Bachelor of Science in Mathematics

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
 URCRPRTotal
Mandatory Core Credits12154572
Mandatory Support Credits--1212
Elective Core Courses12-2133
Elective Support Credits--66
Total241584123


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 TitleCrHrsPrerequisites
1440132Calculus II31440131
1440211Linear Algebra I31440131
1440231Calculus III31440132
1440232Vector Calculus31440231
1440233Foundations of Mathematics31440131
1440241Ordinary Differential Equations I31440132
1440251Geometry31440233; 144033
1440281Introduction to Probability and Statistics31440131
1440381Mathematical Statistics31440281
1440311Abstract Algebra I31440233
1440331Real Analysis I31440132; 1440233
1440332Complex Analysis31440231
1440371Numerical Analysis I31440132; 1440211
1440372Operations Research I31440211
1440492Graduation Project3Senior 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 #TitleCrHrsPrerequisites
1411211Programming II31411116
1411215Data Structures31411211
1411246Object Oriented Design with Java31411211
1411263Introduction to Database Management Systems31411116


C.    ElectiveCourses

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 TitleCrHrsPrerequisites
1440312Linear Algebra II31440211
1440313Number Theory31440132; 1440233
1440341Partial Differential Equations31440231, 1440241
1440373Graph Theory31440211
1440411Abstract Algebra II31440311
1440431Real Analysis II31440331
1440441Ordinary Differential Equations II31440341; 1440331
1440451Topology31440331
1440471Numerical Analysis II31440371
1440472Operations Research II31440372
1440481Stochastic Processes31440381
1440491Selected Topics in Mathematics3Department's Consent

 

Elective Support Courses

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

Course #Course TitleCrHrsPrerequisites
1411319 Programming Languages and Paradigms 31411215
1411352 Operating Systems 3 1411215
1411365Database Design and Implementation31411263
1411366Software Engineering 31411215
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 #TitleCrHrsPrerequisites

0201102

or  0201105

Arabic Language 3 
1410100Introduction to IT3 
1430111Physics I 3 
1430112Physics ILAB1 
1440131Calculus I3 
 University Elec. I 3 

 

Year 1, Semester 2 (17 Credits)
Course #TitleCrHrsPrerequisites
0202105English for Academic Purposes3 
1411116Programming I4 
1420101Chemistry I 3 
1420102Chemistry I LAB1 
1440132Calculus II31440131
1440211Linear Algebra I31440131

 

Year 2, Semester 3 (15 Credits)
CourseTitleCrHrsPrerequisites
0104100Islamic Culture I 3 
1411211Programming II31411116
1440231Calculus III31440132
1440233Found. of Mathematics31440131
1440281Intro. To Prob.& Stat.31440131

 

Year 2, Semester 4 (15 Credits)
Course #TitleCrHrsPrerequisites
1411215Data Structures31411211
1440232Vector Calculus31440231
1440241Ord. Diff. Equation31440132
1411xxxDept. Support Elect. I 3 
 UnivElec (2)3 

 

Year 3, Semester 5 (16 Credits)
Course #TitleCrHrsPrerequisites
1411246Object Oriented Design with Java31411211
1440251Geometry31440233
1440311Abstract Algebra I31440233
1440371Numerical Analysis I31440132; 1440211
 University Elect.  III  3 

 

Year 3, Semester 6 (15 Credits)
Course #TitleCrHrsPrerequisites
1440381Mathematical Statistics 31440281
1440332Complex Analysis31440231
1411xxxDept. Support Elect. (2)3 
1441xxxDept. Core Elect. I 3 
 University Elect. (4)3 
1440461Training Course01440281

 

Year 4, Semester 7 (12 Credits)
Course #TileCrHrsPrerequisites
1411263Introduction to Database31411211
1440372Operations Research I31440211
1441xxxDept. Core Elect. (2)3 
1441xxxDept. Core Elect  III  3 
 Depart. Core Elect. (4)3 

 

Year 4, Semester 8 (15 Credits)
Course #TitleCrHrsPrerequisites
1440331Real Analysis I31440132; 1440233
1440492Graduation Project3Senior Standing
1441xxxDept. Core Elect.(5)3 
1441xxxDept. Core Elect. (6)3 
1441xxxDept. 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:

AYear (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

CCourse sequence in area 

 

1440131  Calculus I3-0:3
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 II3-0:3
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 I3-0:3
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 III3-0:3
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 Calculus3-0:3
 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 Mathematics3-0:3
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.

 

1440241Ordinary Differential Equations I3-0:3
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.

 

1440251Geometry3-0:3
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.

 

1440281Introduction to Probability and Statistics3-0:3
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.

 

1440381Mathematical 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.

 

1440311Abstract Algebra I3-0:3

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

Prerequisite:144023.

 

1440312Linear Algebra II3-0:3
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.

 

1440313Number Theory3-0:3
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.

 

1440331Real Analysis I3-0:3
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.

 

1440332Complex Analysis3-0:3
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.

 

1440341Partial Differential Equations3-0:3
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.

 

1440371Numerical Analysis I3-0:3
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.

 

1440372Operations Research I3-0:3
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 Theory3-0:3
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.

 

1440411Abstract Algebra II3-0:3
Unique factorization domains. Modules and sub-modules. Field extensions. Finite Fields.  Introduction to Galois theory. Applications.Prerequisite:1440311.

 

1440431Real Analysis II3-0:3
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.

 

1440441Ordinary Differential Equations II3-0:3
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.

 

1440451Topology3-0:3
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.

 

1440471Numerical Analysis II3-0:3
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.

 

1440472Operations Research II3-0:3
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.

 

1440481Stochastic Processes3-0:3
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.

 

1440491Selected Topics in Mathematics3-0:3
Senior standing; Consent of the department.Prerequisite: Senior standing; Consent of the department.