**MAT 410 Introduction to General Topology. **(3) A

Topological spaces, metric spaces, compactness, connectedness, and product spaces. Prerequisite: MAT 300 or 371 or instructor approval.

**MAT 415 Combinatorial Mathematics I. **(3) F

Permutations and combinations, recurrence relations, generating functions, graph theory, and combinatorial proof techniques. Prerequisites: MAT 300 and 342 *or* instructor approval.

**MAT 416 Combinatorial Mathematics II. **(3) S

Continuation of MAT 415 considering some advanced aspects of the theory as well as applications. Topics chosen from transport networks, matching theory, block designs, coding theory, Polya's counting theory, and applications to the physical and life sciences. MAT 443 is recommended. Prerequisite: MAT 415 or instructor approval.

**MAT 419 Linear Programming. **(3) S

Linear programming and the simplex algorithm, network problems, quadratic, and nonlinear programming. Prerequisites: MAT 242 or 342; 1 semester of college calculus. *General Studies: N2.*

**MAT 431 Foundations of Mathematics. **(3) N

Topics from mathematical logic and set theory. May be repeated for credit with instructor approval. Prerequisites: MAT 300 and 342 *or* instructor approval.

**MAT 442 Advanced Linear Algebra. **(3) F

Fundamentals of linear algebra, dual spaces, invariant subspaces, canonical forms, bilinear and quadratic forms, and multilinear algebra. Prerequisites: MAT 300 and 342 *or* instructor approval.

**MAT 443 Introduction to Abstract Algebra. **(3) F

Introduction to concepts of abstract algebra. Not open to students with credit in MAT 444. Prerequisites: MAT 300 and 342 *or* instructor approval.

**MAT 444 Intermediate Abstract Algebra. **(3) S

Basic theory of groups, rings, and fields, including an introduction to Galois theory. Appropriate as preparation for MAT 543. Prerequisites: MAT 300 and 342.

**MAT 445 Theory of Numbers. **(3) S

Prime numbers, unique factorization theorem, congruences, Diophantine equations, primitive roots, and quadratic reciprocity theorem. Prerequisites: MAT 300 and 342 *or* instructor approval.

**MAT 451 Mathematical Modeling. **(3) S

A detailed study of 1 or more mathematical models that occur in the physical or biological sciences. May be repeated for credit with instructor approval. Prerequisites: MAT 242 (or 342) and 274 *or* instructor approval. *General Studies: N2.*

**MAT 460 Applied Real Analysis. **(3) S

Vectors, curvilinear coordinates, Jacobians, implicit function theorem, line and surface integrals, Green's, Stokes', and divergence theorems. Not open to students with credit in MAT 372. Prerequisites: MAT 242 (or 342), 272, 274.

**MAT 461 Applied Complex Analysis. **(3) F, SS

Analytic functions, complex integration, Taylor and Laurent series, residue theorem, conformal mapping, and harmonic functions. Prerequisite: MAT 272 or equivalent.

**MAT 462 Partial Differential Equations. **(3) F, S, SS

Second order partial differential equations, emphasizing Laplace, wave, and diffusion equations. Solutions by the methods of characteristics, separation of variables, and integral transforms. Prerequisites: MAT 242 (or 342), 274.

**MAT 464 Numerical Analysis I. **(3) F, S

Theory and methods for numerical solution of algebraic and transcendental equations; iteration methods; eigenvalues and eigenvectors; interpolation; introductory computer arithmetic. Prerequisites: MAT 342 and 371 and fluency in computer programming *or* instructor approval. *General Studies: N3.*

**MAT 465 Numerical Analysis II. **(3) F, S

Theory and methods for numerical solution of analysis problems; differentiation; quadrature; solution of differential equations. Prerequisite: MAT 342 and 371 and fluency in computer programming *or* instructor approval. *General Studies: N3.*

**MAT 466 Applied Computational Methods. **(3) F, S

Numerical methods for quadrature, differential equations, roots of nonlinear equations, interpolation, approximation, linear equations, floating-point arithmetic, and roundoff error. Prerequisites: MAT 271 (or equivalent) and fluency in computer programming (preferably FORTRAN) *or* instructor approval. *General Studies: N3.*

**MAT 467 Computer Arithmetic. **(3) S

Number systems, hardware/software arithmetic, overflow, significance, rounding, multiple precision, and automatic error control; impact on languages, architectures, robust programming, and software development. Prerequisite: CSE 100 or 200 or MAT 464, 465, or 466 or instructor approval. *General Studies: N3.*

**MAT 472 Intermediate Real Analysis. **(3) F

Introduction to analysis in metric spaces with emphasis on the real line. Appropriate as preparation for MAT 570. Prerequisites: MAT 300, 342.

**MAT 475 Differential Equations. **(3) S

Asymptotic behavior of solutions of linear and nonlinear ordinary differential equations, stability, Sturm-Liouville problems, boundary value problems, and singular point behavior of autonomous systems. Prerequisites: MAT 242 (or 342), 274.

**MAT 485 History of Mathematics. **(3) N

Topics from the history of the origin and development of mathematical ideas. Prerequisite: MAT 272 or equivalent.

**MAT 510 Point Set Topology. **(3) F

Topological spaces, metric spaces, compactness, connectedness, local properties, product and decomposition spaces, mappings, covering properties, and separation properties. Prerequisite: MAT 371 or 410 or instructor approval.

**MAT 511 Point Set Topology. **(3) S

Continuation of MAT 510. Prerequisite: MAT 510 or instructor approval.

**MAT 520 Numerical Linear Algebra. **(3) A

Direct solution of linear systems, iterative methods, eigenvalues and eigenvectors, singular value decomposition, the QR algorithm, error propagation, arithmetic, and stability. Prerequisites: MAT 342 and 464 (or 466) *or* instructor approval.

**MAT 521 Iterative Methods. **(3) N

Numerical methods for solving linear/nonlinear systems of equations (symmetric, nonsym
metric). Iterative methods for linear systems, conjugate gradients, multigrid methods, preconditioning, Krylov methods. Prerequisites: MAT 371 and 464 (or 466) *or* instructor approval.

**MAT 523 Numerical Optimization. **(3) N

Linear programming, unconstrained nonlinear minimization, line search algorithms, conjugate gradients, quasi-Newton methods, constrained nonlinear optimization, gradient projection, and penalty methods. Prerequisite: MAT 342 or 371 or 460 or 520 or equivalent or instructor approval.

**MAT 524 Parallel Numerical Algorithms. **(3) N

Algorithms for massively parallel, hypercube architectures; "parallel" FORTRAN; solution of linear, nonlinear systems; partial differential equations; iterative methods; multigrid; domain decomposition. Prerequisites: MAT 371 and 464 (or 466) *or* instructor approval.

**MAT 526 Numerical Solution of Bifurcation Problems.** (3) N

Nonlinear parameter-dependent differential, algebraic equations, numerical solutions; bifurcation, turning points; continuation methods, branch switching; steady-state, time-dependent cases; Hopf Bifurcation. Prerequisites: MAT 371 and 464 (or 466) *or* instructor approval.

**MAT 528 Advanced Numerical Analysis. **(3) N

Finite difference equations, orthogonal polynomials, quadrature, approximation and integration theory, numerical solution of differential equations, and numerical linear algebra. May be repeated for credit with instructor approval. Prerequisite: MAT 464 or instructor approval.

**MAT 529 Advanced Numerical Analysis. **(3) N

Continuation of MAT 528. Prerequisite: MAT 528 or instructor approval.

**MAT 530 Numerical Solution of Ordinary Differential Equations. **(3) N

One step, linear multistep methods; consistency, order, stability, convergence; discretization, round-off errors, error estimation, adaptive strategy; implementation, software for nonstiff equations. Prerequisites: MAT 371 and 464 (or 466) *or* instructor approval.

**MAT 531 Numerical Solution of Stiff Differential Systems.** (3) N

Runge-Kutta methods, order conditions, construction of highly stable methods, order stars, error estimation, stepsize selection, contractivity properties, linear multistep methods. Prerequisites: MAT 371 and 464 (or 466) *or* instructor approval.

**MAT 533 Computational Elliptic and Parabolic Partial Differential Equations. **(3) N

Parabolic and elliptic equations, finite difference, finite element methods, stability, consistency, convergence, practical aspects, applications, software. Prerequisites: MAT 371 and 464 (or 466) *or* instructor approval.

**MAT 534 Computational Hyperbolic Partial Differential Equations.** (3) N

Numerical solutions of hyperbolic PDEs, finite difference methods, well-posedness, stability, consistency, convergence, adaptive grids; Maxwell's equations, elastic wave propagation; Navier-Stokes. Prerequisites: MAT 371 and 464 (or 466) *or* instructor approval.

**MAT 535 Spectral Methods for Partial Differential Equations. **(3) N

Spectral, pseudo-spectral theory; Galerkin, collocation methods; Tau-methods, global approximation properties, stability; convergence; solutions for linear, nonlinear systems. Prerequisites: MAT 371 and 464 (or 466) *or* instructor approval.

**MAT 536 Numerical Solution of Boundary Value Problems. **(3) N

Difference methods, finite element methods, defect correction, irregular meshes, nonlinear problems, bifurcation, boundary layers, and sparse systems. May be repeated for credit with instructor approval. Prerequisites: MAT 371 (or 460 or 462) and 464 (or 466) *or* instructor approval.

**MAT 543 Abstract Algebra. **(3) F

Groups, modules, rings and fields, Galois theory, homological algebra, and the representation theory. Prerequisite: MAT 443 or instructor approval.

**MAT 544 Abstract Algebra. **(3) S

Continuation of MAT 543. Prerequisite: MAT 543 or instructor approval.

**MAT 550 Variational Methods. **(3) F

Calculus of variations and its applications to extremal problems, classical mechanics, and partial differential equations. Prerequisites: MAT 274 and 462 *or* equivalents.

**MAT 551 Linear Operators and Integral Equations. **(3) S

Bounded linear and compact operators on Hilbert spaces. Linear integral equations, Fredholm and Hilbert-Schmidt theory, and approximate methods. Distributions. Prerequisites: MAT 242 and 462 *or* equivalents.

**MAT 570 Real Analysis. **(3) S

Lebesgue integration, selected function spaces, differentiation, abstract measure theory, and elements of functional analysis. Prerequisite: MAT 372 or instructor approval.

**MAT 571 Real Analysis. **(3) F

Continuation of MAT 570. Prerequisite: MAT 570 or instructor approval.

**MAT 572 Complex Analysis. **(3) F

Analytic functions, series and product representations, entire and meromorphic functions, normal families, Riemann mapping theorem, harmonic functions, and Riemann surfaces. Prerequisite: MAT 371 or instructor approval.

**MAT 573 Complex Analysis. **(3) S

Continuation of MAT 572. Prerequisite: MAT 572 or instructor approval.

**MAT 574 Theory of Ordinary Differential Equations. **(3) N

Systems, existence proofs, singularities, asymptotic behavior of solutions, boundedness of solutions, eigenvalues and eigenfunctions, and perturbation theory. Prerequisite: MAT 372 or instructor approval.

**MAT 575 Theory of Ordinary Differential Equations. **(3) N

Continuation of MAT 574. Prerequisite: MAT 574 or instructor approval.

**MAT 576 Theory of Partial Differential Equations. **(3) N

Existence and uniqueness theorems, boundary value and initial value problems, characteristics, Green's functions, maximum principle, distributions, and weak solutions. Prerequisite: knowledge of Lebesgue integration or instructor approval.

**MAT 577 Theory of Partial Differential Equations. **(3) N

Continuation of MAT 576. Prerequisite: MAT 576 or instructor approval.

**MAT 578 Functional Analysis. **(3) N

Locally convex, normed, and Hilbert spaces. Linear operators, spectral theory, and application to classical analysis. Prerequisite: MAT 472 or 571 or instructor approval.

**MAT 579 Functional Analysis. **(3) N

Continuation of MAT 578. Prerequisite: MAT 578 or instructor approval.

**MAT 591 Seminar. **(1–3) N

Topics may be selected from the following:

(a) | Algebra |

(b) | Analysis |

(c) | Applied Mathematics |

(d) | Combinatorial Mathematics |

(e) | Mathematical Logic |

(f) | Numerical Analysis |

(g) | Topology |

**Omnibus Graduate Courses:** See omnibus graduate courses that may be offered.

*1996–98 Graduate Catalog * Table of Contents

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