# Mathematics MS

## Master of Science in Mathematics - Option I Thesis

The MS in Mathematics requires 8 courses and a thesis (6 credit hours), a total of 30 credit hours, as follows:

Thesis | ||

MATH 518 | Thesis | 3-6 |

Only 6 semester hours of credit for 518 per degree will be given upon satisfactory completion of the requirement. | ||

Core Courses: 12 semester hours (4 courses) including one complete sequence from: | ||

MATH 501 | Mathematical Statistics I | 3 |

MATH 502 | Mathematical Statistics II | 3 |

or | ||

MATH 511 | Introduction to Real Analysis I | 3 |

MATH 512 | Introduction to Real Analysis II | 3 |

or | ||

MATH 522 | General Topology I | 3 |

MATH 523 | General Topology II | 3 |

or | ||

MATH 538 | Functions of Complex Variables I | 3 |

MATH 539 | Functions of Complex Variables II | 3 |

or | ||

MATH 543 | Abstract Algebra I | 3 |

MATH 544 | Abstract Algebra II | 3 |

Complete 12 semester hours (4 additional courses) from either the core section or from the following: | ||

MATH 515 | Dynamical Systems | 3 |

MATH 517 | Calculus of Finite Differences | 3 |

MATH 532 | Fourier Analysis and Wavelets | 3 |

MATH 533 | Optimization | 3 |

MATH 536 | CRYPTOGRAPHY | 3 |

MATH 537 | Theory of Numbers | 3 |

MATH 546 | Numerical Analysis | 3 |

MATH 561 | Regression Analysis and Experimental Design | 3 |

MATH 563 | Image Processing with Applications | 3 |

MATH 580 | Topics in the History of Mathematics | 3 |

MATH 597 | Special Topics (approval by the math department) | 3 |

Total Hours | 30 |

## Master of Science in Mathematics Option II Non-Thesis

The MS in Mathematics requires 11 courses and a project (Math 595, 3 credit hours), a total of 36 credit hours, as follows:

Core Courses: 12 semester hours (4 courses) including one complete sequence from: | ||

MATH 501 | Mathematical Statistics I | 3 |

MATH 502 | Mathematical Statistics II | 3 |

or | ||

MATH 511 | Introduction to Real Analysis I | 3 |

MATH 512 | Introduction to Real Analysis II | 3 |

or | ||

MATH 522 | General Topology I | 3 |

MATH 523 | General Topology II | 3 |

or | ||

MATH 538 | Functions of Complex Variables I | 3 |

MATH 539 | Functions of Complex Variables II | 3 |

or | ||

MATH 543 | Abstract Algebra I | 3 |

MATH 544 | Abstract Algebra II | 3 |

Complete 9 semester hours (3 courses) from either the core section or from the following: | ||

MATH 515 | Dynamical Systems | 3 |

MATH 517 | Calculus of Finite Differences | 3 |

MATH 532 | Fourier Analysis and Wavelets | 3 |

MATH 533 | Optimization | 3 |

MATH 536 | CRYPTOGRAPHY | 3 |

MATH 537 | Theory of Numbers | 3 |

MATH 546 | Numerical Analysis | 3 |

MATH 561 | Regression Analysis and Experimental Design | 3 |

MATH 563 | Image Processing with Applications | 3 |

MATH 580 | Topics in the History of Mathematics | 3 |

MATH 597 | Special Topics (approval by the math department) | 3 |

Electives | ||

An additional 12 semester hours (4 graduate level electives) from any graduate courses offered by the mathematics department, or outside courses with the approval of the mathematics department. | 12 | |

Research Project | ||

MATH 595 | Research Literature & Techniques (3 semester hours required) | 3 |

Total Hours | 36 |

**MATH 500 - Discrete Mathematics**

Hours: 4

Discrete Mathematics. Four semester hours. Study of formal logic; sets; functions and relations; principle of mathematical induction; recurrence relations; and introductions to elementary number theory; counting (basic combinatorics); asymptotic complexity of algorithms; graph theory; and NPcompleteness. This course is useful to those taking graduate classes in computer science. It may be taken for graduate credit towards a masters in mathematics only by consent of the department. Prerequisite: Consent of the instructor.

**MATH 501 - Mathematical Statistics I**

Hours: 3

Probability, distributions, moments, point estimation, maximum likelihood estimators, interval estimators, test of hypothesis. Prerequisites: MATH 314.

**MATH 502 - Mathematical Statistics II**

Hours: 3

Probability, distributions, moments, point estimation, maximum likelihood estimators, interval estimators, test of hypothesis. Prerequisites: MATH 501.

**MATH 503 - Introduction to 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 401 or equivalent.

**MATH 511 - Introduction to 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 314 or Consent of Instructor.

**MATH 512 - Introduction to 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

Dynamical Systems. Three semester hours. Iteration of functions; graphical analysis; the linear, quadratic and logistic families; fixed points; symbolic dynamics; topological conjugacy; complex iteration; Julia and Mandelbrot sets. Computer algebra systems will be used. Recommended background; MATH 192 and MATH 331.

**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 314.

**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

General Topology I - Three semester hours Ordinals and cardinals, topological spaces, identification topology, convexity, separation axioms, covering axioms. Pre-requisites : MATH 440 or consent of instructor.

**MATH 523 - General Topology II**

Hours: 3

The course is a continuation of MATH 522. Compact spaces, metric spaces, product spaces, convergence, function spaces, path connectedness, homotopy, fundamental group. Prerequisites: MATH 522.

**MATH 529 - WORKSHOP**

Hours: 3

Workshop in School Mathematics. Three semester hours. 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 - Intro Theory of Matrices**

Hours: 3

Introduction to Theory of Matrices. Three semester hours. Vector spaces, linear equations, matrices, linear transformations, equivalence relations, metric concepts. Prerequisite: MATH 334 or 335.

**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 335 or the Consent of the instructor.

**MATH 533 - Optimization**

Hours: 3

Linear and Nonlinear Optimization - Three semester hours Graphical optimization, linear programming, simplex method, interior point methods, nonlinear programming, optimality conditions, constrained and unconstrained problems, combinatorial and numerical optimization, applications. Pre-requisites : MATH 335 or the consent of the instructor

**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 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**

Hours: 3

The course will include numerical methods for derivatives and integrals approximation, teach Euler's and Runge-Kutta methods for solving ordinary differential equations, and study methods for approximate solution of partial differential equations (PDE), including parabolic PDE. Students will learn also how to program the basic methods in MatLab, improving their skills in working with this software. Prerequisites: Consent of the instructor or MATH 314.

**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. Prerequisites: MATH 331 or MATH 500.

**MATH 560 - Euclidean and NonEuclidean Geometry**

Hours: 3

This course is specifically designed for middle and high-school teachers. The National Council of Teachers of Mathematics (NCTM) explains in its Principles and Standards the geometric skills students should be able to use by the time they finish high school are: (1) analyze characteristics and properties of two and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; (2) specify locations and describe spatial relationships using coordinate geometry and other representational systems; (3) apply transformations and use symmetry to analyze mathematical situations; and (4) use visualization, spatial reasoning, and geometric modeling to solve problems. Prerequisites: MATH 331 or MATH 500.

**MATH 561 - Regression Analysis and Experimental Design**

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.

**MATH 563 - Image Processing with Applications**

Hours: 3

Introduction to image processing, with applications to images from medicine, agriculture, satelite imagery, physics, etc. Students will learn techniques such as edge detection, 2D image enhancement using laplacian and gradient operators, fourier transforms and the FFT, filtering, and wavelets, as time allows. Students will acquire practical skills in image manipulation by implementing the above mentioned algorithms. Prerequisites: MATH 314 or consent of the instructor.

**MATH 569 - Image Analysis and Recognition**

Hours: 3

Description: This class will start with an introduction to color image processing using vector functions. The basics of Wavelets theory will be developed in order to expand a function for the purpose of multiresolution imaging. The following step is the objects representation and description. Then, basic Mathematical Morphology operations will be formulated. Two different set of methods will be taught from the field of Objects/Pattern Recognition: Decision theoretic methods- build up a decision function on the base of a metric; structural methods- based on correlation; and templates with radial and circular lines. The course will end with teaching methods for image segmentation to objects and background. The students program image analysis methods or their components in Java/C++/Matlab. Prerequisites: Instructor approval. Crosslisted with: CSCI 569.

**MATH 571 - Higher Order Approximations**

Hours: 3

This course, specifically for teachers, explores algebra-based techniques for powerful, highly accurate numerical approximations. Graphing calculators and some computer software will be used. Approximations for areas and volumes of regions, solutions to equations and systems of equations, sums of infinite series, values of logarithmic and trigonometric functions and other topics are covered. Prerequisites: MATH 192.

**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 192.

**MATH 573 - Calculus of Real and Complex Functions**

Hours: 3

This course is designed for teachers and explores similarities and differences between functions whose domain and range consist of sets of real numbers and sets of complex numbers. Complex numbers are reviewed with nontraditional applications to plane geometry. Alternate approaches to the meaning of the derivative are given so as to provide links between the notions of f (x) and f (z) (x real, z complex), and ways of understanding derivatives of inverse functions and composite functions. The geometry of functions of a complex number are explored. Cauchy-Riemann equations are derived and utilized. Power series in both the real and complex context are compared. Prerequisites: MATH 192.

**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 331.

**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.