# University of Vermont

## 2012-2013 Catalogue

### Graduate Courses in Mathematics (MATH)

- MATH 207 - Probability Theory
- Distributions of random variables and functions of random variables. Expectations, stochastic independence, sampling and limiting distributions (central limit theorems). Concepts of random number generation. Prerequisites: MATH 121; STAT 151 or STAT 153 recommended. Cross-listed with: STAT 251, BIOS 251.
- Credits: 3.00
- MATH 221 - Deterministic Modls Oper Rsch
- The linear programming problem. Simplex algorithm, dual problem, sensitivity analysis, goal programming. Dynamic programming and network problems. Prerequisites: MATH 124; MATH 121 desirable. Cross-listed with: CSYS 221.
- Credits: 3.00
- MATH 222 - Stochastic Models in Oper Rsch
- Development and solution of some typical stochastic models. Markov chains, queueing problems, inventory models, and dynamic programming under uncertainty. Prerequisite: MATH 207, STAT 151, or Instructor permission.
- Credits: 3.00
- MATH 224 - Algorithm Design & Analysis
- Comprehensive analysis of common algorithmic paradigms including greedy algorithms, divide and conquer, dynamic programming, graph algorithms, and approximation algorithms. Complexity hierarchies. Prerequisites: CS 124, MATH 173 recommended. Cross-listed with: CS 224.
- Credits: 3.00
- MATH 230 - Ordinary Differential Equation
- Solutions of linear ordinary differential equations, the Laplace transformation, and series solutions of differential equations. Prerequisite: MATH 121. Corequisite: MATH 124 or Instructor permission. Credit not granted for more than one of the courses MATH 230 or MATH 271.
- Credits: 3.00
- MATH 235 - Mathematical Models & Analysis
- Techniques of Undergraduate calculus and linear algebra are applied for mathematical analysis of models of natural and human-created phenomena. Students are coached to give presentations. Prerequisites: MATH 121 and any of MATH 124, MATH 230, or MATH 271.
- Credits: 3.00
- MATH 236 - Calculus of Variations
- Necessary conditions of Euler, Legendre, Weierstrass, and Jacobi for minimizing integrals. Sufficiency proofs. Variation and eigenvalue problems. Hamilton-Jacobi equations. Prerequisite: MATH 230.
- Credits: 3.00
- MATH 237 - Intro to Numerical Analysis
- Error analysis, root-finding, interpolation, least squares, quadrature, linear equations, numerical solution of ordinary differential equations. Prerequisite: MATH 121, MATH 124 or MATH 271; Knowledge of computer programming.
- Credits: 3.00
- MATH 238 - Applied Computational Methods
- Direct and iterative methods for solving linear systems; numerical solution of ordinary and partial differential equations. Focus will be on application of numerical methods. Prerequisites: MATH 121; either MATH 124 or MATH 271.
- Credits: 3.00
- MATH 240 - Fourier Series&Integral Trans
- Fourier series, orthogonal functions, integral transforms and boundary value problems. Prerequisite: MATH 230 or MATH 271.
- Credits: 3.00
- MATH 241 - Anyl in Several Real Vars I
- Properties of the real numbers, basic topology of metric spaces, infinite sequences and series, continuity. Prerequisites: MATH 052, MATH 121, MATH 124 or Instructor permission.
- Credits: 3.00
- MATH 242 - Anyl Several Real Variables II
- Differentiation and integration in n-space, uniform convergence of functions, fundamental theorem of calculus, inverse and implicit function theorems. Prerequisite: MATH 241.
- Credits: 3.00
- MATH 251 - Abstract Algebra I
- Basic theory of groups, rings, fields, homomorphisms, and isomorphisms. Prerequisite: MATH 052, MATH 124, or Instructor permission.
- Credits: 3.00
- MATH 252 - Abstract Algebra II
- Modules, vector spaces, linear transformations, rational and Jordan canonical forms. Finite fields, field extensions, and Galois theory leading to the insolvability of quintic equations. Prerequisite: MATH 251.
- Credits: 3.00
- MATH 255 - Elementary Number Theory
- Divisibility, prime numbers, Diophantine equations, congruence of numbers, and methods of solving congruences. Prerequisite: MATH 052 or MATH 054.
- Credits: 3.00
- MATH 257 - Topics in Group Theory
- Topics may include abstract group theory, representation theory, classical groups, Lie groups. Prerequisite: MATH 251.
- Credits: 3.00
- MATH 260 - Foundations of Geometry
- Geometry as an axiomatic science; various non-Euclidean geometries; relationships existing between Euclidean plane geometry and other geometries; invariant properties. Prerequisite: MATH 052 or MATH 054.
- Credits: 3.00
- MATH 264 - Vector Analysis
- Gradient, curl and divergence, Green, Gauss, and Stokes Theorems, applications to physics, tensor analysis. Prerequisite: MATH 121, MATH 124, or MATH 271.
- Credits: 3.00
- MATH 266 - Chaos,Fractals&Dynamical Syst
- Discrete and continuous dynamical systems, Julia sets, the Mandelbrot set, period doubling, renormalization, Henon map, phase plane analysis and Lorenz equations. Co-requisite: MATH 271 or MATH 230 or Instructor permission. Cross-listed with: CSYS 266.
- Credits: 3.00
- MATH 268 - Mathematical Biology&Ecology
- Mathematical modeling in the life sciences. Topics include population modeling, dynamics of infectious diseases, reaction kinetics, wave phenomena in biology, and biological pattern formation. Prerequisite: MATH 124, MATH 230, or Instructor permission. Cross-listed with: CSYS 268.
- Credits: 3.00
- MATH 271 - Adv Engineering Mathematics
- Differential equations and linear algebra, including linear ordinary differential equations, Laplace transforms, matrix theory, and systems of differential equations. Examples from engineering and physical sciences. Prerequisite: MATH 121. Credit not granted for both MATH 230 and MATH 271. No credit for Mathematics majors.
- Credits: 3.00
- MATH 272 - Applied Analysis
- Basics of Fourier series, partial differential equations of mathematical physics, functions of a complex variable, Cauchy's theorem, integral formula. Prerequisites: MATH 230 or MATH 271.
- Credits: 3.00
- MATH 273 - Combinatorial Graph Theory
- Paths and trees, connectivity, Eulerian and Hamiltonian cycles, matchings, edge and vertex colorings, planar graphs, Euler's formula and the Four Color Theorem, networks. Prerequisite: MATH 052 or MATH 054, or Instructor permission.
- Credits: 3.00
- MATH 274 - Numerical Linear Algebra
- Direct and iterative methods for solving linear equations, least square factorization methods, eigenvalue computations, ill-conditioning and stability. Prerequisite: MATH 237.
- Credits: 3.00
- MATH 275 - Adv Engineering Analysis I
- Analytical methods for the solution of partial differential equations in engineering mechanics and physics, including: eigenfunction expansions; Fourier series; Sturm-Liouville theory and special functions. Prerequisites: Graduate standing in Engineering, Mathematics, or physical sciences or permission. Not available for 300-level credit for Mathematics students. Cross-listed with: CE 304 and ME 304.
- Credits: 3.00
- MATH 276 - Adv Engineering Analysis II
- Advanced analytical techniques for problems in engineering mechanics and physics, including: integral transform methods, Green's functions, perturbation methods, and variational calculus. Prerequisites: ME 304 or equivalent. Not available for 300-level credit for Mathematics students. Cross-listed with: CE 305, ME 305.
- Credits: 3.00
- MATH 278 - Intro Wavelets & Filter Banks
- Continuous and discrete-time signal processing. Continuous wavelet transform. Series expansion of continuous and discrete-time signals. Perfect reconstruction, orthogonal and biorthogonal filter banks. Wavelets from filter. Pre/co-requisites: EE 171, or Instructor permission. Cross-listed with: EE 274.
- Credits: 3.00
- MATH 295 - Special Topics
- For advanced students in the indicated fields. Lectures, reports, and directed readings on advanced topics. Prerequisite: Instructor permission. Credit as arranged. Offered as occasion warrants.
- Credits: 1.00 to 18.00
- MATH 300 - Principles of Complex Systems
- Introduction to fundamental concepts of complex systems. Topics include: emergence, scaling phenomena, and mechanisms, multi-scale systems, failure, robustness, collective social phenomena, complex networks. Students from all disciplines welcomed. Pre/co-requisites: Calculus and statistics required; Linear Algebra, Differential Equations, and Computer programming recommended but not required. Cross-listed with: CSYS 300.
- Credits: 3.00
- MATH 303 - Complex Networks
- Detailed exploration of distribution, transportation, small-world, scale-free, social, biological, organizational networks; generative mechanisms; measurement and statistics of network properties; network dynamics; contagion processes. Students from all disciplines welcomed. Pre/co-requisites: MATH 301/CSYS 301, Calculus, and Statistics required. Cross-listed with: CSYS 303.
- Credits: 3.00
- MATH 330 - Adv Ordinary Diff Equations
- Linear and nonlinear systems, approximate solutions, existence, uniqueness, dependence on initial conditions, stability, asymptotic behavior, singularities, self-adjoint problems. Prerequisite: MATH 230.
- Credits: 3.00
- MATH 331 - Theory of Func of Complex Var
- Differentiation, integration, Cauchy-Riemann equations, infinite series, properties of analytic continuation, Laurent series, calculus of residues, contour integration, meromorphic functions, conformal mappings, Riemann surfaces. Prerequisite: MATH 242.
- Credits: 4.00
- MATH 332 - Approximation Theory
- Interpolation and approximation by interpolation, uniform approximation in normed linear spaces, spline functions, orthogonal polynomials. Least square, and Chebychev approximations, rational functions. Prerequisite: MATH 124, MATH 237.
- Credits: 3.00
- MATH 333 - Thry Functions Real Variables
- The theory of Lebesgue integration, Lebesgue measure, sequences of functions, absolute continuity, properties of LP-spaces. Prerequisite: MATH 242.
- Credits: 4.00
- MATH 335 - Advanced Real Analysis
- L2-spaces, LP-spaces; Hilbert, Banach spaces; linear functionals, linear operators; completely continuous operators (including symmetric); Fredholm alternative; Hilbert-Schmidt theory; unitary operators; Bochner's Theorem; Fourier-Plancherel, Watson transforms. Prerequisites: MATH 333.
- Credits: 3.00
- MATH 336 - Advanced Real Analysis
- L2-spaces, LP-spaces; Hilbert, Banach spaces; linear functionals, linear operators; completely continuous operators (including symmetric); Fredholm alternative; Hilbert-Schmidt theory; unitary operators; Bochner's Theorem; Fourier-Plancherel, Watson transforms. Prerequisite: MATH 333 and MATH 335.
- Credits: 3.00
- MATH 337 - Numerical Diff Equations
- Numerical solution and analysis of differential equations: initial-value and boundary-value problems; finite difference and finite element methods. Prerequisites: MATH 237; either MATH 230 or MATH 271 recommended.
- Credits: 3.00
- MATH 339 - Partial Differential Equations
- Classification of equations, linear equations, first order equations, second order elliptic, parabolic, and hyperbolic equations, uniqueness and existence of solutions. Prerequisite: MATH 230; MATH 242.
- Credits: 3.00
- MATH 351 - Topics in Algebra
- Topics will vary each semester and may include algebraic number theory, algebraic geometry, and the arithmetic of elliptic curves. Repeatable for credit with Instructor permission. Prerequisite: MATH 252.
- Credits: 3.00
- MATH 353 - Point-Set Topology
- Topological spaces, closed and open sets, closure operators, separation axioms, continuity, connectedness, compactness, metrization, uniform spaces. Prerequisite: MATH 241.
- Credits: 3.00
- MATH 354 - Algebraic Topology
- Homotopy, Seifert-van Kampen Theorem; simplicial, singular, and Cech homology. Prerequisite: MATH 353.
- Credits: 3.00
- MATH 373 - Topics in Combinatorics
- Topics will vary each semester and may include combinatorial designs, coding theory, topological graph theory, cryptography. Prerequisite: MATH 251 or MATH 273; or Instructor permission.
- Credits: 3.00
- MATH 382 - Seminar
- Topical discussions with assigned reading. Required of M.S. degree candidates.
- Credits: 1.00
- MATH 391 - Master's Thesis Research
- Credits: 1.00 to 18.00
- MATH 395 - Special Topics
- Subject will vary from year to year. May be repeated for credit.
- Credits: 1.00 to 6.00
- MATH 491 - Doctoral Dissertation Research
- Credits: 1.00 to 18.00