Course Descriptions

MATH-I 514 - Numerical Analysis

Credits:

3

Description:

Numerical Analysis is concerned with finding numerical solutions to problems, especially those for which analytical solutions do not exist or are not readily obtainable. This course provides an introduction to the subject and treats the topics of approximating functions by polynomials, solving linear systems of equations, and of solving nonlinear equations. These topics are of great practical importance in science, engineering and finance, and also have intrinsic mathematical interest. The course concentrates on theoretical analysis and on the development of practical algorithms.

Prerequisite:

MATH-I 266 or MATH-I 366, and MATH-I 351 or MATH-I 511, or consent of instructor and familiarity with one of the high-level programming languages: Fortran 77/90/95, C, C++, Matlab.

MATH-I 518 - Advanced Discrete Mathematics

Credits:

3

Description:

This course covers mathematics useful in analyzing computer algorithms. Topics include recurrence relations, evaluation of sums, integer functions, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods.

Prerequisite:

MATH-I 266 or MATH-I 366.

MATH-I 520 - Boundary Value Problems of Differential Equations

Credits:

3

Description:

Sturm-Liouville theory, singular boundary conditions, orthogonal expansions, separation of variables in partial differential equations, and spherical harmonics.

Prerequisite:

MATH-I 261, and MATH-I 266 or MATH-I 366.

MATH-I 522 - Qualitative Theory of Differential Equations

Credits:

3

Description:

Nonlinear ODEs, critical points, stability and bifurcations, perturbations, averaging, nonlinear oscillations and chaos, and Hamiltonian systems.

Prerequisite:

MATH-I 266 and MATH-I 351.

MATH-I 523 - Introduction to Partial Differential Equations

Credits:

3

Description:

Method of characteristics for quasilinear first-order equations, complete integral, Cauchy-Kowalewsky theory, classification of second-order equations in two variables, canonical forms, difference methods of hyperbolic and parabolic equations, and Poisson integral method for elliptic equations.

Prerequisite:

MATH-I 266 or MATH-I 366, and MATH-I 261 or MATH-I 510.

MATH-I 525 - Introduction to Complex Analysis

Credits:

3

Description:

Complex numbers and complex-valued functions; differentiation of complex functions; power series, uniform convergence; integration, contour integrals; and elementary conformal mapping.

Prerequisite:

MATH-I 261, and MATH-I 266 or MATH-I 366. Instructor consent required for any undergraduate student.

MATH-I 526 - Principles of Mathematical Modeling

Credits:

3

Description:

Ordinary and partial differential equations of physical problems, simplification, dimensional analysis, scaling, regular and singular perturbation theory, variational formulation of physical problems, continuum mechanics, and fluid flow.

Prerequisite:

MATH-I 266 or MATH-I 366, and MATH-I 426.

MATH-I 528 - Advanced Mathematics for Engineering and Physics II

Credits:

3

Description:

Divergence theorem, Stokes' Theorem, complex variables, contour integration, calculus of residues and applications, conformal mapping, and potential theory.

Prerequisite:

MATH-I 537.

MATH-I 530 - Functions of a Complex Variable I

Credits:

3

Description:

Complex numbers, holomorphic functions, harmonic functions, and linear transformations. Power series, elementary functions, Riemann surfaces, contour integration, Cauchy's theorem, Taylor and Laurent series, and residues. Maximum and argument principles. Special topics.

Prerequisite:

or C: MATH-I 544.

MATH-I 531 - Functions of a Complex Variable II

Credits:

3

Description:

Compactness and convergence in the space of analytic functions, Riemann mapping theorem, Weierstrass factorization theorem, Runge's theorem, Mittag-Leffler theorem, analytic continuation and Riemann surfaces, and Picard theorems.

Prerequisite:

MATH-I 530.

MATH-I 535 - Theoretical Mechanics

Credits:

3

Description:

Continuum mechanics deals with the analysis of the motion of materials modeled as a continuous mass rather than as discrete particles. Applications of continuum mechanics are ubiquitous in science and engineering, and are getting more and more popular in medicine too. The goal of this course is to study the basic principles of continuum mechanics for deformable bodies, including conservation laws and constitutive equations, while discussing the mathematical challenges in solving these equations analytically and/or numerically.

Prerequisite:

MATH-I 266 or MATH-I 366, and MATH-I 351 or MATH-I 511, or consent of the instructor.

MATH-I 537 - Applied Mathematics for Scientists and Engineers I

Credits:

3

Description:

Covers theories, techniques, and applications of partial differential equations, Fourier transforms, and Laplace transforms. Overall emphasis is on applications to physical problems.

Prerequisite:

MATH-I 261, and MATH-I 266 or MATH-I 366.

MATH-I 544 - Real Analysis and Measure Theory

Credits:

3

Description:

Algebras of sets, real number system, Lebesgue measure, measurable functions, Lebesgue integration, differentiation, absolute continuity, Banach spaces, metric spaces, general measure and integration theory, and Riesz representation theorem.

Prerequisite:

MATH-I 444.

MATH-I 545 - Principles of Analysis II

Credits:

3

Description:

Continues the study of measure theory begun in MATH-I 544.

Prerequisite:

MATH-I 544.

MATH-I 546 - Introduction to Functional Analysis

Credits:

3

Description:

Banach spaces, Hahn-Banach theorem, uniform boundedness principle, closed graph theorem, open mapping theorem, weak topology, and Hilbert spaces.

Prerequisite:

MATH-I 545.

MATH-I 547 - Analysis for Teachers I

Credits:

3

Description:

Set theory, logic, relations, functions, Cauchy's inequality, metric spaces, neighborhoods, and Cauchy sequence.

Prerequisite:

MATH-I 261.

MATH-I 549 - Applied Mathematics for Secondary School Teachers

Credits:

3

Description:

Applications of mathematics to problems in the physical sciences, social sciences, and the arts. Content varies. May be repeated for credit with the consent of the instructor. Course is offered on an as needed basis.

Prerequisite:

MATH-I 266 or MATH-I 366, and MATH-I 351.

MATH-I 552 - Applied Computational Methods II

Credits:

3

Description:

The first part of the course focuses on numerical integration techniques and methods for ODEs. The second part concentrates on numerical methods for PDEs based on finite difference techniques with brief surveys of finite element and spectral methods.

Prerequisite:

MATH-I 559 and consent of instructor.

MATH-I 553 - Introduction to Abstract Algebra

Credits:

3

Description:

Group theory: finite abelian groups, symmetric groups, Sylow theorems, solvable groups, Jordan-Holder theorem. Ring theory: prime and maximal ideals, unique factorization rings, principal ideal domains, Euclidean rings, and factorization in polynomial and Euclidean rings. Field theory: finite fields, Galois theory, and solvability by radicals.

Prerequisite:

MATH-I 453.

MATH-I 554 - Linear Algebra

Credits:

3

Description:

Review of basics: vector spaces, dimension, linear maps, matrices, determinants, and linear equations. Bilinear forms, inner product spaces, spectral theory, and eigenvalues. Modules over principal ideal domain, finitely generated abelian groups, and Jordan and rational canonical forms for a linear transformation.

Prerequisite:

MATH-I 351.