**Program Overview**

Established in 2007, the Department of Mathematics provides students at the University of Sharjah with the opportunity to learn fundamental scientific and mathematical concepts in an atmosphere that is friendly, conducive to learning and encourages intellectual curiosity, exploration and independent thinking, and high ethics.

The Department offers a wide array of courses in pure and applied mathematics for all types of learners in addition to applications. More adventurous student can study advanced courses in mathematics and its applications.

Faculty members are active professionals in the fields they teach. All are graduates of prestigious universities and are active in research and self -development. The faculty, through their dedication to teaching and guidance help students develop meaningful and lasting bonds with science and mathematics, while providing invaluable skills for leading a more interesting and productivelives.

A student undertaking the BS program in Mathematics should complete a total of 123 credit hours distributed as follows:

BS in Mathematics | | | | |

| UR | CR | PR | Total |

Mandatory Core Credits | 12 | 15 | 45 | 72 |

Mandatory Support Credits | - | - | 12 | 12 |

Elective Core Courses | 12 | - | 21 | 33 |

Elective Support Credits | - | - | 6 | 6 |

Total | 24 | 15 | 84 | 123 |

**I.** **University Requirements**

The list of the University required courses and their descriptions are presented in the introductory pages of the College of Sciences section in this Bulletin.

**II.** **College Requirements**

The list of the College required courses and their descriptions are presented in the introductory pages of the College of Sciences section in this Bulletin.

**III.** **Program Requirements**

**A.** **Mandatory Core Courses**

The Department of Mathematics core courses (45 credit hours) are listed below:

Course # | Course Title | CrHrs | Prerequisites |

1440132 | Calculus II | 3 | 1440131 |

1440211 | Linear Algebra I | 3 | 1440131 |

1440231 | Calculus III | 3 | 1440132 |

1440232 | Vector Calculus | 3 | 1440231 |

1440233 | Foundations of Mathematics | 3 | 1440131 |

1440241 | Ordinary Differential Equations I | 3 | 1440132 |

1440251 | Geometry | 3 | 1440233; 144033 |

1440281 | Introduction to Probability and Statistics | 3 | 1440131 |

1440381 | Mathematical Statistics | 3 | 1440281 |

1440311 | Abstract Algebra I | 3 | 1440233 |

1440331 | Real Analysis I | 3 | 1440132; 1440233 |

1440332 | Complex Analysis | 3 | 1440231 |

1440371 | Numerical Analysis I | 3 | 1440132; 1440211 |

1440372 | Operations Research I | 3 | 1440211 |

1440492 | Graduation Project | 3 | Senior Standing |

**B.** **Mandatory Support Courses **

All Mathematics major students are required to take the following four courses (12 credits) of mandatory computer science courses.

Course # | Title | CrHrs | Prerequisites |

1411211 | Programming II | 3 | 1411116 |

1411215 | Data Structures | 3 | 1411211 |

1411246 | Object Oriented Design with Java | 3 | 1411211 |

1411263 | Introduction to Database Management Systems | 3 | 1411116 |

**C.** **Elective****Courses**

The program includes 27 credit hours of elective courses chosen from various categories; 21 credits are Mathematics core electives and 6 credits of Computer Science courses.

**Elective Core Courses**

The following courses are offered by the Mathematics Department as electives although all may not be available in a particular semester. Additional courses may be developed in the future, based on changes in the discipline and demand.

Course # | Course Title | CrHrs | Prerequisites |

1440312 | Linear Algebra II | 3 | 1440211 |

1440313 | Number Theory | 3 | 1440132; 1440233 |

1440341 | Partial Differential Equations | 3 | 1440231, 1440241 |

1440373 | Graph Theory | 3 | 1440211 |

1440411 | Abstract Algebra II | 3 | 1440311 |

1440431 | Real Analysis II | 3 | 1440331 |

1440441 | Ordinary Differential Equations II | 3 | 1440341; 1440331 |

1440451 | Topology | 3 | 1440331 |

1440471 | Numerical Analysis II | 3 | 1440371 |

1440472 | Operations Research II | 3 | 1440372 |

1440481 | Stochastic Processes | 3 | 1440381 |

1440491 | Selected Topics in Mathematics | 3 | Department's Consent |

**Elective Support Courses**

The required six-credit electives encompass two Computer Science courses selected from the following list:

Course # | Course Title | CrHrs | Prerequisites |

1411319 | Programming Languages and Paradigms | 3 | 1411215 |

1411352 | Operating Systems | 3 | 1411215 |

1411365 | Database Design and Implementation | 3 | 1411263 |

1411366 | Software Engineering | 3 | 1411215 |

1411440 | Introduction to Computer Graphics | 3 | 1411215 |

**Study Plan **

The BS program in Mathematics encompasses 123 credits hours that are spread over eight semesters and could be completed in four years. The following distribution of courses by semester facilitates student's normal progression through the study plan.

Year I, Semester 1 (16 Credits) | | | |

Course # | Title | CrHrs | Prerequisites |

0201102 or 0201105 | Arabic Language | 3 | |

1410100 | Introduction to IT | 3 | |

1430111 | Physics I | 3 | |

1430112 | Physics ILAB | 1 | |

1440131 | Calculus I | 3 | |

University Elec. I | 3 |

Year 1, Semester 2 (17 Credits) | | | |

Course # | Title | CrHrs | Prerequisites |

0202105 | English for Academic Purposes | 3 | |

1411116 | Programming I | 4 | |

1420101 | Chemistry I | 3 | |

1420102 | Chemistry I LAB | 1 | |

1440132 | Calculus II | 3 | 1440131 |

1440211 | Linear Algebra I | 3 | 1440131 |

Year 2, Semester 3 (15 Credits) | | | |

Course | Title | CrHrs | Prerequisites |

0104100 | Islamic Culture I | 3 | |

1411211 | Programming II | 3 | 1411116 |

1440231 | Calculus III | 3 | 1440132 |

1440233 | Found. of Mathematics | 3 | 1440131 |

1440281 | Intro. To Prob.& Stat. | 3 | 1440131 |

Year 2, Semester 4 (15 Credits) | | | |

Course # | Title | CrHrs | Prerequisites |

1411215 | Data Structures | 3 | 1411211 |

1440232 | Vector Calculus | 3 | 1440231 |

1440241 | Ord. Diff. Equation | 3 | 1440132 |

1411xxx | Dept. Support Elect. I | 3 | |

UnivElec (2) | 3 |

Year 3, Semester 5 (16 Credits) | | | |

Course # | Title | CrHrs | Prerequisites |

1411246 | Object Oriented Design with Java | 3 | 1411211 |

1440251 | Geometry | 3 | 1440233 |

1440311 | Abstract Algebra I | 3 | 1440233 |

1440371 | Numerical Analysis I | 3 | 1440132; 1440211 |

University Elect. III | 3 |

Year 3, Semester 6 (15 Credits) | | | |

Course # | Title | CrHrs | Prerequisites |

1440381 | Mathematical Statistics | 3 | 1440281 |

1440332 | Complex Analysis | 3 | 1440231 |

1411xxx | Dept. Support Elect. (2) | 3 | |

1441xxx | Dept. Core Elect. I | 3 | |

University Elect. (4) | 3 | ||

1440461 | Training Course | 0 | 1440281 |

Year 4, Semester 7 (12 Credits) | | | |

Course # | Tile | CrHrs | Prerequisites |

1411263 | Introduction to Database | 3 | 1411211 |

1440372 | Operations Research I | 3 | 1440211 |

1441xxx | Dept. Core Elect. (2) | 3 | |

1441xxx | Dept. Core Elect III | 3 | |

Depart. Core Elect. (4) | 3 |

Year 4, Semester 8 (15 Credits) | | | |

Course # | Title | CrHrs | Prerequisites |

1440331 | Real Analysis I | 3 | 1440132; 1440233 |

1440492 | Graduation Project | 3 | Senior Standing |

1441xxx | Dept. Core Elect.(5) | 3 | |

1441xxx | Dept. Core Elect. (6) | 3 | |

1441xxx | Dept. Core Elect. (7) | 3 |

**Course Description**

Courses in the proposed program that are offered in the Department of Mathematics start with (1440). The program of study contains courses that are offered by other departments as well as from outside the College. Consistent with the University policies, mathematics courses in the program will be assigned numbers of the form (1440ABC) where:

A | Year (level) | |

B | Areas (as follows): 1: Algebra 3: Calculus and Analysis 4: Differential Equations | 5: Geometry 7: Applied Mathematics 8: Statistics 9: Projects and Selected Topics |

C | Course sequence in area |

** **

1440131 | Calculus I | 3-0:3 |

Functions, domain and range, examples of functions. Limits and continuity. Derivatives, applications of derivatives in optimization, linearization and graphing, the Mean Value Theorem. Integration, the Fundamental Theorem of Calculus, areas, volumes of solids of revolution, arc length. Conic sections.Prerequisite: None. | | |

1440132 | Calculus II | 3-0:3 |

Functions, Inverse functions. Transcendental functions. L'Hopital's rule. Techniques of integration. Improper integrals. Sequence and infinite series of real numbers. Polar coordinates. Parametric curves in the plane.Prerequisite: 1440131. | | |

1440211 | Linear Algebra I | 3-0:3 |

Systems of linear equations, Gauss and Gauss-Jordan elimination processes. Matrix algebra, determinants, Cramer's rule. Vector spaces, subspaces, basis and dimension, rank, change of basis. Characteristic polynomial, eigenvalues and eigenvectors of square matrices, diagonalization. Inner product spaces, orthogonal projections, Gram-Schmidt process. Computer applications. Introduction to linear transformation.Prerequisite: 1440131, 1440131. | | |

1440231 | Calculus III | 3-0:3 |

Vectors and analytic geometry in space. Graphing surfaces in three dimensions. Vector–valued functions and motion in space. Functions of several variables. Extreme values and Lagrange multipliers. Multiple integrals. Areas and volumes.Prerequisite: 1440131, 1440132. | | |

1440232 | Vector Calculus | 3-0:3 |

Integration in vector fields. Line integrals, circulation and flux, path independence and conservative fields. Green's Theorem in the plane. Surface area and surface integrals. Parameterized surfaces. Stocke's and Divergence Theorems. Curvilinear coordinates. Transformation of coordinates. Introduction to Cartesian tensors.Prerequisite: 1440231. | | |

1440233 | Foundations of Mathematics | 3-0:3 |

Logic, propositional logic, truth tables, propositional formulas, logical implication and equivalence, tautologies and contradictions, quantifiers. Methods of proof. Sets, applications of sets, Venn diagrams, Cartesian product, the power set. Cardinality. Mathematical Induction. Relations and partitions, functions. Zorn's Lemma and Axiom of Choice.Prerequisite: 1440131. | | |

1440241 | Ordinary Differential Equations I | 3-0:3 |

This course covers first and higher order ordinary differential equations (ODE) with applications in various fields. It contains: Basic concepts. First order ODE's, initial value problems, an existence and uniqueness theorem. Higher order ode's with constant coefficients. Laplace transform and inverse. Power series solutions, Frobenius theorem. Introduction to Linear systems of ODE's.Prerequisite:1440132. | | |

1440251 | Geometry | 3-0:3 |

The axiomatic Systems, Finite geometry. Finite Projective Plane, Non-Euclidean geometry. Hyperbolic geometry (Sensed Parallels, Asymptotic Triangles. Saccheri Quadrilaterals, Area of Triangles, Ultraparallels, Transformation of the Euclidean Plane.Prerequisite:1440233. | | |

1440281 | Introduction to Probability and Statistics | 3-0:3 |

Descriptive statistics; Axiomatic probability; Random variables and their moments; Special discrete and continuous distributions; Sampling distributions; Estimation; Hypothesis testing; Linear regression; Analysis of variance. Prerequisite:1440131. | | |

** **

1440381 | Mathematical Statistics | 3-0:3 |

Review of basic concepts of probability, random variables and distribution theory. Distribution of functions of random variables. Expectation and moment generating functions. Unbiased and Sufficient estimators. Point estimation, optimal properties of estimators. Interval estimation. Hypotheses testing.Prerequisite:1440281. | | |

** **

1440311 | Abstract Algebra I | 3-0:3 |

Groups. Subgroups. Quotient groups and homomorphisms. Introduction to rings and fields. Ideals. Ring homomorphisms and quotient rings. Applications.
| | |

1440312 | Linear Algebra II | 3-0:3 |

Linear transformations. Change of basis, transition matrix and similarity. Nilpotent linear transformations and matrices. Canonical representation of matrices, Jordan canonical forms. Linear functionals and the dual space. Bilinear forms. Quadratic forms and real symmetric bilinear forms. Complex inner product spaces. Normal operators. Unitary operators. The spectral theorem.Prerequisites:1440211 and 1440233. | | |

1440313 | Number Theory | 3-0:3 |

Divisibility. Prime numbers. Euclidean algorithm. Linear congruences. The Chinese remainder theorem. Fermat's little theorem. Wilson's theorem. Euler's theorem. Quadratic residues and reciprocity laws. Diophantine equations. Fermat's last theorem. Applications to cryptology and primality tests. Other possible applications.Prerequisites:1440132 and 1440233. | | |

1440331 | Real Analysis I | 3-0:3 |

Sequences and Cauchy sequences of real numbers. Topology of the real line. The Bolzano-Weierstrass theorem. The Heine-Borel theorem. Limits, continuity, uniform continuity and differentiability of real-valued functions. The Mean Value Theorem. L'Hopital's rule. The Riemann integral.Prerequisites: 1440132 and 1440233. | | |

1440332 | Complex Analysis | 3-0:3 |

Complex numbers; Analytic functions; Derivatives; Differentiation; Cauchy-Riemann equations; Polar coordinates; Harmonic functions; Elementary functions; Integrals; Complex-valued functions; Antiderivatives; Cauchy-Goursat theorem; Cauchy–integral formulas; Morera's theorem; Liouville's theorem; Fundamental Theorem of algebra; Series; Taylor and Laurent series; Power series, Integration and differentiation of power series; Residues and poles.Prerequisite:1440231. | | |

1440341 | Partial Differential Equations | 3-0:3 |

First order partial differential equations, the method of characteristics. Classification of second order pde's: parabolic, elliptic, and hyperbolic. The canonical form. Boundary value problems with applications to physical sciences and engineering. Detailed analysis of the wave, heat and Laplace equations; Separation of variables. Application of Fourier theory.Prerequisite: 1440241. | | |

1440371 | Numerical Analysis I | 3-0:3 |

Error analysis. Roots of nonlinear equations: bisection, fixed point, secant and Newton's methods. Systems of linear equations: direct methods, iterative methods. Systems of nonlinear equations: Newton's method. Interpolation: Lagrange, Newton's formulas, Gaussian quadrature. Approximation theory: orthogonal polynomials (Legendre, Laguerre, Chebychev, Hermite), Gram-Schmidt process, LS approximation. Numerical differentiation and integration: trapezoidal, Simpson, Newton-Cotes formulas. Prerequisites: 1440132; 1440211. | | |

1440372 | Operations Research I | 3-0:3 |

Linear Programming. The simplex method, duality, sensitivity analysis, various versions of the simplex method. Transportation models. Network models. Nonlinear programming. Constrained and unconstrained optimization, KKT conditions.Prerequisites:1440211, 1440231. | | |

** **

1440373 | Graph Theory | 3-0:3 |

Introduction to graphs. Representation of graphs. Graph isomorphism, connectivity. Euler and Hamilton paths. Shortest path problems. Planarity, graph coloring. Trees, tree traversal, sorting, spanning trees, matching. Networks, max flow.Prerequisite: 1440211. | | |

** **

1440411 | Abstract Algebra II | 3-0:3 |

Unique factorization domains. Modules and sub-modules. Field extensions. Finite Fields. Introduction to Galois theory. Applications.Prerequisite:1440311. | | |

1440431 | Real Analysis II | 3-0:3 |

The Riemann-Stieltjes integral and functions of bounded variation. Metric spaces. Pointwise and uniform convergence of sequences of functions in metric spaces. Completeness of the space C(X,Y) of continuous functions. Pointwise and uniform convergence of infinite series of real-valued functions.Prerequisite:1440331. | | |

1440441 | Ordinary Differential Equations II | 3-0:3 |

Existence and uniqueness of solutions. Some fixed point theorems. Matrix analysis of differential equations. Second order differential equations in phase plane. Lyapunov functions. Stability of equilibria. Qualitative theory. Autonomous systems in one and two dimensions. Phase portraits, stability. Sturm-Liouville theory: eigenvalues and eigenfunctions.Prerequisites:1440241 and 1440331. | | |

1440451 | Topology | 3-0:3 |

Topological spaces. Open and closed sets. Bases and sub-bases. Interior, exterior and boundary points. The closure of a set. Continuous functions. Homeomorphisms. Product spaces. Axioms of countability and separability. Compact spaces. Connected spaces. Metric spaces.Prerequisite:1440331. | | |

1440471 | Numerical Analysis II | 3-0:3 |

Numerical solution of ordinary differential equations. One-step methods: Euler, Taylor, Runge-Kutta. Multistep methods.The eigenvalue problem: power and inverse power methods. Numerical solution of boundary value problems: finite difference and shooting methods. Numerical solution of partial differentialequations: Difference methods.Prerequisite:1440371. | | |

1440472 | Operations Research II | 3-0:3 |

Dynamic programming. Integer programming. Inventory models. Introduction to Game Theory. Queuing theory. Simulation models. Markov chains. Nonlinear programming algorithms: unconstrained optimization, constrained optimization. Prerequisite:1440372. | | |

1440481 | Stochastic Processes | 3-0:3 |

Revision of probability. Bernoulli processes and sum of independent random variables. Poisson processes. Markov chains and their application to queuing theory and branching process. Markov processes. Renewal process.Prerequisite: 1440381. | | |

1440491 | Selected Topics in Mathematics | 3-0:3 |

Senior standing; Consent of the department.Prerequisite: Senior standing; Consent of the department. | | |

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