# University of Vermont

## 2012-2013 Catalogue

### Courses in Mathematics (MATH)

- MATH 001 - Elementary College Algebra
- Review of fundamental operations and a more extensive study of fractions, exponents, radicals, linear and quadratic equations, ratio, proportion, variation, progressions, and the binomial theorem. Topics normally included in intermediate algebra in high school. Students who have satisfactorily completed two years of high school algebra, or the equivalent, receive no credit for this course. Offered only in Evening Division and Summer Session. Prerequisite: One year of high school algebra.
- Credits: 3.00
- MATH 009 - College Algebra
- Sets, relations, functions with particular attention to properties of algebraic, exponential, logarithmic functions, their graphs and applications in preparation for MATH 019. May not be taken for credit concurrently with, or following receipt of, credit for any mathematics course numbered MATH 019 or above. Pre/co-requisites: Two years of secondary school algebra; one year of secondary school geometry.
- Credits: 3.00
- MATH 010 - Pre-Calculus Mathematics
- Skills in working with numerical, algebraic, and trigonometric expressions are developed in preparation for MATH 021. May not be taken for credit concurrently with, or following receipt of, credit for any mathematics course numbered MATH 019 or above. Prerequisite: Two years of secondary school algebra; one year of secondary school geometry.
- Credits: 3.00
- MATH 015 - Elementary School Math
- Operations with real numbers: decimals, fractions, percents, integers. Set operations, Venn diagrams, algebra, and problem solving provide background for future instruction in elementary/middle school mathematics. Prerequisite: Three years of secondary school math.
- Credits: 3.00
- MATH 016 - Fund Concepts Elem School Math
- Topics include geometry, measurement, probability, statistics, algebra, number theory, and problem solving to provide background for future instruction in elementary and middle school mathematics. Prerequisite: Three years of secondary school math.
- Credits: 3.00
- MATH 017 - Applications of Finite Math
- Introduction to mathematics of finite systems with applications, such as probability, statistics, graph theory, fair division and apportionment problems, voting systems. Prerequisites: Two years of secondary school algebra or MATH 009 or MATH 010.
- Credits: 3.00
- MATH 018 - Basic Mathematics
- Data, statistics, modeling, algebra, word problems, calculus. Students who do well in the algebra section may continue with MATH 019 or MATH 021. Prerequisite: three years of high school math. No credit for CEMS students.
- Credits: 3.00
- MATH 019 - Fundamentals of Calculus I
- Introduction to limits and differential calculus with a wide variety of applications. Students interested in intensive use of mathematics should take MATH 021. Credit not given for more than one of the courses MATH 019, MATH 021 unless followed by MATH 022. See MATH 023. Prerequisite: MATH 009 or MATH 010, or sufficiently strong background in secondary school algebra and geometry.
- Credits: 3.00
- MATH 020 - Fundamentals of Calculus II
- Introduction to integral calculus with a wide variety of applications. A student who completes MATH 020 may be admitted to MATH 022; however, MATH 019, MATH 023 is preferable to MATH 019, MATH 021, MATH 022 or MATH 019, MATH 020, MATH 022. Prerequisite: MATH 019.
- Credits: 3.00
- MATH 021 - Calculus I
- Introduction to calculus of functions of one variable including: limits, continuity, techniques and applications of differentiation and integration. Prerequisites: MATH 010, or strong background in secondary school algebra and trigonometry. Credit not given for more than one course in the pair MATH 019, MATH 021 unless followed by MATH 022.
- Credits: 4.00
- MATH 022 - Calculus II
- Techniques and applications of integration. Polar coordinates, Taylor polynomials, sequences and series, power series. Prerequisite: MATH 021. Credit will not be given for both MATH 022 and MATH 023.
- Credits: 4.00
- MATH 023 - Transitional Calculus
- Intended to make the transition from a B or better in MATH 019 to MATH 121. Topics are similar to MATH 022 but recognizing different backgrounds of students in MATH 019 versus MATH 021. Prerequisite: B or better in MATH 019. Credit will not be given for both MATH 022 and MATH 023.
- Credits: 5.00
- MATH 052 - Fundamentals of Mathematics
- Emphasizing proofs, fundamental mathematical concepts and techniques are investigated within the context of number theory and other topics. Co-requisite: MATH 021. Credit not given for both MATH 052 and MATH 054.
- Credits: 3.00
- MATH 054 - Fund of Math of Computation
- Introduction to mathematical theory and techniques underlying computer science. Co-requisite: MATH 019 or MATH 021.
- Credits: 3.00
- MATH 095 - Special Topics
- Introductory courses or seminars on topics beyond the scope of existing departmental offerings. See Schedule of Courses for specific titles. Prerequisite: Instructor permission.
- Credits: 5.00
- MATH 121 - Calculus III
- Vectors, vector-valued functions. Calculus of functions of several variables: partial derivatives, gradient, divergence, curl, multiple integrals, line integrals, Stokes' and Green's theorems. Prerequisite: MATH 022.
- Credits: 4.00
- MATH 124 - Linear Algebra
- Matrices, linear dependence, vector spaces, linear transformations, characteristic equations and applications. Prerequisite: MATH 022 or Instructor permission. Co-requisite: MATH 121 recommended but not required. .
- Credits: 3.00
- MATH 141 - Real Analysis in One Variable
- Principles of analysis in one variable. Heine-Borel and Bolzano-Weierstrass theorems; rigorous development of differential and integral calculus; infinite sequences and series of functions. May not be taken concurrently with or after MATH 241. Pre/co-requisite: MATH 052.
- Credits: 3.00
- MATH 151 - Groups and Rings
- An introduction to the basic concepts of abstract algebra emphasizing examples, including modular arithmetic, symmetric groups, cyclic groups, polynomial rings, homomorphisms, and isomorphisms. May not be taken concurrently with or after MATH 251. Prerequisite: MATH 052.
- Credits: 3.00
- MATH 161 - Development of Mathematics
- Historical development of mathematical sciences emphasizing interrelations among them. Individual assignments correspond to background and interests of students. Prerequisite: Nine hours of college mathematics.
- Credits: 3.00
- MATH 167 - Physical Chemistry Preparation
- Review of relevant mathematical and physical concepts as applied to physical chemistry. Credit cannot be obtained for both MATH 167 and MATH 121. Not available for credit for E&M students. Prerequisite: MATH 022; CHEM 032 or CHEM 036. Cross-listed with: CHEM 167.
- Credits: 1.00
- MATH 168 - Mathematics of Biology
- Discrete biological processes: nonlinear difference equations. Continuous processes: ordinary differential equations, phase plane methods, quantitative solutions. Applications: population dynamics, epidemiology, Michaelis-Menten kinetics, autocatalysis, muscle contraction. Includes a lab. May not be taken concurrently with or after MATH 268. Pre/co-requisites: MATH 022 or MATH 023, MATH 124. Cross-listed with: BIOL 168.
- Credits: 3.00
- MATH 173 - Basic Combinatorial Theory
- Introduction to basic combinatorial principles emphasizing problem-solving techniques. Enumeration, generating functions, Fibonacci numbers, pigeonhole principle, inclusion-exclusion, and graph theory. Prerequisites: MATH 052 or MATH 054 or CS 064.
- Credits: 3.00
- MATH 191 - Special Topics
- An approved project under guidance of a staff member and culminating in a written report. Involvement with off-campus groups permitted. Prerequisite: Junior/ Senior standing; approval of Department Chair.
- Credits: 3.00
- MATH 193 - College Honors
- Credits: 3.00
- MATH 195 - Special Topics
- See Schedule of Courses for specific titles.
- Credits: 4.00
- 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 243 - Theory of Computation
- Introduction to theoretical foundations of computer science. Models of computation. Church's thesis and noncomputable problems. Formal languages and automata. Syntax and semantics. Prerequisite: CS 104 or CS 124. Cross-listed with: CS 243.
- 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 283 - Junior-Senior Seminar
- Students required to give presentations on selected topics. Prerequisite: Instructor permission.
- Credits: 1.00
- MATH 293 - Undergraduate Honors Thesis
- Program of reading and research culminating in written thesis and oral presentation. Honors notation appears on transcript and Commencement Program. Contact department chairperson for procedures.
- Credits: 8.00
- MATH 294 - Undergraduate Honors Thesis
- Program of reading and research culminating in written thesis and oral presentation. Honors notation appears on transcript and Commencement Program. Contact department chairperson for procedures.
- 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: 4.00
- 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 300/CSYS 300, 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 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: 5.00
- MATH 395 - Special Topics
- Subject will vary from year to year. May be repeated for credit.
- Credits: 6.00
- MATH 491 - Doctoral Dissertation Research
- Credits: 12.00