MATH 104: Linear algebra and applications
Description: Systems of linear equations and elementary row operations,
Euclidean n-space and subspaces, linear transformations
and matrix representations, Gram-Schmidt orthogonalization
process, determinants, eigenvectors and eigenvalues;
applications.
Prerequisites: Math 32, 32L, or 41
MATH 104C: Linear algebra/scientific computing
Description:Introductory linear algebra developed from the
perspective of computational algorithms. Similar to Mathematics 104, but
emphasizes matrix factorizations and includes the programming of basic
algorithms and the use of software packages.
Prerequisites: Math 32, 32L, or 41.
MATH 111: APPLIED MATH ANALYSIS I
Description: First and second order differential equations with
applications; matrices, eigenvalues, and eigenvectors; linear systems of
differential equations; Fourier series and applications to partial differential
equations. Intended primarily for engineering and science
students with emphasis on problem solving. Students taking Mathematics 104,
especially mathematics majors, are urged to take Mathematics 131 instead.
Mathematics 111 is not open to students who have had Mathematics 131.
Prerequisites: Math 103.
MATH 114: APPLIED MATH ANALYSIS II
Description: Boundary value problems, complex variables, Cauchy's
theorem, residues, Fourier transform, applications to partial differential
equations. Not open to students who have had Mathematics 133, 181, or 211.
Prerequisites: Math 111, 131.
MATH 131: Elementary differential equations
Description: First and second order differential equations with
applications; linear systems of differential equations;
Fourier series and applications to partial differential
equations. Additional topics may include stability, nonlinear
systems, bifurcations, or numerical methods. Not open to
students who have had Mathematics 111.
Prerequisites: Math 103; corequisite: Math 104.
MATH 135: Probability
Description: Probability models, random variables with discrete and
continuous distributions. Independence, joint distributions, conditional
distributions. Expectations, functions of random variables, central limit
theorem.
Prerequisites: Math 103.
MATH 139: ADVANCED CALCULUS I
Description: Algebraic and topological structure of the real number
system; rigorous development of one-variable calculus including continuous,
differentiable, and Riemann integrable functions and the Fundamental Theorem
of Calculus; uniform convergence of a sequence
of functions; contributions of Newton, Leibniz, Cauchy, Riemann, and
Weierstrass. Not open to students who have had Mathematics 203.
Prerequisites: Math 103.
MATH 160: Mathematical Numerical Analysis
Description: Development of numerical techniques for accurate,
efficient solution of problems in science, engineering, and mathematics
through the use of computers. Linear systems, nonlinear equations,
optimization, numerical integration, differential equations, simulation of
dynamical systems, error analysis. Not open to students who have had
Computer Science 150 or 250.
Prerequisites: Mathematics 103 and 104 and basic knowledge of a
programming
language (at the level of Computer Science 6), or consent of instructor.
MATH 181: COMPLEX ANALYSIS
Description: Complex numbers, analytic functions, complex
integration, Taylor and Laurent series, theory of residues, argument and
maximum principles, conformal mapping. Not open to students who have had
Mathematics 114 or 212.
Prerequisites: Math 103.
MATH 196S: Seminar in Mathematical Modeling
Description: Introduction to techniques used in the construction,
analysis, and evaluation of mathematical models. Individual modeling projects
in biology, chemistry, economics, engineering, medicine, or physics.
Prerequisites: Math 111 or 131.
MATH 203: Basic Analysis I
Description:Topology of R<^>n, continuous functions, uniform
convergence, compactness, infinite series, theory of differentiation, and
integration. Not open to students who have had Mathematics 139.
Prerequisites: Math 104.
MATH 204: Basic Analysis II
Description: Differential and integral calculus in R<^>n. Inverse
and implicit function theorems. Further topics in multivariable analysis.
Prerequisites: Math 104 and 203.
MATH 224: SCIENTIFIC COMPUTING I
Description: Well-posedness of ODEs; method, order, and stability.
Methods for hyperbolic, parabolic, and elliptic PDEs. Structured
programming and graphical user interfaces. Programming in C++, C, and Fortran.
Prerequisites: Math 103, plus some familiarity with
ODEs and PDEs.
MATH 225: SCIENTIFIC COMPUTING II
Description:
Prerequisites: Math 224.
MATH 226: NUMERICAL PARTIAL DIFFERENTIAL
EQUATIONS I
Description:Numerical solution of hyperbolic conservation laws.
Conservative difference schemes, modified equation analysis and Fourier
analysis, Lax-Wendroff process. Gas dynamics and Riemann problems. Upwind
schemes for hyperbolic systems. Nonlinear stability, monotonicity and entropy;
TVD, MUSCL, and ENO schemes for scalar laws. Approximate Riemann solvers and
schemes for hyperbolic systems. Multidimensional schemes. Adaptive mesh
refinement.
Prerequisites: Math 224, 225.
MATH 227: NUMERICAL PARTIAL DIFFERENTIAL
EQUATIONS II
Description:Numerical solution of parabolic and elliptic equations.
Diffusion equations and stiffness, finite difference methods and operator
splitting (ADI). Convection-diffusion equations. Finite element methods for
elliptic equations. Conforming elements, nodal basis functions, finite element
matrix assembly and numerical quadrature. Iterative linear algebra; conjugate
gradients, Gauss-Seidel, incomplete factorizations and multigrid. Mixed and
hybrid methods. Mortar elements. Reaction-diffusion problems, localized
phenomena, and adaptive mesh refinement.
Prerequisites: Math 224, 225.
MATH 231: ORDINARY DIFFERENTIAL EQUATIONS
Description:Existence and uniqueness theorems for nonlinear systems,
well-posedness, two-point boundary value problems, phase plane diagrams,
stability, dynamical systems, and strange attractors.
Prerequisites: Math 104, 111 or 131 and 139 or 203.
MATH 232: Partial differential equations I
Description:Fundamental solutions of linear partial differential
equations, hyperbolic equations, characteristics, Cauchy-Kowalevski theorem,
propagation of singularities.
Prerequisites: Math 204.
MATH 233: Asymptotic analysis/Perturbation
methods
Description:Asymptotic solution of linear and nonlinear ordinary
and partial differential equations. Asymptotic evaluation of integrals.
Singular perturbation. Boundary layer theory. Multiple scale analysis.
Prerequisites: Math 114.
MATH 238: Topics in Applied Mathematics
Description:
Prerequisites:
MATH 239: Mathematical Finance
Description:
Prerequisites:
MATH 241: Real Analysis I
Description:This will be a course in measure theory. We develop
simultaneously the theory of Lebesgue measure/integration on the reals and the
measure theory needed to do probability. Standard results about convergence and
inequalities involving integrals will be done. We will use Real Analysis by
H.L. Royden as a text. There will also be an introduction to Fourier analysis
including a proof of the central limit theorem using Fourier analysis; for
this purpose we will very likely use the notes of Greg Lawler on this subject.
Prerequisites: Math 204.
MATH 242: Real Analysis II
Description:Metric spaces, fixed point theorems, Baire category
theorem, Banach spaces, fundamental theorems of functional analysis,
Fourier transform.
Prerequisites: Math 241.
MATH 245: Complex analysis
Description:Complex calculus, conformal mapping, Riemann mapping
theorem, Riemann surfaces.
Prerequisites: Math 204.
MATH 278: Complex analysis
Description:Geometric function theory, function algebras,
several complex variables, uniformization, or analytic number theory.
Prerequisites: Math 245.
MATH 281: Partial Differential Equations II
Description:Linear wave motion, dispersion, stationary phase,
foundations of continuum mechanics, characteristics, linear hyperbolic
systems, and nonlinear conservation laws.
Prerequisites: Math 232.
MATH 282: Partial Differential Equations III
Description:Fourier transforms, distributions, elliptic equations,
singular integrals, layer potentials, Sobolev spaces, regularity of elliptic
boundary value problems.
Prerequisites: Math 232 and 241.