# Mathematics Department

## Course Description

**41101 Calculus 1**

This course reviews the concepts of functions, limits, continuity and derivatives, tangent and normal lines, local extreme, concavity, related rates, vertical and horizontal asymptotes, the mean value theorem of differentiation and its applications, the definite integral, the fundamental theorem of calculus, the indefinite integral, applications of the definite integral, and area.

**41102 Calculus II **

This course treats solids of revolutions, volumes using cylindrical shells, disks and washers, arc length and surface area of revolution, the transcendental functions, the general exponential and logarithmic functions, the L-Hopital rule, the hyperbolic functions, the inverse function of the trigonometric and hyperbolic function, techniques of integration, integration by parts, trigonometric substitutions, partial fractions, quadratic expressions, improper integrals, sequences and infinite series, convergence and divergence, positive term series, alternating series, absolute and conditional convergence, power series, and Taylor and Maclaurin series

**41104 General Mathematics**

This course examines rectangular coordinates, linear and quadratic functions, slope of curve, the derivative and implicit differentiation, maxima and minima, curve sketching, exponential and logarithmic functions, the definite integral, matrices, determinants, solution of systems of linear equations, and linear programming problems (graphical method ).

**41201 Calculus III**

This course treats conic sections, parametrizations of plane curves, polar coordinates, vectors in R2 and R3: Lines, planes, surfaces, space curves, cylindrical and spherical coordinate systems, vector-valued functions, the calculus of vector-valued functions, curvature, tangential and normal components of acceleration, functions of several variables, limits, continuity, differentiation, the chain rule, gradient, tangent planes, extremes of functions of two variables, Lagrange multipliers, double integrals, and triple integrals.

**41202 Ordinary Differential Equations I**

This course includes introduction and classification, solutions of first order differential equations and their applications, growth and decay problems and linear motion problems, solutions of higher-order linear differential equations and their applications, series solution near ordinary point, Laplace transform, and linear systems of differential equations.

**41203 Selected Topics in Mathematics**

This course includes study of transcendental functions, general exponential and logarithmic functions, inverse trigonometric functions, their derivatives and integrals, techniques of integration, functions of several variables, multiple integrates, first-order differential equations and applications and higher-order differential equations.

**41211 Real Analysis I**

This course examines real numbers, order, absolute value, bounded subsets, completeness property, Archimedean property supremum and infimum, sequence limits, Cauchy sequence, recurrence sequence, increasing and decreasing sequences, function limits, continuity, uniform continuity, the relationship between continuity and uniform continuity, differentiability, the definition of derivative, the relationship between differentiability and continuity, Rolle’s theorem, mean value theorems, and Darboux’s theorem.

**41230 Statistical Methods**

This course treats the concept of inferential statistics, the null and alternative hypothesis, test function, the power of test, statistical errors, parametric tests, nonparametric test, the analysis of Variances (ANOVA), and index numbers.

**41231 Biostatistics**

This course reviews topics such as types of data, vital statistics, plots, measures of location and variation, correlation and association, probability, binomial, poisson and multinomial distributions, probit analysis, chi-square test of independence, sign and rank tests, normal and T distributions, and tests for means and proportions (ANOVA).

**41232 Theory of Probability ** This course examines subjects such as distributions of random variable, conditional probability and stochastic independence, some special distributions (discrete and continuous distributions), univariate, bivariate and multivariate distributions, distributions of functions of random variables (distribution function method, moment generating function method, and the Jacobian transformation method), and limiting distributions.

**41241 Linear Algebra 1 **

This course examines systems of linear equations, Gaussian elimination, homogeneous systems, matrices and matrix arithmetic, the inverse of a matrix, determinants, evaluating determinants, properties of determinants, cofactor expansion, Cramer's rule, vector spaces, subspaces, linear dependence, bases and dimension, row and column spaces, null space, rank and nullity, Eigen values and Eigen vectors, linear transformations, kernel and range, and invertible transformations.

**41251 Foundation of Mathematics**

This course introduces mathematical logic, methods of proof, the concept of a set, relations and functions, finite and infinite sets, denumerable sets and non-denumerable sets, cardinal numbers, the Schroder-Bernstein theorem, and the axiom of choice and some of its equivalent forms.

**41252 Discrete Mathematics**

This course treats topics such as logic, methods of proof, Boolean algebra, sets, relations, functions, ordered relations, groups and semigroups (definitions and examples), counting principles, mathematical induction, recursive relations, permutations, graphs and trees.

**41261 Non-Euclidean Geometry**

This course includes study of the axiomatic approach to modern mathematics, with emphasis on geometry, Eucid’s postulates, incidence geometry, Hilbert geometry, neutral geometry, and hyperbolic and elliptic geometries.

**41301 Partial Differential Equations **This course includes classification, some physical models (heat, wave, Laplace equations), separation of variables, Sturm-Liousville BVP, Fourier series and Fourier transform, BVP involving rectangular and circular regions, and BVP involving cylindrical and spherical regions.

**41304 Vector Analysis **This course treats vector differential calculus, gradient, divergence, curl, curvilinear coordinates, vector integral calculus, line integral, surface integral volume integral, Green’s theorem, Stocke’s theorem, divergence theorem, general curve linear coordinates, and applications.

**41311 Real Analysis II **This course includes study of Riemann integrals, sequence of functions, convergence and uniform convergence, approximation theorems (Stone, Weierstrass theorems), series of functions, absolute and uniform convergence, Gauchy criterion, Weierstrass M-test, Dirichlet test and Abel test, and metric spaces.

**41312 Complex Analysis I **

This course examines complex numbers, geometric interpretation, polar form, exponential form, powers and roots, regions in the complex plane, analytic functions, functions of complex variables, exponential and logarithmic functions, trigonometric and hyperbolic functions, definite integrals, Cauchy theorem, Cauchy integral formula, series such as the Taylor series and Laurrent series, integration and differentiation of power series, zeros of analytic functions, singularity, principle part, residues, poles, residue theorem of a function, residues at poles, evaluation of improper integrals, integration through a branch cut.

**41318 Linear Programming**

This course includes the linear programming model, convex sets, graphic method, the simplex model, matrix representations of the simplex table, duality, the dual simplex algorithm, integer linear programming, sensevity analysis, the transportation problem , and computational techniques.

**41320 Methods of Numerical Analysis **

This course covers numerical solutions of non-linear equations, interpolation, numerical differentiation and integration, direct methods for solving systems of linear equations, and the use of sophisticated software for numerical problems.

**41321 Numerical Analysis I **This course introduces numerical solution of non-linear equations, bisection method, fixed point iterations, the Newton-Raphson method, secant method, false position method, interpolation, numerical

differentiation and integration, andnumerical solution of ordinary differential equation of the first order.

**41325 Numerical Analysis Lab **This course includes study of underlying mathematical principles and the use of sophisticated software for numerical problems such as spline interpolation, ordinary differential equations, nonlinear equations, differentiation, and integration.

**41332 Mathematical Statistics **This course includes topics such as point estimation, confidence interval, statistical tests, the UMP test, likelihood ratio tests, chi-square tests, SPRT, non-parametric methods, sufficient statistics and its properties, complete statistics exponential family, Fisher information and the Rao-Cramer inequality.

**41335 Operation Research **This course introduces the basics of O.R. linear programming, transportation model, assignment model, sequencing models, decision theory and investment analysis, and queuing models.

**41341 Abstract Algebra I **

This course addresses binary operations, groups, subgroups and cyclic subgroups, symmetric and alternating groups, cyclic groups and their classification, isomorphism and Cayley's theorem, external and internal direct products, Cosest's and Lagrange's theorem, normal subgroups and factor groups, the fundamental theorem of homomorphisms, statements and some simple applications of the Sylow theorems.

**41342 Linear Algebra II **

This course reviews vector spaces and linear transformations, matrix representations of linear transformation, change of basis, similarity, characteristic and minimal polynomials of a linear operator. Cayley-Hamilton theorem, eigenvalues, eigenvectors and diagonalization, canonical forms. Inner product spaces. Gram-Schmidt orthogonalization process. Normal, orthogonal and unitary operators. Linear functionals and the dual spaces.

**41343 Number Theory **

This course covers divisibility, the division algorithm, greatest common divisor and least common multiple, diophentine equations, prime numbers and their distribution, the fundamental theorem of arithmetic, congruences, linear congruence equations, Chinese remainder theorem, tests of divisibility, Fermat’s little theorem, Wilson's theorem, number theoretic functions, perfect numbers , and cryptography as an application of number theory

**41344 Coding Theory **

This course introduces Shanon theorem, error detection, correction and decoding, finite fields, linear codes bounds on parameters of linear codes, constructions of linear codes, cyclic codes, and some special cyclic codes.

**41351 History of Mathematics **

This course gives the student a brief historical introduction to ancient mathematics (Indian, Egyptian, Babylonian), through its main mathematical operations. It includes Greek math such as the school of Pythagorus, and Euclid and his system of axioms, and a brief biography of three to four Greek mathematicians (Pythagoras, Euclid, Archimedes, Ptolemy). We will also discuss mathematics of the world of Islam, its main contributions and salient characteristics. Concise biographies of Al.Khowarizmi, Thabit bin Qurrah, Omar Al-Khayyam, and Al- Bayrouni will be given, along with selected topics from their writings, including the algebra of Khowarizmi, the determination of Qibla of Bayrouni, and Khayyam and his geometric method of solving cubic equations.

**41371 Special Functions**

This course discusses integrals depending on parameter, beta and gama functions, series solutions near regular singular points (Bessel functions), Legendre and associated Legendre functions, hyper geometric functions, Hermite polynomials. Laguerre polynomials, and the orthogonal set of functions.

**41404 Ordinary Differential Equations II **This course covers proof of existence and uniqueness theorem, and continuous dependence on initial conditions, phase plane for autonomous linear systems and their critical points, properties of solutions of nth-order linear systems, and stability of solutions of linear systems.

**41406 Calculus of Variations**

This course includes the study of variational problems with fixed boundaries, variational problems with movable boundaries and natural boundary conditions, variational problems with constraints, and direct methods for variational problems.

**41408 Integral Equations **This course introduces Abel’s problem classification of integral equations, the relationship between differential and integral equations, non-linear integral equations, solution of integral equations, Volterra’s and Fredholm’s integral equations, Hilpert-Schmidit theory, and applications of the Fredholm theory.

**41412 Complex Analysis II **This course includes discussion of residues and poles, evaluation of improper real integrals, improper integrals involving sines and cosines, definite integrals involving sines and cosines, integration through a branch cut, logarithmic residues and Rouche's theorem, mapping by elementary functions, conformal mappings and transformations of harmonic functions, and singularities and argument principles.

**41413 Functional Analysis **

This course provides an introduction to metric spaces, linear spaces, Normed and Banach spaces (with concrete examples of Normed and Banach spaces such as R, C,Rn,Cn ,lp , l, co , C[a,b]...), equivalent norms, finite dimensional Normed spaces and compactness, bounded linear operators, spaces of bounded linear operators , bounded linear functional, the Hahn-Banach theorem, the closed graph theorem, theopen mapping theorem, dual spaces, and Hilbert spaces.

**41418 Measure Theory **This course treats Lebesgue measure on the real line, outermeasure and measurability, measurable functions, the Lebesgue integral of a function, convergence of sequences of measurable functions, and Lp-spaces.

**41441 Abstract Algebra II **

This course examines rings, fields, ideals, prime ideal, maximal ideal, ring homeomorphisms, polynomial rings, factorization of polynomials, reducibility and irreducibility tests, divisibility in integral domains, principal ideal domains, and unique factorization domains, and algebraic extension of fields. It includes an introduction to Galois theory.

**41462 General Topology I**

This course treats topological spaces, open sets, closed sets, closure, interior and boundary of a set, cluster points and the derived set, isolated points, relative topology and subspaces, bases and sub-bases, separable spaces, first countable spaces, second countable spaces, finite product of topological spaces. Continuous functions, open functions, closed functions, homomorphism, separation axioms, convergence, compact spaces, and connected spaces

**41463 General Topology II **

This course examines topics such as local bases and first countable spaces, sequences in topological spaces, second countable spaces, separable spaces, connected spaces and their properties, components and path-wise components, locally connected spaces, compact spaces and their properties, compactness in Rn, countably compact spaces, locally compact spaces, metric spaces, metric topologies, equivalent metrics, continuity and uniform continuity of functions on metric spaces, and compactness of metric spaces.

**41464 Graph Theory**

This courses introduces the basic concepts of paths and circuits, Eulerian and Hamiltonian graphs, infinite graphs, trees, planar graphs and dual graphs, chromatic numbers, chromatic polynomials, and diagraphs.

**41491 Seminar in Mathematics **

Fourth-year level course.

**41492 Special Topics**

Fourth-year level course.

**41493 Mathematical Software Packages **

In this course, we will use mathematical software packages such as Matlab, Mathematica, Mathcad, and Maple to solve problems in calculus, differential equations, linear algebra and numerical analysis.

**41103 Math for Science Students **

This course introduces integration and its definition, integration by substitution, integration by parts, integration of trigonometric functions, areas under curves by integration, exponential and logarithmic functions, polar coordinate sequences and series, partial differentiation, and solving first-order and second-order differential equations.