including linear codes and cyclic codes. These codes are important for the digital transmission of data. Finite fields, polynomial rings, and ideals play central roles. Prer., MATH 4140.

- 3 Credits

Careful theoretical study of topology of Euclidean space, metric spaces, sequences and series of functions, calculus of several variables. Prer., MATH 4310. Meets with MATH 4320.

- 3 Credits

Lebesgue measure, measurable and nonmeasurable sets, sigma algebras. Lebesgue integral, comparison with Riemann integration, Monotone and Dominated Convergence Theorems, Fatou's Lemma. Differentiation, functions of bounded variation, absolute continuity integration in product spaces, Fubini's Theorem. Prer., MATH 4320/5320. Meets with MATH 6330. Graduate students only.

- 3 Credits

Basic concepts, methods, and applications of functional analysis. Complete metric spaces, contraction mapping, and applications. Banach spaces and linear operators. Inner product and Hilbert spaces, orthonormal bases and expansions, approximation, and applications. Spectral theory of compact operators, including self adjoint and normal operators. Prer., MATH 4320 or MATH 5320. Meets with MATH 6350. Graduate students only.

- 3 Credits

Topics selected from linear and nonlinear programming, the simplex algorithm and other approaches to linear optimization, minimax theorems, convex functions, introduction to calculus of variations. Prer., MATH 3130 and MATH 3400. Meets with MATH 4420.

- 3 Credit

Existence and uniqueness theorem, linear systems, linear systems with periodic coefficients (Floquet theory), stability analysis of planar systems, series solutions at regular singular points, Sturm-Liouville problems. Prer., MATH 3130, MATH 3400. Meets with MATH 4430.

- 3 Credits

Approximate solutions of differential equations by asymptotic expansions, asymptotic expansion of integrals, regular and singular perturbation methods, boundary layer analysis, WKB methods, and multiple-scale techniques. Prer., MATH 5430/6430 and MATH 5610/6610. Graduate students only. Meets with MATH 6440.

- 3 Credits

Theory of functions of one complex variable, including integrals, powering series, residues, conformal mapping and special functions. Prer., MATH 2350. Meets with MATH 4450.

- 3 Credits

Boundary value problems for the wave, heat, and Laplace equations, separation of variables methods, eigenvalue problems, Fourier series, orthogonal systems. Prer., MATH 2350, MATH 3130 and MATH 3400. Meets with MATH 4470.

- 3 Credits

The use of diverse mathematical techniques to analyze and solve problems from science and engineering, particularly problems likely to arise in a nonacademic setting such as industry or government. Converting a problem to a mathematical model. Commonly encountered classes of mathematical models, including optimization problems, dynamical systems, probability models, and computer simulations. Communication of results of mathematical analysis. Prer., MATH 3130, MATH 3400, and MATH 3100 or MATH 3810 or ECE 3610. MATH 2650 or adequate experience in computer programming. Meets with MATH 4480.

- 3 Credits

An introduction to Point-set topology and elements of geometric or algebraic topology. Prer., MATH 4310. Meets with MATH 4510

- 3 Credits

Perturbation methods including Lagrange and Hamiltonian mechanics and the generalized method of averaging. Gravitational and atmosphere modeling. Prer., MAE 4410/5410 or PHYS 5510.

- 3 Credits

Complex numbers, Cauchy-Reimann equations, harmonic functions. Elementary functions and conformal mapping. Contour integrals, Cauchy integral representation. Uniform convergence and power series. Residues. Prer., MATH 4310/5310. Graduate students only. Meets with MATH 6610.

- 3 Credits

Argument principle, Rouche's Theorem. Homotopy and countour integrals. Compact sets of functions and uniform convergence. Conformal mappings and the Riemann Mapping Theorem. Infinite products, analytic continuation, special topics. Prer., MATH 5610/6610. Graduate students only. Meets with MATH 6620.

- 3 Credits

Error analysis, root finding, numerical integration and differentiation, numerical methods for ordinary differential equations, numerical linear algebra and eigenvalue problems. Prer., MATH 3130, MATH 3400, and CS 1120 or CS 1150 or MATH 3670. Meets with MATH 4650.

- 3 Credits

Numerical solutions of initial-value problems, two-point boundary-value problems for ordinary differential equations, and applications. Numerical methods for solving linear partial differential equations, including finite difference and finite element method. Laplace, heat, and wave equation. Prer., MATH 3670 or MATH 4650/5650, and MATH 4470/5470. Meets with MATH 4670.

- 3 Credits

Advanced numerical methods for solving linear and nonlinear partial differential equations, including spectral and pseudo-spectral methods. Iterative methods for solving large linear systems. Prer., MATH 4670 or MATH 5670. Graduate students only. Meets with MATH 6680.

- 3 Credits

Exponential, Beta, Gamma, Student, Fisher and Chi-square distributions are covered in this course, along with joint and conditional distributions, moment generating techniques, transformations of random variables and vectors. Prer., MATH 2350 and MATH 3130. Meets with MATH 4810.

- 3 Credits

Point and confidence interval estimation, principles of maximum likelihood, sufficiency and completeness; tests of simple and composite hypotheses. Linear models, and multiple regression analysis. Other topics will be included. Prer., MATH 3100 or MATH 3810. Meets with MATH 4820.

- 3 Credits

Methods and results of linear algebra are developed to formulate and study a fundamental and widely applied area of statistics. Topics include generalized inverses, multivariate normal distribution and the general linear model. Applications focus on model building, design models and computing methods. The "Statistical Analysis System" (software) is introduced as a tool for doing computations. Prer., MATH 3810 or ECE 3610, or MATH 3100 and MATH 3130. Meets with MATH 4830.

- 3 Credits

Representation and manipulation of digital images; Fourier analysis of images; enhancement techniques in spatial and frequency domain; segmentation procedures; digital geometry, region and boundary representation; texture processing; pattern recognition and application to robotics. Prer., Graduate standing in mathematics, engineering or computer science. Meets with C S 5840.

- 3 Credits

Mathematical development of continuous and discrete time Markov chains, queuing theory, reliability theory and Brownian motion with applications to engineering and computer science. Prer., MATH 3810 or ECE 3610. MATH 2650 or adequate experience with computer programming. Meets with MATH 4850.

- 3 Credits

Various topics in mathematics at the graduate level. Prer., Consent of instructor. Meets with MATH 4900.

- 1 Credits (Minimum); 3 Credits (Maximum)

Measure theory is given form within a large body of probabilistic examples, ideas, and applications. Weak and strong laws of large numbers, central limit theory, and random walk in the context of independent random variables. Prer., MATH 4310. Graduate students only. Meets with MATH 6910.

- 3 Credits

Probability theory for sequences of dependent random variables, with the major focus on martingale theory and its applications. Prer., MATH 5910/6910. Graduate students only. Meets with MATH 6920.

- 3 Credits

Fundamentals of ring and module theory, including simple and semisimple rings and modules, projective and injective modules, chain conditions on ideals, Jacobson radical, von Neumann regular rings, group rings. Meets with MATH 5170. Prer., MATH 4140.

- 3 Credits

Further topics in ring and module theory, including division rings, perfect and semiperfect rings. Prer., MATH 5170 or MATH 6170.

- 3 Credits

Designed to help secondary school mathematics teachers present and facilitate mathematical problem-solving activities with their students. Mathematical approaches to problem-oriented questions will be discussed, as well as teaching methodologies regarding optimal use of such classroom activities.

- 2 Credits

Designed to help secondary math teachers follow up on the educational and pedagogical benefits and issues of using a problem-solving approach to teaching mathematics in their classrooms. Prer., MATH 6210.

- 1 Credits

The basic ideas, examples, and applications of the theory of error-correcting codes are presented, including linear codes and cyclic codes. These codes are important for the digital transmission of data. Finite fields, polynomial rings, and ideals play central roles. Meets with MATH 5270. Prer., MATH 4140.

- 3 Credits

Designed to provide K-12 teachers with various methods and concepts from mathematics and economics which can be incorporated into K-12 mathematics or economics curricula. Not an option for MATH majors or graduate students. Meets with ECON 6310.

- 0.5 Credits (Minimum); 3 Credits (Maximum)

Lebesgue measure, measurable and nonmeasurable sets, sigma algebras. Lebesgue integral, comparison with Riemann integration, Monotone and Dominated Convergence Theorems, Fatou's Lemma. Differentiation, functions of bounded variation, absolute continuity integration in product spaces, Fubini's Theorem. Prer., MATH 4320/5320. Meets with MATH 5330. Graduate students only.

- 3 Credits

Basic concepts, methods, and applications of functional analysis. Complete metric spaces, contraction mapping, and applications. Banach spaces and linear operators. Inner product and Hilbert spaces, orthonormal bases and expansions, approximation, and applications. Spectral theory of compact operators, including self adjoint and normal operators. Prer., MATH 4320 or MATH 5320. Meets with MATH 5350. Graduate students only.

- 3 Credits

Approximate solutions of differential equations by asymptotic expansions, asymptotic expansion of integrals, regular and singular perturbation methods, boundary layer analysis, WKB methods, and multiple-scale techniques. Prer., MATH 5430/6430 and MATH 5610/6610. Graduate students only. Meets with MATH 5440.

- 3 Credits

Complex numbers, Cauchy-Reimann equations, harmonic functions. Elementary functions and conformal mapping. Contour integrals, Cauchy integral representation. Uniform convergence and power series. Residues. Prer., MATH 4310/5310. Graduate students only. Meets with MATH 5610.

- 3 Credits

Argument principle, Rouche's Theorem. Homotopy and countour integrals. Compact sets of functions and uniform convergence. Conformal mappings and the Riemann Mapping Theorem. Infinite products, analytic continuation, special topics. Prer., MATH 5610/6610. Graduate students only. Meets with MATH 5620.

- 3 Credits

Advanced numerical methods for solving linear and nonlinear partial differential equations, including spectral and pseudo-spectral methods. Iterative methods for solving large linear systems. Prer., MATH 4670 or MATH 5670. Graduate students only. Meets with MATH 6680.

- 3 Credits

Measure theory is given form within a large body of probabilistic examples, ideas, and applications. Weak and strong laws of large numbers, central limit theory, and random walk in the context of independent random variables. Prer., MATH 4310. Graduate students only. Meets with MATH 5910.

- 3 Credits

Probability theory for sequences of dependent random variables, with the major focus on martingale theory and its applications. Prer., MATH 5910/6910. Graduate students only. Meets with MATH 5920.

- 3 Credits

Masters Thesis

- 1 Credits (Minimum); 6 Credits (Maximum)

Enrollment is limited to those students who are in the PhD program in Engineering, and have primary thesis advisor in the Department of Mathematics. Prer., Consent of instructor.

- 1 Credits (Minimum); 10 Credits (Maximum)

A review of basic algebra and arithmetic, including algebra of polynomials, factorization of simple polynomials, arithmetic operations on fractions and rational expressions, laws of exponents, linear equations and inequalities in one variable, quadratic equations using factoring. Administered through the Department of Mathematics. Pass/fail grading only. Does not count toward BA or BS degree.

- 2 Credits

Independent Study Math, Graduate.

- 1 Credits (Minimum); 3 Credits (Maximum)

Candidate for Degree.

- 0 Credits

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