# Courses - Applied Mathematics

Courses

# Course Catalog Descriptions

MA Courses MA1010-MA2300

## MA1010 Algebra and Trigonometry (4-0) As Required

Real number system, complex numbers, exponents and radicals, algebraic expressions and operations, linear and quadratic equations, inequalities, functions and graphs, polynomials and their zeros, rational functions, exponential and logarithmic functions, systems of equations, matrices, trigonometry and unit circles, trigonometric identities and functions. MA1010 syllabus.

Prerequisites: None.

## MA1025 Introduction to Mathematical Reasoning (4-0) As Required

An introductory course in logic and elementary discrete mathematics to be taken by students in the Operations Research curriculum. Considerable emphasis is placed on propositional and predicate logic, and on techniques of proof in mathematics. Mathematical topics include sets, functions, and relations. Coverage of combinatorics includes an introduction to permutations, combinations, the pigeon-hole principle, and the principle of inclusion/exclusion. No previous experience with this material is assumed. MA1025 syllabus.

Prerequisites: None.

## MA1113 Single Variable Calculus (4-0) Spring/Summer/Fall/Winter

Review of analytic geometry and trigonometry, functions of one variable, limits, derivatives, continuity and differentiability; differentiation of algebraic, trigonometric, logarithmic and exponential functions with applications to maxima and minima, rates, differentials; product rule, quotient rule, chain rule; antiderivatives, integrals and the fundamental theorem of calculus; definite integrals, areas. Taught at the rate of nine hours per week for five weeks. MA1113 syllabus. Prerequisites: Pre-Calculus mathematics.

## MA1114 Single Variable Calculus II with Matrix Algebra (4-0) Spring/Summer/Fall/Winter

Topics in calculus include applications of integration, special techniques of integration, infinite series, convergence tests, and Taylor series. Matrix algebra topics covered are: the fundamental algebra of matrices including addition, multiplication of matrices, multiplication of a matrix by a constant and a column (vector) by a matrix; elementary matrices and inverses, together with the properties of these operations; solutions to mxn systems of linear algebraic equations using Gaussian elimination and the LU decomposition (without pivoting); determinants, properties of determinants; and a brief introduction to the arithmetic of complex numbers and DeMoivre's theorem. Taught at the rate of nine hours per week for five weeks. MA1114 syllabus. Prerequisites: MA1113.

## MA1115 Multi-variable Calculus (4-0) Spring/Summer/Fall/Winter

Vector algebra and calculus, directional derivative, gradient, polar coordinates and parametric equations, functions of several independent variables, limits, continuity, partial derivatives, chain rule, maxima and minima, double and triple integrals, cylindrical and spherical coordinate systems. Taught at the rate of nine hours per week for five weeks. MA 1115 syllabus. Prerequisites: MA1114.

## MA1116 Vector Calculus (3-0) Spring/Summer/Fall/Winter

The calculus of vector fields; directional derivative, gradient, divergence, curl; potential fields; Green's, Stokes', and the divergence integral theorems. Applications in engineering and physics. Taught at the rate of seven hours per week for five weeks. MA1116 syllabus. Prerequisites: MA1115.

## MA1118 Multivariable Calculus for Operations Research (4-0) Fall/Spring

First-order linear differential equations, curves and surfaces, polar coordinates, vector algebra and calculus, functions of several independent variables, partial derivatives, Taylor series, chain rule, maxima and minima, directional derivatives and gradient, Lagrange multipliers, double integrals. MA1118 syllabus. Prerequisite: MA1114.

## MA2025 Logic and Discrete Mathematics I (4-1) Summer/Winter

MA2025 is a first course in discrete mathematics for students of mathematics and computer science. Topics include propositional and predicate logic up to the deduction theorem, methods of mathematical proof, naive set theory, properties of functions, sequences and sums, mathematical induction, an introduction to divisibility and congruences, and an introduction to enumerative combinatorics. MA2025 syllabus. Prerequisites: None, although a review of algebra skills is recommended.

## MA2043 Introduction to Matrix and Linear Algebra (4-0) As Required

The fundamental algebra of vectors and matrices including addition, scaling, and multiplication. Block operations with vectors and matrices. Algorithms for computing the LU (Gauss) factorization of an MxN matrix, with pivoting. Matrix representation of systems of linear equations and their solution via the LU factorization. Basic properties of determinants. Matrix inverses. Linear transformations and change of basis. The four fundamental subspaces and the fundamental theorem of linear algebra. Introduction to eigenvalues and eigenvectors. MA2043 syllabus. Prerequisites: Students should have mathematical background at the level generally expected of someone with a B.S. in Engineering, i.e., familiarity with Calculus and solid algebra skills. EC1010 (May be taken concurrently.)

## MA2121 Differential Equations (4-0) Spring/Summer/Fall/Winter

Ordinary differential equations: linear and nonlinear (first order) equations, homogeneous and non-homogeneous equations, linear independence of solutions, power series solutions, systems of differential equations, Laplace transforms. Applications include radioactive decay, elementary mechanics, mechanical and electrical oscillators, forced oscillations and resonance. MA 2121 syllabus. Prerequisites: MA1114.

## MA2300 Mathematics for Management (5-0) Winter/Spring/Summer

Mathematical basis for modern managerial tools and techniques. Elements of functions and algebra; differential calculus of single- and multi-variable functions; integration (antidifferentiation) of single-variable functions. Applications of the derivative to rates of change, curve sketching, and optimization, including the method of Lagrange multipliers. MA 2300 syllabus. Prerequisite: College algebra.

## MA Courses MA3001-MA3730

MA3001 Incremented Directed Study (Variable 1-0 or 2-0) (V-0) As Required

Provides the opportunity for a student who is enrolled in a 3000 level mathematics course to pursue the course material and its applications in greater depth by directed study to the extent of one additional hour beyond the normal course credit. Prerequisites: Enrollment in a 3000 level mathematics course and consent of instructor.

## MA3025 Logic and Discrete Mathematics II (4-1) As Required

Provides a rigorous foundation in logic and elementary discrete mathematics to students of mathematics and computer science. Topics from logic include modeling English propositions, propositional calculus, quantification, and elementary predicate calculus. Additional mathematical topics include elements of set theory, mathematical induction, relations and functions, and elements of number theory. MA3025 syllabus. Prerequisites: MA2025 (preferable) or MA1025.

## MA3030 Introduction to Combinatorics and Its Applications (4-1) As Required

Provides a thorough grounding in elementary combinatorics and its applications to computer science and discrete probability theory to students of computer science who concurrently take MA3025, Logic and Discrete Mathematics. Topics from combinatories include fundamental counting rules, binomial and multinomial theorems, the pigeonhole and inclusion/exclusion principles, and homogeneous recurrence relations. Elementary discrete probability is covered, up to the expectation of a discrete random variable. Corequisite: MA3025.

## MA3042 Linear Algebra (4-0) As Required

Finite-dimensional vector spaces, linear dependence, basis and dimension, change of basis. Linear transformations and similarity. Scalar product, inner product spaces. Orthogonal subspaces and least squares. LU (with pivoting), Cholesky, and QR factorizations. Eigenvalues/eigenvectors, diagonalization. Hermitian matrices, quadratic forms, definite matrices. Vector and matrix norms, orthogonal transformations, condition numbers. Prerequisite: MA1114.

## MA3046 Matrix Analysis (4-1) As Required

This course provides students in the engineering and physical sciences curricula with an applications-oriented coverage of major topics of matrix and linear algebra. Matrix factorizations (LU, QR, Cholesky), the Singular Value Decomposition, eigenvalues and eigenvectors, the Schur form, subspace computations, structured matrices. Understanding of practical computational issues such as stability, conditioning, complexity, and the development of practical algorithms. Prerequisites: MA2043 and EC1010.

## MA3110 Intermediate Analysis (4-0) Summer/Winter

Multi-variable calculus integrated with linear algebra. Functions of several variables, continuous transformations, Jacobians, chain rule, implicit function theorem, inverse function theorem, extreme, optimization and Lagrange multiplier technique. Applications in Operations Research. Prerequisites: MA1115 and MA3042.

## MA3132 Partial Differential Equations and Integral Transforms (4-0) Spring/Summer/Fall/Winter

Solution of boundary value problems by separation of variables; Sturm-Liouville problems; Fourier and Bessel series solutions, Fourier transforms; classification of second-order equations; applications, method of characteristics. Applications to engineering and physical science. Satisfies the ESR in differential equations for the Applied Mathematics program. Prerequisites: MA2121 and MA1116.

## MA3139 Fourier Analysis and Partial Differential Equations (4-0) Summer/Winter

Fourier series; solution of the one and two-dimensional wave equations, D'Alembert's solution, frequency and time domain interpretations; Fourier integral transforms and applications to ordinary and partial differential equations and linear systems; Convolution theorems. Course covers basic material essential for signal processing, filtering, transmission, waveguides, and other related problems. Applications include spectral analysis of electronic signals, e.g., radar or sonar. Designed for UW and EW/IW students. Prerequisites: MA1115 and MA2121.

## MA3185 Tensor Analysis (3-0) Fall

Definition and algebra of tensors. Dyadic representation in Cartesian and general components. Calculus of tensor fields in curvilinear coordinates. Derivation and application of the basic equations of heat conduction, rigid body mechanics, elasticity, fluid mechanics, electromagnetism, Newtonian and Einsteinian orbital mechanics. Prerequisites: MA1116.

## MA3232 Numerical Analysis (4-0) Spring/Summer/Fall/Winter

Provides the basic numerical tools for understanding more advanced numerical methods. Topics for the course include: Sources and Analysis of Computational Error, Solution of Nonlinear Equations, Interpolation and Other Techniques for Approximating Functions, Numerical Integration and Differentiation, Numerical Solution of Initial and Boundary Value Problems in Ordinary Differential Equations, and Influences of Hardware and Software. Prerequisites: MA1115, MA2121 and ability to program in MATLAB and MAPLE.

## MA3243 Numerical Methods for Partial Differential Equations (4-1) Winter

Course designed to familiarize the student with analytical techniques as well as classical finite difference techniques in the numerical solution of partial differential equations. In addition to learning applicable algorithms, the student will be required to do programming. Topics covered include: Implicit, Explicit, and Semi-Implicit methods in the solution of Elliptic and Parabolic PDE's, iterative methods for solving Elliptic PDEs (SOR, Gauss-Seidel, Jacobi), the Lax-Wendroff and Explicit methods in the solution of 1st and 2nd order Hyperbolic PDEs. Prerequisites: MA3132 and the ability to program in a high level language such as Fortran, C, or MATLAB.

## MA3261 Basic Parallel Computation (3-0) As Required

The course has two goals: First, to introduce fundamental issues such as shared vs. distributed memory, connection topologies, communication algorithms, speedup, efficiency, storage requirements, granularity, pipelining, problem scaling, and useful paradigms for algorithm development. Second, to develop working proficiency by designing, implementing, and evaluating the performance of several parallel algorithms. These include, but are not limited to, numerical quadrature, matrix computations, sorting, network analysis, and dynamic programming. Prerequisites: MA1115 or MA3025 and ability to program in a high-level language.

## MA3301 Linear Programming (Same as OA3201) (4-0) As Required

See OA3201 for course description.

## MA3393 Topics in Applied Mathematics (4-0) Winter

A selection of topics in applied mathematics. The course content varies and the credit varies. This course is intended to reflect study for the beginning graduate student in an area for which no formal course is taught. Credit for this course may be granted more than one time to an individual student. Prerequisites: Consent of instructor.

## MA3560 Applied Modern Algebra and Number Theory (4-0) As Required

This course is devoted to aspects of modern algebra and number theory that directly support applications, principally in communication. The algebraic emphasis is on ring and field theory, with special emphasis on the theory of finite fields, as well as those aspects of group theory that are important in the development of coding theory. Elements of number theory include congruences and factorization. Applications are drawn from topics of interest to DoN/DoD. These include error correcting codes and cryptography. Prerequisites: MA3025.

## MA3607 Introduction to Real Analysis (4-0) Summer

The objective of this course is for students to achieve a solid understanding of the basic concepts, theorems, and proofs in introductory real analysis, including: limits, sequences, series, continuity, uniform convergence and uniform continuity, differentiation, and Riemann integration. This is a mathematics course in the pure sense. Proofs will be emphasized, and the student will learn how to reproduce, understand, create and enjoy mathematical proofs. Prerequisites: MA1114.

## MA3610 Topology, Fractals, and Chaotic Dynamics (3-0) As Required

An introductory course on chaotic dynamics systems and fractals. Topics covered include: flows on the line, bifurcations, linear systems, phase plane, limit cycles, the Lorenz equations, fractals, and one-dimensional maps. Applications include population growth, laser threshold, the pendulum, relaxation oscillations, and synchronized chaos. Prerequisites: MA1115 and MA2121.

## MA3677 Theory of Functions of a Complex Variable I (4-0) As Required

Selected topics from the theory of functions of a complex variable; analytic functions, power series, Laurent series. Singularities of analytic functions; contour integration and residues; applications of residues to real integrals and Laplace transforms, zeros of analytic functions, infinite product representation for analytic functions; maximum modulus theorems for analytic and harmonic functions; conformal mapping. Applications include interference effects in optics and problems from heat flow and fluid flow. Prerequisites: MA1115.

## MA3730 Theory of Numerical Computation (3-0) As Required

Analysis of computational methods used for the solution of problems from the areas of algebraic equations, polynomial approximation, numerical differentiation and integration, and numerical solutions of ordinary differential equations. Prerequisites: MA2121.

## MA Courses MA4026-MA4693

MA4026 Combinatorial Mathematics (4-0) As Required

Advanced techniques in enumerative combinatorics and an introduction to combinatorial structures. Topics include generating functions, recurrence relations, elements of Ramsey theory, theorems of Burnside and Polya, and balanced incomplete block designs. Application areas with DoD/DoN relevance range from mathematics to computer science and operations research, including applications in probability, game theory, network design, coding theory, and experimental design. Prerequisites: MA3025.

## MA4027 Graph Theory and Applications (4-0) Fall

Advanced topics in the theory of graphs and digraphs. Topics include graph coloring, Eulerian and Hamiltonian graphs, perfect graphs, matching and covering, tournaments, and networks. Application areas with DoD/DoN relevance range from mathematics to computer science and operations research, including applications to coding theory, searching and sorting, resource allocation, and network design. MA4027 syllabus. Prerequisites: MA3025.

## MA4103 Thesis Topics Seminar (3-0) As Required

Explores in depth discrete dynamical systems and the thesis topics of students enrolled in the Applied Mathematics degree program. Fulfills the ESR to provide students with the experience of organizing and presenting applied mathematical ideas to students and faculty, including a classroom environment. Prerequisites: Consent of instructor. Graded on a Pass/Fail basis only.

## MA4237 Advanced Topics in Numerical Analysis (V-0)

The subject matter will vary according to the abilities and interest of those enrolled. Applications of the subject matter to DoD/DoN are discussed. Prerequisites: Consent of instructor.

## MA4242 Numerical Solution of Ordinary Differential Equations (4-0) As Required

Adams formulas, Runge-Kutta formulas, extrapolation methods, implicit formulas for stiff equations; convergence and stability, error estimation and control, order and stepsize selection, applications. Prerequisites: MA3232.

## MA4243 Numerical Solution of Partial Differential Equations (3-1) As Required

Finite difference methods for parabolic, elliptic, and hyperbolic equations, multi-grid methods; convergence and stability, error estimation and control, numerical solution of finite difference equations, applications. Prerequisites: MA3132, MA3232 suggested.

## MA4245 Mathematical Foundations of Galerkin Methods (4-0) As Required

Variational formulation of boundary value problems, finite element and boundary element approximations, types of elements, stability, eigenvalue problems. Prerequisites: MA3132, MA3232 or equivalent.

## MA4248 Computational Linear Algebra (4-1) As Required

Development of algorithms for matrix computations. Rounding errors and introduction to stability analysis. Stable algorithms for solving systems of linear equations, linear least squares problems and eigen problems. Iterative methods for linear systems. Structured problems from applications in various disciplines. Prerequisites: MA3046, or consent of instructor, advanced MATLAB programming.

## MA4261 Distributed Scientific Computing (3-2) As Required

General principles of parallel computing, parallel techniques and algorithms, solution of systems of linear equations, eigenvalues and singular value decomposition, domain decomposition and application (e.g., satellite orbit determination and shallow water fluid flow). Prerequisites: MA3042 or MA3046, MA3132, and MA3232.

## MA4301 Nonlinear Programming (Course Taught by OR Staff, Same as OA4201) (4-0) As Required

See OA4201 for course description.

## MA4302 Design of Experiments (Course Taught by OR Staff, Same as OA4101) (3-1) As Required

See OA4101 for course description.

## MA4303 Regression Analysis (Course Taught by OR Staff, Same as OA4102) (4-0) As Required

See OA4102 for course description.

## MA4304 Time Series Analysis (Course Taught by OR Staff, Same as OA4308) (4-0) As Required

See OA4308 for course description.

## MA4305 Stochastic Models II (Course Taught by OR Staff, Same as OA4301) (4-0) As Required

See OA4301 for course description.

## MA4311 Calculus of Variations (4-0) As Required

Euler equation, Weierstrass condition, Legendre condition, numerical procedures for determining solutions, gradient method, Newton method, Transversability condition, Rayleigh Ritz method, conjugate points. Concepts are related to geometric principles whenever possible. Prerequisites: MA2121 (programming experience desirable).

## MA4321 Stability, Bifurcation and Chaos (3-0) As Required

Differential equations and dynamical systems, equilibrium of autonomous systems, stability, Liapunov's method, examples of chaos, local bifurcations of vector fields and maps, chaotic dynamical systems. Prerequisites: MA3610.

## MA4322 Principles and Techniques of Applied Mathematics I (4-0) Fall

Selected topics from applied mathematics to include: Dimensional Analysis, Scaling, Stability and Bifurcation, Perturbation Methodsâ€” regular and singular with boundary layer analysis, as well as, asymptotic expansions of integral, integrals equations, Green's functions of boundary value problems, and distribution theory. Prerequisites: MA3042 and MA3132; MA3232 strongly recommended.

## MA4323 Principles and Techniques of Applied Mathematics II (4-0) Winter

Continuation of MA4322. Selected topics include: calculus of variations, Hamiltonian Mechanics, distribution theory and Green's Functions in two and three dimensions, and discrete models. Prerequisites: MA4322

## MA4332 Partial Differential Equations (4-0) As Required

This course provides an introduction to the theory of partial differential equations. It includes the following topics: classification of second order equations; initial value and boundary value problems for hyperbolic, parabolic, and elliptic partial differential equations; existence and uniqueness of linear elliptic and parabolic PDEs; nonlinearparabolic and elliptic PDEs; Hamilton-Jacobi equations; systems of conservation laws and nonlinear wave equations; transform methods and Green's functions. Prerequisites: MA3132, and MA3232 strongly recommended.

## MA4335 Linear and Nonlinear Waves (3-0) As Required

Analysis of the two main classes of wave motion, hyperbolic waves and linear dispersive waves. Topics covered include: kinematic waves, shock waves, shock structure and shock fitting, Burger's equation, the wave equation, linear dispersive waves, wave patterns and water waves. Prerequisite: MA3132.

## MA4362 Astrodynamics (3-0) As Required

Review of the two-body problem. The effects of a third point mass and a distributed mass. Expansion of the disturbing potential in series of Legendre functions. Variation of parameter equations for osculating orbital elements. Perturbation and numerical solution techniques. Statistical orbit determination. Codes used by the military to maintain the catalog of artificial satellites and space debris. Prerequisites: SS3500 or equivalent.

## MA4372 Integral Transforms (3-0) As Required

The Laplace, Fourier and Hankel transforms and their inversions; Asymptotic behavior. Applications to problems in engineering and physics. Prerequisites: MA3132.

## MA4377 Asymptotic and Perturbation Methods I (3-0) As Required

Advanced course in the application of approximate methods to the study of integrals and differential equations arising in physical problems. Topics covered include: asymptotic sequences and expansions, integrals of a real variable, contour integrals, limit process expansions applied to ordinary differential equations, multiple variable expansion procedures and applications to partial differential equations. Prerequisites: MA3132.

## MA4378 Asymptotic and Perturbation Methods Ii (3-0) As Required

Continuation of MA4377. Prerequisites: MA4377.

## MA4391 Analytical Methods for Fluid Dynamics (4-0) As Required

The basic fluid dynamic equations will be derived, and a variety of analytical methods will be applied to problems in viscous flow, potential flow, boundary layers, and turbulence. Applications in aeronautics will be discussed. Prerequisites: MA3132 or MA3139.

## MA4392 Numerical Methods for Fluid Dynamics (4-0) As Required

Numerical methods exclusively will be applied to fluid dynamics problems in viscous flow, potential flow, boundary layers, and turbulence. Applications in aeronautics will be discussed. Prerequisites: MA3232 and MA4391.

## MA4393 Topics in Applied Mathematics (4-0)

The course content varies but applications of interest to the DoN/DoD will be discussed. Credit may be granted for taking this course more than once. Prerequisites: Consent of instructor.

## MA4394 Topics in Network Science (4-0) As Required

Offered any quarter that there are enough requests. This course focuses on the emerging field of network science. The students will learn about the ongoing research in the field by listening to conference presentations (prerecorded or in person if possible to attend conferences), and applying the learned knowledge to a research project. If the research project is used towards a funded research, then (1) students' visit to sponsors will be included during the quarter (if feasible) or virtual conversations with sponsors are facilitated, and (2) a final presentation to the sponsor is anticipated upon the completion of the project. There is no set content for the course, as students will be exposed to current topics in the network science field at the time. Prerequisites: consent of instructor (Prof. Ralucca Gera).

## MA4400 Cooperation and Competition (4-0) Spring

The course will develop game theoretic concepts in evaluations of the importance of players in bargaining situations and of elements in networks. Topics covered include cooperative and noncooperative games, bargaining, the Shapley Value, and coalitions. The course will study applications to military problems and applications to economics, political science, and biology. There will be extensive reading from the literature. MA 4400 syllabus. Prerequisites: MA3042, OA3201, and an introductory course in probability.

## MA4404 Structure and Analysis of Complex Networks (4-0) Winter

The course focuses on the emerging science of complex networks and their applications, through an introduction to techniques and models for understanding and predicting their behavior. The topics discussed will be building mainly on graph theory concepts, and they will address the mathematics of networks, their applications to the computer networks and social networks, and their use in research. The students will learn the fundamentals of dynamically evolving complex networks, study current research in the field, and apply their knowledge in the analysis of real network systems through a final project. DoD applications include security of critical communication infrastructure. MA 4404 syllabus. Prerequisites: MA3025 or 4027.

## MA4550 Combinatorial and Cryptographic Properties of Boolean Functions (4-0) As Required

The course will discuss the Fourier analysis of Boolean functions and the relevant combinatorics with an eye toward cryptography and coding theory. Particular topics will include avalanche features of Boolean functions, correlation immunity and resiliency, bentness, trade-offs among cryptographic criteria and real-life applications in the designs of stream and block ciphers. Prerequisite: MA3025 or a similar combinatorial/discrete mathematics course (and recommended, but not required, an introductory course in probability).

## MA4560 Coding and Information Theory (4-0) Fall

Mathematical analysis of the codes used over communication channels is made. Techniques developed for efficient, reliable and secure communication are stressed. Effects of noise on information transmission are analyzed and techniques to combat their effects are developed. Linear codes, finite fields, single and multiple error-correcting codes are discussed. Codes have numerous applications for communication in the military, and these will be addressed. Prerequisites: MA3560.

## MA4565 Advanced Modern Algebra (3-0) As Required

A continuation of MA3560. Rings, ring homomorphism, integral domains and Euclidean domains. Unique factorization rings, polynomial rings. Modules and ideals. Noetherian rings, Field extension and Galois theory. Prerequisites: MA3560.

## MA4570 Cryptography - Foundations and Practice (4-1) Summer

The methods of secret communication are addressed. Simple cryptosystems are described and classical techniques of substitution and transposition are considered. The public-key cryptosystems, RSA, Discrete Logarithm and other schemes are introduced. Applications of cryptography and cryptanalysis. Prerequisites: MA3560.

## MA4593 Topics in Algebra (3-0) Fall

A selection of topics in algebra. Content of the course varies. Credit for taking the course more than once is allowed. Students may select a topic of interest to the DoN/DoD, so the course can support the MERs in a variety of curricula. Prerequisite: MA3560.

## MA4620 Theory of Dynamical Systems (4-0) As Required

This course provides an introduction to the theory of dynamical systems providing a basis for the analysis and design of systems in engineering and applied science. It includes the following topics: Second order linear systems; contraction mapping, existence and uniqueness of solutions; continuous dependence on initial conditions; comparison principle; Lyapunov stability theorems; LaSalle's theorem; linearization methods; nonautonomous systems; converse theorems; center manifold theorems; and stationary bifurcations of nonlinear systems. Prerequisites: MA2121.

## MA4635 Functions of Real Variables I (3-0) As Required

Semi-continuous functions, absolutely continuous functions, functions of bounded variation; classical Lebesgue measure and integration theory, convergence theorems and Lp spaces. Abstract measure and integration theory, signed measures, Radon-Nikodym theorem; Lebesgue decomposition and product measure; Daniell integrals and integral representation of linear functionals. Prerequisites: MA3606.

## MA4636 Functions of Real Variables II (3-0) As Required

Continuation of MA4635. Prerequisites: MA4635.

## MA4675 Complex Analysis (4-0) As Required

A continuation of MA3677. Differential equations in the complex plane, transform methods, the Wiener-Hopf method, integral equations, discrete Fourier analysis. Prerequisite: MA3677.

## MA4693 Topics in Analysis (3-0) Spring

Content of the course varies. Students will be allowed credit for taking the course more than once. Prerequisites: Consent of instructor.

MO Courses

MO designated courses are intended for students in operational curricula only. They do not satisfy the mathematics course requirements for accredited engineering curricula, nor do they satisfy the prerequisites for any of the MA designated courses.

## MO1180 Topics in Mathematics for Systems Analysis (3-2) Spring/Fall

A one quarter course in logic, elementary mathematics, combinatorics, and matrix algebra, plus a review of selected topics from single variable calculus with extensions to two variables. This course is intended for first-quarter students in the distributed learning Master of Systems Analysis curriculum. Logic places emphasis on the Propositional and Predicate Calculus. Elementary mathematical topics include sets, functions, and relations. Coverage of combinatorics includes an introduction to basic principles of counting (sum and product rules), permutations, and combinations. The fundamental algebra of matrices includes addition, multiplication of matrices, and multiplication of a matrix by a constant, and a column (vector) by a matrix; elementary matrices and inverses, together with the properties of these operations; solutions to m x n systems of linear algebraic equations using Gaussian elimination. Selected topics from single-variable calculus are extended to functions of two-variables, including double integrals over rectangles and general regions. (This course may not be taken for credit by students in an engineering or science degree program, nor may it be used as a prerequisite for any other mathematics course). Prerequisite: Single-variable calculus.

## MO1901 Mathematics for ISSO (3-0) As Required

A brief survey of selected calculus and post-calculus topics--single variable derivatives and integrals, infinite series and sequences, complex numbers, and Fourier series and transforms. (This course may not be taken for credit by students in an engineering or science degree program, nor may it be used as a prerequisite for any other mathematics course.) Prerequisites: None.

## MO1903 Mathematics for ISSO Space Systems Operations Specialization (3-0) Fall

To be taken concurrently with MA1114. The course consists of a brief survey of the following topics: Complex numbers, Fourier series and transforms, and Ordinary Linear Differential Equations. (This course may not be taken for credit by students in an engineering or science degree program, nor may it be used as a prerequisite for any other mathematics course.) Taught at the rate of seven hours per week for five weeks. Prerequisites: MA1113.