**MATH 500 - Discrete Mathematics**

Hours: 3

Study of formal logic; sets; functions and relations; principle of mathematical induction; recurrence relations; and introductions to elementary number theory and graph theory; counting (basic combinatorics); asymptotic complexity of algorithms; and NPcompleteness. This course is useful to those taking graduate classes in computer science. It also helpful to secondary teachers by giving them a better understanding of the terms and ideas used in modern mathematics. This is an elective course, eligible for the non-thesis option of the MS degree in math only. The maximum credit hours can be earned towards the MS degree in math among MATH 500, 550, 560 is six. Prerequisites: A grade of C or better on MATH 2414.

**MATH 501 - Mathematical Statistics I**

Hours: 3

A graduate level course in Probability Theory intended to provide the theoretical background for a course in statistical inference. Topics covered include: probability, random variables, distributions, moments, convergence of random variables, probability inequalities, random samples. Prerequisites: MATH 2415 with a minimum grade of C.

**MATH 502 - Mathematical Statistics II**

Hours: 3

A graduate level course in statistical inference. Topics covered include: point estimation, interval estimation, hypothesis testing, Bayesian inference. Prerequisites: MATH 501 with a minimum grade of C.

**MATH 503 - Actuarial Mathematics**

Hours: 3

A course in business/financial mathematics designed as an introduction to actuarial science and as preparation for the Exam P/1 and Exam FM actuarial exams. Encounters appropriate topics from analysis, linear algebra, probability and statistics, and financial mathematics. Prerequisites: MATH 402 with a minimum grade of C or MATH 403 with a minimum grade of C.

**MATH 511 - Real Analysis I**

Hours: 3

Properties of real numbers, continuity, differentiation, integration, sequences and series of functions, differentiation and integration of functions of several variables. Prerequisites: MATH 2415, or Math 314, or Consent of Instructor.

**MATH 511A - Real Analysis I**

Hours: 3

(Same as MATH 511) Properties of real numbers, continuity, differentiation, integration, sequences and series of functions, differentiation and integration of functions of several variables. Prerequisites: MATH 2415, or Math 314, or Consent of Instructor. Crosslisted with: MATH 511.

**MATH 512 - Real Analysis II**

Hours: 3

Properties of real numbers, continuity, differentiation, integration, sequences and series of functions, differentiation and integration of functions of several variables. Prerequisites: MATH 511.

**MATH 515 - Dynamical Systems**

Hours: 3

Topics can be chosen from discrete or/and continuous dynamical systems such as linear systems and linear algebra, local theory for nonlinear systems, local existence-uniqueness theorem, the Hartman-Grobman theorem, Liapunov functions, the stable manifold theorem, limit sets of trajectories, the Poincare-Bendixson theorem, bifurcation theory, center manifold and normal form, chaotic dynamics, iteration of functions, graphical analysis, the linear, quadratic and logistic families, fixed points, symbolic dynamics, topological conjugacy, complex iteration, Julia and Mandelbrot sets. Prerequisites: MATH 2414 and MATH 2318.

**MATH 517 - Calculus of Finite Differences**

Hours: 3

Finite differences, integration, summation of series, Bernoulli and Euler Polynomials, interpolation, numerical integration, Beta and Gamma functions, difference equations. Prerequisites: MATH 2415 or Math 314 with a minimum grade of C.

**MATH 518 - Thesis**

Hours: 3-6

This course is required of all graduate students who have an Option I degree plan. Graded on a (S) satisfactory or (U) unsatisfactory basis. Prerequisite: Consent of the instructor.

**MATH 522 - General Topology I**

Hours: 3

Ordinals and cardinals, topology of the real line, metric spaces, topological spaces, sequences, continuity and homeomorphisms on topological spaces. Prerequisites: MATH 2414 with a minimum grade of C. Crosslisted with: MATH 440.

**MATH 523 - General Topology II**

Hours: 3

The course is a continuation of MATH 522. Compact spaces, connectedness and path connectedness, separation axioms, topological manifolds, further topics on metric spaces, product spaces. Prerequisites: MATH 522 with a minimum grade "C" or consent of the instructor.

**MATH 529 - Workshop in School Mathematics**

Hours: 3

This course may be taken twice for credit. A variety of topics, taken from various areas of mathematics, of particular interest to elementary and secondary school teachers will be covered. Consult with instructor for topics.

**MATH 531 - Theory of Matrices**

Hours: 3

Covers real and complex vector spaces, subspaces, linear operators, eigenvalues and eigenvectors, inner product spaces, orthogonality, operators on inner product spaces, singular value, LU and range-null-space decompositions, pseudo-inverse, canonical forms. Prerequisites: MATH 2305 and 2414 with minimum grades of C or MATH 2318 or MATH 332 with a minimum grade of C. Crosslisted with: MATH 333.

**MATH 532 - Fourier Analysis and Wavelets**

Hours: 3

Inner Product Spaces; Fourier Series; Fourier Transform; Discrete Fourier Analysis; Haar Wavelet Analysis; Multiresolution Analysis; The Daubechies Wavelets; Applications to Signal Processing; Advanced Topics. Prerequisites: MATH 2414 with a minimum grade of C.

**MATH 533 - Linear and Nonlinear Optimization**

Hours: 3

Graphical optimization, linear programming, simplex method, interior point methods, nonlinear programming, optimality conditions, constrained and unconstrained problems, combinatorial and numerical optimization, applications. Prerequisites: MATH 333 with a minimum grade of C.

**MATH 536 - Cryptography**

Hours: 3

The course begins with some classical cryptanalysis (Vigenere ciphers, etc). The remainder of the course deals primarily with number-theoretic and/or algebraic public and private key cryptosystems and authentication, including RSA, DES, AES and other block ciphers. Some cryptographic protocols are described as well. Prerequisites: MATH 437, or MATH 537, or consent of the instructor.

**MATH 537 - Theory of Numbers**

Hours: 3

Factorization and divisibility, diophantive equations, congruences, quadratic reciprocity, arithmetic functions, asymptotic density, Riemann's zeta function, prime number theory, Fermat's Last Theorem. Prerequisites: MATH 437 or Consent of instructor.

**MATH 538 - Functions of Complex Variables I**

Hours: 3

Geometry of complex numbers, mapping, analytic functions, Cauchy-Riemann conditions, complex integration. Taylor and Laurent series, residues. Prerequisites: MATH 436, or MATH 438, Consent of Instructor.

**MATH 538A - Functions of Complex Variables I**

Hours: 3

(Same as 538) This course covers the elements of one-dimensional complex analysis. The complex numbers (their algebra, geometry, and topology); Analytic functions of a complex variable (definition, examples, properties); complex functions (complex exponential and trigonometric functions, complex logarithm and power functions), Mobius transformations, stereographic projection; Integration in the complex plane, particularly Cauchy's integral formula and its consequences; infinite series of complex numbers and complex variables, including Taylor series and Laurent series; the residue theorem and the computation of real integrals by complex methods; and conformal mapping. Prerequisites: MATH 2415 Calculus III (Min Grade C) or MATH 314 (Min Grade C). Crosslisted with: MATH 438, MATH 538.

**MATH 539 - Functions of Complex Variables II**

Hours: 3

Geometry of complex numbers, mapping, analytic functions, Cauchy-Riemann conditions, complex integration. Taylor and Laurent series, residues. Prerequisites: MATH 538.

**MATH 543 - Abstract Algebra I**

Hours: 3

Groups, isomorphism theorems, permutation groups, Sylow Theorems, rings, ideals, fields, Galois Theory. Prerequisites: MATH 334 or MATH 550, or Consent of Instructor.

**MATH 544 - Abstract Algebra II**

Hours: 3

Groups, isomorphism theorems, permutation groups, Sylow Theorems, rings, ideals, fields, Galois Theory. Prerequisites: MATH 543.

**MATH 546 - Numerical Analysis and Elements of Machine Learning**

Hours: 3

The course will include numerical methods for derivatives approximation; will teach data approximation and interpolation by Fourier series; Euler's and Runge-Kutta's methods for solving ordinary differential equations (ODE) and systems of ODE. Also, the students will study methods to approximate solutions of partial differential equations (PDE) and will learn the basics of optimization (minimization of functions) for machine learning (ML). The students will develop the important skills of knowledge and methods generalization for their computer implementation and will program the basic methods in MatLab. Some programming skills would be of help. Prerequisites: MATH 2414. Crosslisted with: CSCI 546.

**MATH 550 - Foundations of Abstract Algebra**

Hours: 3

This course will cover the fundamental properties of algebraic structures such as properties of the real numbers, mapping, groups, rings, and fields. The emphasis will be on how these concepts can be related to the teaching of high school algebra. Note: This course will be helpful to secondary teachers by giving them a better understanding of the terms and ideas used in modern mathematics. This is an elective course, eligible for the non-thesis option only. The maximum credit hours can be earned towards the MS degree in math among MATH 500, 550, 560 is six. Prerequisites: MATH 332 or or MATH 500 with a minimum grade of C. Crosslisted with: MATH 334.

**MATH 560 - Euclidean and NonEuclidean Geometry**

Hours: 3

The National Council of Teachers of Mathematics (NCTM) in its Principles and Standards states the geometric skills that students should be able to use when they finish high school.This course trains students, particularly, middle and high-school teachers for understanding and mastering these geometric skills. This is an elective course, eligible for the non-thesis option of the MS degree in math only. The maximum credit hours can be earned towards the MS degree in math among MATH 500, 550, 560 is six. Prerequisites: MATH 332 or MATH 500.

**MATH 561 - Regression Analysis**

Hours: 3

A computer oriented statistical methods course which involves concepts and techniques appropriate to design experimental research and the application of the following methods and techniques: methods of estimating parameters and testing hypotheses about them; analysis of variance, multiple regression methods, orthogonal comparisons, experimental designs with applications. Prerequisites: MATH 401 or 502, or 402 and 403.

**MATH 563 - Image Processing with Elements of Learning**

Hours: 3

This class will provide the students with an introduction to image processing, with applications to science, medicine, and industry. Students will learn methods for 2D image enhancement, sharpening, blurring, noise detection, modeling, and cleaning, as well as edge detection in gray-level images. The students will learn and will be able to implement methods like local statistics, convolution, Laplacian and Gradient operators, Fourier transforms, and the Fast Fourier Transform. Further, the teacher will introduce basic elements of neural networks (NN) and machine learning(ML). At the end of the class, the students will know which gray-level image methods apply to color images. The students will develop skills in independent study, program, experiment, report, and present advanced methods from the field. Some programming skills would be of help. Prerequisites: MATH 2414. Crosslisted with: CSCI 567.

**MATH 569 - Image Analysis and Recognition with Learning**

Hours: 3

This class will start with a study of the basic color image models. Next, the students will learn about scaling functions and calculus with them. Further will study the basics of wavelets and how to decompose a function to wavelets. Next, the students will learn about convolution-correlation, convolutional neural networks (CNN), and the fundamentals of machine learning (ML) and deep ML(DML). In the following stage, the students will learn basic image segmentation methods based on active contours (In case of time permission deep active contours). Further, they will learn about image and object representation and description, mainly boundary and region description. The following methods will be taught from the field of Recognition: Decision making; feature extraction. The students will develop skills in independent study, program, experiment, report, and present advanced methods from the field. Some programming skills would be of help. Prerequisites: MATH 2414. Crosslisted with: CSCI 569.

**MATH 572 - Modern Applications of Mathematics**

Hours: 3

This course, specifically designed for teachers, covers a range of applications of mathematics. Topics may include classical encryption, data compression ideas, coding theory, private and public key cryptography, data compression including wavelets, difference equations, GPS systems, computer tomography, polynomial interpolation/Belier curves, construction and use of mathematical models, probability theory, Markov chains, network analysis, linear programming, differentiation and integration, linear algebra, complex variables, Fourier-series, Fourier and Laplace transforms and their applications, differential equations, integral equations, calculus of variations, and topics from student presentations. Prerequisites: MATH 2414 or MATH 192 with a minimum grade of C.

**MATH 580 - Topics in the History of Mathematics**

Hours: 3

A chronological presentation of historical mathematics. The course presents historically important problems and procedures. Prerequisites: MATH 332 or MATH 500.

**MATH 589 - Independent Study**

Hours: 1-4

Individualized instruction/research at an advanced level in a specialized content area under the direction of a faculty member. Note: May be repeated when the topic varies. Prerequisites: Consent of department head.

**MATH 595 - Research Literature & Techniques**

Hours: 3

This course provides a review of the research literature pertinent to the field of mathematics. The student is required to demonstrate competence in research techniques through literature investigation and formal reporting of a problem. Graded on a (S) satisfactory or (U) unsatisfactory basis. Prerequisites: Consent of instructor.

**MATH 597 - Special Topics**

Hours: 3

Organized class. May be repeated when topics vary. Prerequisites: Consent of instructor.