Course Curriculum for Master Degree in Mathematics The Master Degree in Mathematics, is awarded by the Faculty of Graduate Studies at Jordan University of Science and Technology (JUST) upon the fulfillment of the following requirements: 1. Compliance with the J.U.S.T. Master Degree regulations approved by the Dean Council (No. 492/2006), dated 8/8/2006. 2. Successful completion of (34) credit hours in one of the following tracks: First: Thesis Track 1. Compulsory Requirements: (16) credit hours as follows: MATH 701 Advanced Methods in Applied Mathematics 3 MATH 707 Real Analysis I 3 MATH 721 Numerical Analysis I 3 MATH 731 Probability Theory 3 MATH 741 Abstract Algebra I 3 MATH 791 Seminar 1 2. Elective Requirements: (9) credit hours from the following * : MATH 708 Complex Analysis I 3 MATH 709 Real Analysis II 3 MATH 722 Numerical Analysis II 3 MATH 732 Applied Statistics 3 MATH 733 Mathematical Statistics 3 MATH 742 Abstract Algebra II 3 MATH 761 Topology 3 MATH 771 Linear Programming 3 MATH 777 Simulations 3 MATH 792 Advanced Topics in Mathematics 3 MATH 793 Advanced Topics in Statistics 3 MATH 794 Advanced Topics in Operation Research 3 MATH 795 Independent Studies 3 * The student may study not more than 3 credit hours from courses of 700 or 800 level offered by other programs related to his field of study upon approval of the Dean of Graduate Studies based on the recommendation of the departmental graduate studies committee. 1
3. Master Thesis (MATH 799): Total of 9 credit hours as follows: MATH 799 A Master Thesis 9 MATH 799 B Master Thesis 6 MATH 799 C Master Thesis 3 MATH 799 D Master Thesis 0 Second: Comprehensive Exam Track 1. Compulsory Requirements: (25) credit hours as follows: Course Number Course Name Credit Hours MATH 701 Advanced Methods in Applied Mathematics 3 MATH 707 Real Analysis I 3 MATH 708 Complex Analysis I 3 MATH 721 Numerical Analysis I 3 MATH 731 Probability Theory 3 MATH 741 Abstract Algebra I 3 MATH 761 Topology 3 MATH 771 Linear Programming 3 MATH 791 Seminar 1 MATH 798 Comprehensive Exam 0 2. Elective Requirements: (9) credit hours from the following * : MATH 709 Real Analysis II 3 MATH 722 Numerical Analysis II 3 MATH 732 Applied Statistics 3 MATH 733 Mathematical Statistics 3 MATH 742 Abstract Algebra II 3 MATH 777 Simulations 3 MATH 792 Advanced Topics in Mathematics 3 MATH 793 Advanced Topics in Statistics or Operation Research 3 MATH 794 Advanced Topics in Operation Research 3 MATH 795 Independent Studies 3 * The student may study 6 credit hours from courses of 700 or 800 level offered by other programs related to his field of study upon approval of the Dean of Graduate Studies based on the recommendation of the departmental graduate studies committee. 3. Passing the Comprehensive Exam (Math 798): zero credit hour. 2
Course Description MATH 701 Advanced Methods in Applied Mathematics: (3 Credit Hours) Integral equations: Introduction, transformation to a DE, solution of IE, Fredholm theorems, methods of solution of IE. Calculus of variations: variations of a functional, Euler's equations, the case of several variables, fixed end point theorem for n unknown functionals, functionals depending on high-order derivatives, variations problems with subsidiary conditions, the general variation functional, Weierstrass-Erdmann conditions. The canonical form of Euler equation, Legendre transformation, principle of least action, conservation laws, Hamilton- Jacobi equations, Jacobi's theorem. Perturbation theory. MATH 707 Real Analysis I: (3 Credit Hours) Introduction, Lebesgue measure and Lebesgue integrals, differentiation, classical Banach spaces (-spaces), bounded linear functional on - spaces. MATH 708 Complex Analysis I: (3 Credit Hours) Stereographic projections, power series, analytic functions, Mobius transformation. Complex integration, power series representation of analytic functions, singularities, residues, the argument principle, maximum modulus theorem. Analytic continuation, Schwarz reflection principle, Schwarz lemma. Harmonic functions. MATH 709 Real Analysis II: (3 Credit Hours) Algebra, δ-algebra, measure spaces, measurable functions, integration, P )µ( L classes, convex functions, outer measure, sign measure, Hahn-Banach and Jordan theorems, absolutely continuously, Radon-Nickodem theorem, derivative of sign measure, Fubini theorem. MATH 721 Numerical Analysis I: (3 Credit Hours) Approximation of functions, Numerical Integration and Differentiation, Numerical Solution of Ordinary and Partial Differential Equations, Approximation of Eigenvalues of Matrices. MATH 722 Numerical Analysis II: (3 credit hours) Numerical Solution of Partial Differential Equations. Finite-Difference methods, Weighted remainder method, Stability and convergence of solutions, Monte-Carlo method, Finite- Element method. MATH 730 Advanced Biostatistics: (2 Credit Hours) Screening tests and disease diagnostics, probability distributions, estimation, testing hypothesis, comparison of two means, inference on proportions, inference on variances, analysis of variance, analysis of contingency tables, regression analysis. MATH 731 Probability Theory: (3 Credit Hours) Kolmogorov-axioms, probability spaces, random variables, distribution functions, distributions of functions of random variables, order statistics, expected values, probability inequalities, independence, conditional expectation, Borel-Cantelli lemma, characteristic functions, modes of convergence, laws of large numbers, central limit theorem. 3
MATH 732 Applied Statistics: (3 Credit Hours) Simple and multiple regression, estimation, testing hypothesis, and model diagnostics. Introduction to experimental design, single factor experiments, completely randomized design, block designs, Latin square designs, factorial experiments, nested and crossed designs, and split-plot designs, fixed, random, and mixed effect models. MATH 733 Mathematical Statistics: (3 Credit Hours) Transformation of random variables, point estimation, unbiased estimators, uniformly minimum variance unbiased estimators, sufficient and minimum sufficient statistics, completeness, Cramer-Rao inequality, moment estimators, maximum likelihood estimators, Bayes estimators, asymptotic properties of estimators. MATH 741 Abstract Algebra I: (3 Credit Hours) Rings, Subrings, Ideals and Ring Homomorphisms, Integral Domains, Quotient Fields and Localization, Polynomial Rings, UFD s, PID s and ED s. Modules, Module Homomorphisms and Quotients, Direct Sums, Exact Sequences, Free Modules, Finitely Generated and Free Modules over a PID, Complemented Submodules. Matrices Representation of Homomorphisms, Canonical Forms: Rational Canonical Forms, Jordan Form. MATH 742 Abstract Algebra II: (3 Credit Hours) Fields, Extensions: Algebraic and Transcendental Elements, Degree of an Extension, Algebraic Closure, Separable Extensions, Normal Extensions, Normal Closure, Splitting Fields. Field Automorphisms, Galois Group of an Extension, Galois Group of a Polynomial. The Fundamental Theorem of Galois Theory, Solvable Groups, applications: Solvability by radicals, Ruler and Compass Constructions, Finite Fields. MATH 761 Topology: (3 Credit Hours) Metric Spaces, Homeomorphisms, Subspace Topology, Quotient Topology, Compact and Hausdorff Spaces, Connected and Path Connected Spaces, Pancake Problems (Applications), Manifolds and Surfaces, Homotopic Functions, Homotopy Equivalence, The Fundamental Group, The Fundamental Group of the Circle, Computation of the Fundamental Group, Brower's Fixed Point Theorem. MATH 771 Linear Programming: (3 Credit Hours) Topics will include formulation of models, the simplex method, geometry of the simplex method, duality, sensitivity analysis, network models, transportation and assignment problems. MATH 777 Simulations: (3 Credit Hours) Basic simulation modeling, review of basic probability and statistics, selecting input probability distributions, random number and random variates generation, output analysis. Comparing alternative systems configuration, variance reduction techniques. MATH 792 Advanced Topics in Mathematics: (3 Credit Hours) Different topics per department needs in mathematics which is not covered by the courses offered in the department such as time series, sampling and testing hypotheses. MATH 793 Advanced Topics: (3 Credit Hours) Different topics per department needs in Statistics which is not covered by the courses offered in the department such as time series, sampling and testing hypotheses. 4
MATH 794 Advanced Topics in Operation Research: (3 Credit Hours) Different topics per department needs in Operation Research which is not covered by the courses offered in the department such as programming of natural numbers. MATH 795 Independent Studies: (3 Credit Hours) This course introduces new aspects in different branches of mathematics. MATH 798 Comprehensive Exam: (Zero Credit Hour) In this course the student will set for an exam that includes all topics addressed throughout his academic program. Comprehensive exam will be held inside faculty of science/department of Mathematics and Statistics under the supervision of specialized faculty members. MATH 799: Master Thesis: (9 Credit Hours) A topic for the thesis will be chosen for the student where he is expected to work under the supervision of one or more academic staff. The student is expected to conduct and write up the thesis and successfully defense his work at the end of his study. 5