# ENME Masters Approved Math Courses

**CMSC 460: Computational Methods (3 credits) Permission Required.**

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH341, MATH461); and 1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (CMSC106, CMSC131); and minimum grade of C- in MATH246.**Also offered as:** AMSC460. **Credit only granted for:** AMSC460, AMSC466, CMSC460, or CMSC466. Basic computational methods for interpolation, least squares, approximation, numerical quadrature, numerical solution of polynomial and transcendental equations, systems of linear equations and initial value problems for ordinary differential equations. Emphasis on methods and their computational properties rather than their analytic aspects. Intended primarily for students in the physical and engineering sciences.

**CMSC 467: Intro to Numerical Analysis II**

**MATH 403: Intro to Abstract Algebra (3 Credits)**

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH461, MATH340); and 1 course with a minimum grade of C- from (MATH341, MATH241); and minimum grade of C- in MATH310. Or students who have taken courses with comparable content may contact the department. **Credit only granted for:** MATH402 or MATH403. Integers; groups, rings, integral domains, fields.

**MATH 404: Field Theory (3 Credits)**

**Prerequisite:** Minimum grade of C- in MATH403. Algebraic and transcendental elements, Galois theory, constructions with straight-edge and compass, solutions of equations of low degrees, insolubility of the quintic equation, Sylow theorems, fundamental theorem of finite Abelian groups.

**MATH 405: Linear Algebra (3 Credits)**

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH461, MATH341); and minimum grade of C- in MATH310. An abstract treatment of finite dimensional vector spaces. Linear transformations and their invariants.

**MATH 432: Intro to Point Set Topology (3 Credits)**

**Prerequisite:** Minimum grade of C- in MATH410. Metric spaces, topological spaces, connectedness, compactness (includin Heine-Borel and Bolzano-Weierstrass theorems), Cantor sets, continuous maps and homeomorphisms, fundamental group (homotopy, covering spaces, the fundamental theorem of algebra, Brouwer fixed point theorem), surfaces (e.g., Euler characteristic, the index of a vector field, hairy sphere theorem), elements of combinatorial topology (graphs and trees, planarity, coloring problems).

**MATH 436: Differential Geometry I (3 Credits)**

**Prerequisite:** 1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (MATH461, MATH240, MATH341); and must have completed two 400-level MATH courses with a minimum grade of C- (not including MATH461, 478, and 480’s). Curves in the plane and Euclidean space, moving frames, surfaces in Euclidean space, orientability of surfaces; Gaussian and mean curvatures; surfaces of revolution, ruled surfaces, minimal surfaces, special curves on surfaces, “Theorema Egregium”; the intrinsic geometry of surfaces.

**MATH 437: Differential Geometry II (3 Credits)**

**Prerequisite:** 1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (MATH240, MATH341, MATH461). **Recommended:** MATH405, MATH403, MATH436, MATH410, or MATH432. Introduction to differential forms and their applications, and unites the fundamental theorems of multivariable calculus in a general Stokes Theorem that is valid in great generality. It develops this theory and technique to perform calculations in analysis and geometry. Topics include an introduction to topological spaces, the Gauss-Bonnet Theorem, Gauss’s formula for the linking number, and the Cauchy Integral Theorem. Applications include Maxwell’s equations of electromagnetism, connections and gauge theory, and symplectic geometry and Hamiltonian dynamics.

**MATH 452: Introduction to Dynamics and Chaos (3 Credits)**

**Prerequisite:** MATH341. Or MATH246; and (MATH240 or MATH461). **Also offered as:** AMSC452. **Credit only granted for:** AMSC452 or MATH452. An introduction to mathematical dynamics and chaos. Orbits, bifurcations, Cantor sets and horseshoes, symbolic dynamics, fractal dimension, notions of stability, flows and chaos. Includes motivation and historical perspectives, as well as examples of fundamental maps studied in dynamics and applications of dynamics.

**MATH 462: PDEs for Scientists and Engineers (3 Credits)**

**Prerequisite:** 1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (MATH246, MATH341). Linear spaces and operators, orthogonality, Sturm-Liouville problems and eigenfunction expansions for ordinary differential equations. Introduction to partial differential equations, including the heat equation, wave equation and Laplace’s equation. Boundary value problems, initial value problems and initial-boundary value problems.

**MATH 463: Complex Variables for Scientists and Engineers (3 Credits)**

**Prerequisite:** 1 course with a minimum grade of C- from (MATH241, MATH340). The algebra of complex numbers, analytic functions, mapping properties of the elementary functions. Cauchy integral formula. Theory of residues and application to evaluation of integrals. Conformal mapping.

**MATH 464: Transform Methods for Scientists and Engineers (3 Credits)**

**Prerequisite:** 1 course with a minimum grade of C- from (MATH246, MATH341). Fourier transform, Fourier series, discrete fast Fourier transform (DFT and FFT). Laplace transform. Poisson summations, and sampling. Optional Topics: Distributions and operational calculus, PDEs, Wavelet transform, Radon transform and applications such as Imaging, Speech Processing, PDEs of Mathematical Physics, Communications, Inverse Problems.

**MATH 475: Combinatorics and Graph Theory (3 credits). Permission Required**

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH341, MATH461); and 1 course with a minimum grade of C- from (MATH241, MATH340); and permission of CMNS-Mathematics department. **Also offered as:** CMSC475. **Credit only granted for:** MATH475 or CMSC475. General enumeration methods, difference equations, generating functions. Elements of graph theory, matrix representations of graphs, applications of graph theory to transport networks, matching theory and graphical algorithms.

**STAT 410: Introduction to Probability Theory (3 Credits).**

**Prerequisite:** 1 course with a minimum grade of C- from (MATH240, MATH461, MATH341); and 1 course with a minimum grade of C- from (MATH340, MATH241). **Also offered as:** SURV410. **Credit only granted for:** STAT410 or SURV410. Probability and its properties. Random variables and distribution functions in one and several dimensions. Moments. Characteristic functions. Limit theorems.

**STAT 420: Introduction to Statistics (3 Credits). Offered Spring only.**

**Prerequisite:** 1 course with a minimum grade of C- from (SURV410, STAT410). **Also offered as:** SURV420. **Credit only granted for:** STAT420 or SURV420. Point estimation, sufficiency, completeness, Cramer-Rao inequality, maximum likelihood. Confidence intervals for parameters of normal distribution. Hypothesis testing, most powerful tests, likelihood ratio tests. Chi-square tests, analysis of variance, regression, correlation. Nonparametric methods.

**ENME 605: Advanced Systems Control (3 Credits) ***Prerequisite: ENME462; or permission of instructor.*

*Prerequisite: ENME462; or permission of instructor.*

Modern control theory for both continuous and discrete systems. State space representation is reviewed and the concepts of controllability and observability are discussed. Design methods of deterministic observers are presented and optimal control theory is formulated. Control techniques for modifying system characteristics are discussed.

**ENME 610: Engineering Optimization (3 Credits).**

Overview of applied single- and multi- objective optimization and decision making concepts and techniques with applications in engineering design and/or manufacturing problems. Topics include formulation examples, concepts, optimality conditions, unconstrained/constrained methods, and post-optimality sensitivity analysis. Students are expected to work on a semester-long real-world multi-objective engineering project.

**ENME 625: Multidisciplinary Optimization (3 Credits).**

Overview of single- and multi-level design optimization concepts and techniques with emphasis on multidisciplinary engineering design problems. Topics include single and multilevel optimality conditions, hierarchic and nonhierarchic modes and multilevel post optimality sensitivity analysis. Students are expected to work on a semester-long project.

**ENME 700: Advanced Mechanical Engineering Analysis (3 Credits)**

An advanced, unified approach to the solution of mechanical engineering problems, emphasis is on the formulation and solution of equilibrium, eigenvalue and propagation problems. Review and extension of undergraduate material in applied mathematics with emphasis on problems in heat transfer, vibrations, fluid flow and stress analysis which may be formulated and solved by classical procedures.

**ENME 725: Probabilistic Optimization (3 Credits).**

**Prerequisite:** An advanced undergraduate course in probability and a graduate course in optimization or permission of the instructor required. **Also offered as:** ENCE725. **Credit only granted for:** ENME725 or ENCE725. Provide an introduction to optimization under uncertainty. Chance-constrained programming, reliability programming, value of information, two stage problems with recourse, decomposition methods, nonlinear and linear programming theory, probability theory. The objectives of this course are to provide understanding for studying problems that involve optimization under uncertainty, learn about various stochastic programming formulations (chance constrained programs, two stage methods with recourse, etc.) relevant to engineering and economic settings, present theory for solutions to such problems, and present algorithms to solve these problems.

**ENME 745: Computational Methods in Science and Engineering (3 Credits)**

**Credit only granted for:** ENME745 and ENME808B. **Formerly:** ENME808B. Fundamental aspects of how to apply analytical mathematical concepts to discrete data. The course is aimed at graduate students in any area of engineering, and treats the material in a general manner that is not specific to any application or field of specialization.

**ENRE 620: Mathematical Techniques of Reliability Engineering (3 Credits). ***Prerequisites: MATH 246 or permission of department.*

*Prerequisites: MATH 246 or permission of department.*

Basic probability and statistics Application of selected mathematical techniques to the analysis and solution of reliability engineering problems. Applications of matrices, vectors, tensors, differential equations, integral transforms, and probability methods to a wide range of reliability-related problems

**ENRE 655: Advanced Methods in Reliability Modeling (3 Credits)**

**Prerequisite:** ENRE602. **Credit only granted for:** ENRE655 or ENRE665. **Formerly:** ENRE665. Bayesian methods and applications, estimation of rare event frequencies uncertainty analysis and propagation methods, reliability analysis of dynamic systems, analysis of dependent failures, reliability of repairable systems, human reliability analysis methods and theory of logic diagrams and application to systems reliability

**Any MATH, STAT, or AMSC course at the 600 level or above**