Functions of one variable. Limits and continuity. Differentiation and its applications. The Mean Value Theorem and its applications. Definite integral and the Fundamental Theorem of Calculus. Exponential functions, their derivatives and integrals. Logarithmic functions and their derivatives.**103102 Calculus (2)**** **** **** ****(3:3-0)**

**Prerequisite: 103101**

Inverse functions. Inverse trigonometric functions and their derivatives. Hyperbolic functions. Techniques of integration. Applications of the definite integral to volumes, areas, arc lengths and surface areas. Improper integrals. L’Hopital’s Rule. Infinite sequences and series. Power series.

**103201 Calculus (3)**** **** **** ****(3:3-0)**

**Prerequisite: 103102**

Conic sections. Parametric equations. Polar coordinates and graphs. Derivatives and integrals of functions in polar coordinates. Vectors and analytic geometry in space. Functions of several variables and partial differentiation. Multiple Integrals.

**103202 Engineering Mathematics (1)**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

Systems of linear equations. Matrices. Determinants. Vector spaces, subspaces, bases and dimension. Invertible matrices. Similar and diagnosable matrices. First order differential equations. Linear differential equations of second and higher order. Homogenous and nonhomogeneous differential equations. Series solutions of differential equations. Laplace transforms. Systems of differential equations.

**103212 Introduction to Real Analysis **** **** ****(3:3-0)**

**Prerequisite: 103201**

Topology of real numbers: Ordering, bounded and connected sets. Sequences: Limits, Cauchy sequences, increasing and decreasing sequences. Functions: Limit of a function, continuity at a point and on an interval, uniform continuity. Differentiation: Rolle's Theorem, Mean Value Theorem.

**103222 Differential Equations**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

First order and first-degree equations. The homogeneous differential equation with constant coefficients. The methods of undetermined coefficients, reduction of order and variation of parameters. Infinite series solutions. Laplace transforms.

**103231 Principles of Statistics**** **** ****(3:3-0)**

**Prerequisite: 103101**

Introduction to statistics; Descriptive statistics; Measures of centrality and variation; Percentiles; Chebyshev's inequality and empirical rules; Introduction to probability; Probability laws; Counting rules, conditional probability and independence of events; Discrete and continuous random variables; Probability distribution; Expected and standard deviation of random variable; Binomial and normal probability distribution; Sampling distributions; Hypothesis testing; Simple linear regression and correlation.

**103241 Linear Algebra (1)**** **** ****(3:3-0)**

**Prerequisite: 103102**

Systems of linear equations. Matrices and matrix inverses. Row echelon forms. Determinants and Cramer rule. Vector spaces and subspaces. Basis and orthogonal basis. Linear transformations. Eigenvalues and eigenvectors.

**103242 Abstract Algebra (1) **** **** ****(3:3-0)**

**Prerequisite: 103251**

Binary operations. Groups and subgroups. Normal, cyclic, permutation, symmetric, isomorphic groups. Isomorphisms and homomorphisms of groups. Isomorphism theorems. Rings and subrings. Integral domains. Factor rings. Ideals.

**103250 Discrete Mathematics (1)**** **** **** ****(3:3-0)**

**Prerequisite: 103101**

Propositional and Predicate Calculus: Propositions, truth tables, connectives tautologies, contradiction (fallacies) and quantifiers, valid and invalid arguments. Proofs, Sets, functions, Cardinality of sets and countable and uncountable sets. Divisibility, congruence. Mathematical induction. Relations, equivalence relation, partial order relations, equivalence classes, upper and lower bounds, supremum and infimum. Graphs and graph terminology, special types of graphs, connected graphs and graphs isomorphism. Introduction to trees.

**103251 Fundamental of Mathematics**** **** **** **** ****(3:3-0)**

**Prerequisite: 103102**

Logic and proofs; Quantifiers; Rules of inference mathematical proofs, sets: Set operations, extended set operations and indexed families of sets; Relations; Cartesian Products and relations; Equivalence relations; Partitions; Functions onto functions, one-to-one functions; Induced set functions; Cardinality; Equipotent of sets; finite and infinite sets; Countable sets, topology of R, cardinal numbers.

**103253 Discrete Mathematics (2)**** **** **** ****(3:3-0)**

**Prerequisite: 103250**

Prime integers, Composite integer, greatest common divisor, least common multiple, recursive definition. Basics of counting, Pigeonhole Principle. Permutations and Combinations. Binomial coefficients. Generating Functions. Linear recurrence relation. Euler path and circuit, Hamilton path and circuit, planar graph, graph coloring. Rooted trees, spanning trees. Boolean algebra, Boolean function and models.

**103262 Euclidean Geometry **** **** ****(3:3-0)**

**Prerequisite: 103201**

A study of the origin of geometry. The method of axiomatic reasoning. Euclid's and the connection postulates. The postulate of parallel lines. Congruence and similarity in the Euclidean plane. Introduction to ordered and affine geometry.

**103313 Real Analysis (1)**** **** **** **** **** ****(3:3-0)**

**Prerequisite: 103212**

Real and complex number systems. Metric spaces, connected and compact sets. Sequences and series, tests of convergence, absolute and conditional convergence, power series. Continuity. Differentiation, the Mean Value Theorem. Taylor Theorem. Riemann Integrals: definition, existence and properties.

**103314 Real Analysis (2) **** **** **** ****(3:3-0)**

**Prerequisite: 103313**

Sequences and series of functions. Continuity at a point and uniform continuity. Functions of several variables. Inverse and implicit function theorems. Integration of functions of several variables: Fubini’s Theorem. Line and surface integrals. Green's, divergence and Stokes theorems.

**103322 Partial Differential Equations **** **** **** ****(3:3-0)**

**Prerequisite: 103222**

Classification. Some physical models (heat, wave and Laplace equations). Separation of variables. Sturm-Liouville BVP. Fourier series and transforms. Integral transforms. Applications.

**103332 Probability Theory**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

Distributions of random variables; Conditional probability and 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).

**103333 Biostatistics**** **** **** ****(2:2-0)**

**Prerequisite 103101**

Frequency distributions. Measures of centrality and dispersion. Binomial distribution, Poisson distribution, normal distribution. Confidence intervals, testing hypothesis about the mean and the population based on large samples. Pearson and Spearman correlation coefficients. Chi square tests and tests of independence. Vital statistics.

**103344 Number Theory (3:3-0)**

**Prerequisite: 103251**

Division. Diophantine equations. Prime numbers. The Fundamental Theorem of Arithmetic. Congruence, linear congruence equations. Fermat's and Wilson's Theorems. Arithmetic functions, Euler's Theorem.

**103365 General Topology**** **** **** ****(3:3-0)**

**Prerequisite: 103251 **

Topological spaces, closed sets. Bases and products. Continuous functions. Separation axioms and Hausdorff spaces. Metric spaces. Compact spaces.

**103373 Numerical Analysis (1)**** **** **** ****(3:3-0)**

**Prerequisite: 103102**

Taylor’s Theorem and its applications. Errors. Root findings. Interpolation. Numerical differentiation and integration. Solving systems of linear equations numerically.

**103376 Graph Theory **** **** **** ****(3:3-0)**

**Prerequisite: 103241**

Graphs and subgraphs. Matrices, trees and girth. Eulerian and Hamiltonian problems. Planar and nonplanar graphs. Connected graphs. Matchings, factorization and coverings. Networks.

**103381 Linear Programming **** **** **** ****(3:3-0)**

**Prerequisite: 103241**

Formulation of linear programs and their basic properties. Basic solutions. Graphic solutions. The simplex method. Duality. Sensitivity Analysis. Applications to the transportation model and networks.

**103401 History of Mathematics**** **** **** **** **** ****(3:3-0)**

**Prerequisite: Fourth Year**

Introduction to the history of ancient mathematics: Egyptians, Hindu and Babylonians. Greek math. The school of Pythagoras. A brief biography of Euclid, Archimedes and Ptolemy. Math. In the Arab and Islamic world. Contributions of Arabs in Algebra, geometry and analysis. A brief biography of Al-Khawarismi, Ibn Qurra and Al-Bayrouni. A brief account of the contributions of: Newton, Leibniz, Gauss, Cauchy and Laplace.

**103402 Teaching Mathematics **** **** **** **** ****(3:3-0)**

**Prerequisite: Fourth Year**

The special nature of mathematics, learning and teaching it. Basis for various approaches in mathematics teaching, especially in schools of different levels: Elementary, intermediate and secondary. Preparation and analysis of teaching materials, plans and tests for effective math teaching.

**103413 Theory of Special Functions**** **** **** ****(3:3-0)**

**Prerequisite: 103314**

Gamma and Beta functions. Legendre polynomials and functions. Bessel functions. Other special functions (incomplete gamma functions, the error functions, Riemann’s zeta function). Elliptic integrals.

**103414 Complex Analysis**** **** **** ****(3:3-0)**

**Prerequisite: 103313**

Complex numbers. Analytic functions. Limits and continuity. Differentiability. Cauchy- Riemann conditions. Complex integration. Residues and poles. Evaluation of improper integrals. Basic properties of conformal mapping.

**103422 Advanced Differential Equations**** **** **** ****(3:3-0)**

**Prerequisite: 103222**

Linear systems: Homogeneous, nonhomogeneous, constant coefficients and autonomous. Stability. Linear and almost linear systems. Lyapunov’s method. Existence and uniqueness theorems.

**103431 Mathematical Statistics (1) **** **** **** **** ****(3:3-0)**

**Prerequisite: 103332**

Limiting distribution, central limit theorem; Estimation: Point estimation, MLE, MME; Sufficient statistics and its properties; Complete statistics; Exponential family; Fisher Information and the Cramér–Rao inequality; Interval estimation; Pivotal quantity method; Statistical test: Uniformly most powerful test.

**103432 Mathematical Statistics (2) **** **** **** **** ****(3:3-0)**

**Prerequisite: 103431**

Estimation, some types of estimators; Sufficient statistics; Minimal sufficient statistics; Completeness; Methods of point estimation and properties of point estimators; Bayesian estimator confidence intervals, testing hypotheses; Neman-Pearson theorem; Randomized tests; Likelihood ratio tests.

**103433 Applied Statistics **** **** **** **** **** ****(3:2-1)**

**Prerequisite: 103431**

This course offers a broad treatment of statistics, concentrating on specific statistical techniques used in science. Topics include: Hypothesis testing and estimation, confidence intervals, chi-square tests goodness of fit Tests, analysis of variance, regression, and correlation.

**103442 Abstract Algebra (2)**** **** ****(3:3-0)**

**Prerequisite: 103242**

Ring homomorphisms, polynomial rings. Factorization of polynomials. Reducibility and irreducibility tests. Divisibility in integral domains. Principal ideal domains and unique factorization domains. Algebraic extension of fields. Introduction to Galois Theory.

**103443 Linear Algebra (2)**** **** ****(3:3-0)**

**Prerequisite: 103241**

Abstract treatment of finite dimensional vector spaces. Linear transformations and matrices. Direct sums and factor vector space. Minimal polynomials and Jordan canonical forms. Inner product. Nonnegative and irreducible matrices.

**103445 Matrix Theory**** **** ****(3:3-0)**

**Prerequisite: 103241**

Review of the theory of linear systems. Eigenvalues and eigenvectors. The Jordan canonical form. Bilinear and quadratic forms. Matrix analysis of differential equations. Variational principles and perturbation theory: The Courant minimax theorem, Weyl's inequalities, Gershgorin's theorem, perturbations of the spectrum, vector norms and related matrix norms, the condition number of a matrix.

**103451 Set Theory**** **** **** **** ****(3:3-0)**

**Prerequisite: 103251**

Axioms of set theory: Zermelo-Fraenkel axioms. Equipollence, finite sets, and cardinal numbers. Finite ordinals and denumerable sets. Transfinite induction and ordinal arithmetic. The axiom of choice: Zorn’s lemma and other equivalences.

**103474 Numerical Analysis (2)**** **** **** ****(3:3-0)**

**Prerequisite: 103373**

Introductory numerical integration, Gauss and Romberg integration. Numerical solutions for ordinary differential equations. Initial value problems, Single step and multistep methods. Boundary value problems and finite difference method.

**103470 Math. Software Packages**** **** **** ****(3:3-0)**

**Prerequisite: 9600101**

Training on Mathematica: Build algorithms for problem solving, do numerical and analytical computations and plot specific graphs. Applications on calculus, differential equations, linear algebra, statistics, number theory, programming, calculus of variations, optimal control and graph theory. Writing programs to solve specific problems.

**103472 Applied Mathematics**** **** ****(3:3-0)**

**Prerequisite 103201**

Diffusion-type problems. Hyperbolic-type problems. Elliptic-type problems.

**103479 Mathematical Modeling**** **** **** ****(3:3-0)**

**Prerequisite: 103241**

Introduction to mathematical classification of Models, constraints and terminology on Models, modeling process, population dynamics models for single species, stability analysis of growth models, Fishing management models, scaling variables, bifurcation analysis of the ODE y’ = f(y, c); Saddle-node, transcritical and Pitchfork bifurcations, models from science and finance, Newton’s law of cooling or heating, Chemical Kinetic reactions, modeling by systems of equations, modeling interacting species; Model building, different types of interactions models.

**103481 Nonlinear Programming**** **** ****(3:3-0)**

**Prerequisite: 103381**

Introduction to nonlinear programming. Necessary and sufficient conditions for unconstrained problems. Minimization of convex functions. A numerical method for solving unconstrained problems. Equality and inequality constrained problems. The Lagrange multipliers theorem. The Kuhn-Tucker condition. A numerical method for solving constrained problems.

**103482 Calculus of Variation**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

The simplest problem of calculus of variation and examples. Necessary conditions for an extremum to include: Euler-Lagrange, Weierstrass, Jacobi and corner conditions. Sufficient conditions for an extremum.

**103483 Optimal Control Theory**** **** ****(3:3-0)**

**Prerequisite: 103222**

Statement of the optimal control problem and examples. The Pontryagin maximum principle. Transversality conditions. Dynamic programming in continuous-time and the Hamilton-Jacobi theory. The linear regulator problem.

**103492 Special Topics in Mathematics **** **** ****(3:3-0)**

**Prerequisite: Fourth Year**

Variable contents. Open for Fourth Year Students interested in studying an advanced topic in mathematics with a departmental faculty member. A student can take this course for credit only once.

**103498 Seminar **** **** **** ****(1:1-0)**

**Prerequisite: Fourth Year**

This course aims at enriching students’ ability of independent readings, writing a report about these readings and presenting a seminar. Each student selects a topic offered by the instructor(s) of the course under his/her (their) guidance.

**103101 Calculus (1)**** **** **** **** ****(3:3-0)**

**Prerequisite: None**

Functions of one variable. Limits and continuity. Differentiation and its applications. The Mean Value Theorem and its applications. Definite integral and the Fundamental Theorem of Calculus. Exponential functions, their derivatives and integrals. Logarithmic functions and their derivatives.

**103102 Calculus (2)**** **** **** ****(3:3-0)**

**Prerequisite: 103101**

Inverse functions. Inverse trigonometric functions and their derivatives. Hyperbolic functions. Techniques of integration. Applications of the definite integral to volumes, areas, arc lengths and surface areas. Improper integrals. L’Hopital’s Rule. Infinite sequences and series. Power series.

**103201 Calculus (3)**** **** **** ****(3:3-0)**

**Prerequisite: 103102**

Conic sections. Parametric equations. Polar coordinates and graphs. Derivatives and integrals of functions in polar coordinates. Vectors and analytic geometry in space. Functions of several variables and partial differentiation. Multiple Integrals.

**103202 Engineering Mathematics (1)**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

Systems of linear equations. Matrices. Determinants. Vector spaces, subspaces, bases and dimension. Invertible matrices. Similar and diagnosable matrices. First order differential equations. Linear differential equations of second and higher order. Homogenous and nonhomogeneous differential equations. Series solutions of differential equations. Laplace transforms. Systems of differential equations.

**103212 Introduction to Real Analysis **** **** ****(3:3-0)**

**Prerequisite: 103201**

Topology of real numbers: Ordering, bounded and connected sets. Sequences: Limits, Cauchy sequences, increasing and decreasing sequences. Functions: Limit of a function, continuity at a point and on an interval, uniform continuity. Differentiation: Rolle's Theorem, Mean Value Theorem.

**103222 Differential Equations**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

First order and first-degree equations. The homogeneous differential equation with constant coefficients. The methods of undetermined coefficients, reduction of order and variation of parameters. Infinite series solutions. Laplace transforms.

**103231 Principles of Statistics**** **** ****(3:3-0)**

**Prerequisite: 103101**

Introduction to statistics; Descriptive statistics; measures of centrality and variation; percentiles; Chebyshev's inequality and empirical rules; Introduction to probability; Probability laws; Counting rules, conditional probability and independence of events; Discrete and continuous random variables; Probability distribution; expected and standard deviation of random variable; Binomial and normal probability distribution; Sampling distributions; Hypothesis testing; Simple linear regression and correlation.

**103241 Linear Algebra (1)**** **** **** ****(3:3-0)**

**Prerequisite: 103102**

Systems of linear equations. Matrices and matrix inverses. Row echelon forms. Determinants and Cramer rule. Vector spaces and subspaces. Basis and orthogonal basis. Linear transformations. Eigenvalues and eigenvectors.

**103242 Abstract Algebra (1) **** **** ****(3:3-0)**

**Prerequisite: 103251**

Binary operations. Groups and subgroups. Normal, cyclic, permutation, symmetric, isomorphic groups. Isomorphisms and homomorphisms of groups. Isomorphism theorems. Rings and subrings. Integral domains. Factor rings. Ideals.

**103250 Discrete Mathematics (1)**** **** **** ****(3:3-0)**

**Prerequisite: 103101**

Propositional and Predicate Calculus: Propositions, truth tables, connectives tautologies, contradiction (fallacies) and quantifiers, valid and invalid arguments. Proofs, Sets, functions, Cardinality of sets and countable and uncountable sets. Divisibility, congruence. Mathematical induction. Relations, equivalence relation, partial order relations, equivalence classes, upper and lower bounds, supremum and infimum. Graphs and graph terminology, special types of graphs, connected graphs and graphs isomorphism. Introduction to trees.

**103251 Fundamental of Mathematics**** **** **** ****(3:3-0)**

**Prerequisite: 103102**

Logic and proofs; Quantifiers; Rules of inference mathematical proofs, sets: Set operations, extended set operations and indexed families of sets; Relations; Cartesian Products and relations; Equivalence relations; Partitions; Functions onto functions, one-to-one functions; Induced set functions; Cardinality; Equipotent of sets; finite and infinite sets; Countable sets, topology of R, cardinal numbers.

**103253 Discrete Mathematics (2)**** **** **** ****(3:3-0)**

**Prerequisite: 103250**

Prime integers, Composite integer, greatest common divisor, least common multiple, recursive definition. Basics of counting, Pigeonhole Principle. Permutations and Combinations. Binomial coefficients. Generating Functions. Linear recurrence relation. Euler path and circuit, Hamilton path and circuit, planar graph, graph coloring. Rooted trees, spanning trees. Boolean algebra, Boolean function and models.

**103262 Euclidean Geometry **** **** ****(3:3-0)**

**Prerequisite: 103201**

A study of the origin of geometry. The method of axiomatic reasoning. Euclid's and the connection postulates. The postulate of parallel lines. Congruence and similarity in the Euclidean plane. Introduction to ordered and affine geometry.

**103313 Real Analysis (1)**** **** **** **** ****(3:3-0)**

**Prerequisite: 103212**

Real and complex number systems. Metric spaces, connected and compact sets. Sequences and series, tests of convergence, absolute and conditional convergence, power series. Continuity. Differentiation, the Mean Value Theorem. Taylor Theorem. Riemann Integrals: definition, existence and properties.

**103314 Real Analysis (2) **** **** **** **** ****(3:3-0)**

**Prerequisite: 103313**

Sequences and series of functions. Continuity at a point and uniform continuity. Functions of several variables. Inverse and implicit function theorems. Integration of functions of several variables: Fubini’s Theorem. Line and surface integrals. Green's, divergence and Stokes theorems.

**103322 Partial Differential Equations **** **** **** ****(3:3-0)**

**Prerequisite: 103222**

Classification. Some physical models (heat, wave and Laplace equations). Separation of variables. Sturm-Liouville BVP. Fourier series and transforms. Integral transforms. Applications.

**103332 Probability Theory**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

Distributions of random variables; Conditional probability and 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).

**103333 Biostatistics**** **** **** ****(2:2-0)**

**Prerequisite 103101**

Frequency distributions. Measures of centrality and dispersion. Binomial distribution, Poisson distribution, normal distribution. Confidence intervals, testing hypothesis about the mean and the population based on large samples. Pearson and Spearman correlation coefficients. Chi square tests and tests of independence. Vital statistics.

**103344 Number Theory**** **** **** **** **** **** ****(3:3-0)**

**Prerequisite: 103251**

Division. Diophantine equations. Prime numbers. The Fundamental Theorem of Arithmetic. Congruence, linear congruence equations. Fermat's and Wilson's Theorems. Arithmetic functions, Euler's Theorem.

**103365 General Topology**** **** **** ****(3:3-0)**

**Prerequisite: 103251**

Topological spaces, closed sets. Bases and products. Continuous functions. Separation axioms and Hausdorff spaces. Metric spaces. Compact spaces.

**103373 Numerical Analysis (1)**** **** ****(3:3-0)**

**Prerequisite: 103102**

Taylor’s Theorem and its applications. Errors. Root findings. Interpolation. Numerical differentiation and integration. Solving systems of linear equations numerically.

**103376 Graph Theory **** **** **** ****(3:3-0)**

**Prerequisite: 103241**

Graphs and subgraphs. Matrices, trees and girth. Eulerian and Hamiltonian problems. Planar and nonplanar graphs. Connected graphs. Matchings, factorization and coverings. Networks.

**103381 Linear Programming **** **** **** ****(3:3-0)**

**Prerequisite: 103241**

Formulation of linear programs and their basic properties. Basic solutions. Graphic solutions. The simplex method. Duality. Sensitivity Analysis. Applications to the transportation model and networks.

**103401 History of Mathematics**** **** **** **** **** ****(3:3-0)**

**Prerequisite: Fourth Year**

Introduction to the history of ancient mathematics: Egyptians, Hindu and Babylonians. Greek math. The school of Pythagoras. A brief biography of Euclid, Archimedes and Ptolemy. Math. In the Arab and Islamic world. Contributions of Arabs in Algebra, geometry and analysis. A brief biography of Al-Khawarismi, Ibn Qurra and Al-Bayrouni. A brief account of the contributions of: Newton, Leibniz, Gauss, Cauchy and Laplace.

**103402 Teaching Mathematics **** **** **** **** ****(3:3-0)**

**Prerequisite: Fourth Year**

The special nature of mathematics, learning and teaching it. Basis for various approaches in mathematics teaching, especially in schools of different levels: Elementary, intermediate and secondary. Preparation and analysis of teaching materials, plans and tests for effective math teaching.

**103413 Theory of Special Functions**** **** **** ****(3:3-0)**

**Prerequisite: 103314**

Gamma and Beta functions. Legendre polynomials and functions. Bessel functions. Other special functions (incomplete gamma functions, the error functions, Riemann’s zeta function). Elliptic integrals.

**103414 Complex Analysis**** **** **** ****(3:3-0)**

**Prerequisite: 103313**

Complex numbers. Analytic functions. Limits and continuity. Differentiability. Cauchy- Riemann conditions. Complex integration. Residues and poles. Evaluation of improper integrals. Basic properties of conformal mapping.

**103422 Advanced Differential Equations**** **** **** ****(3:3-0)**

**Prerequisite: 103222**

Linear systems: Homogeneous, nonhomogeneous, constant coefficients and autonomous. Stability. Linear and almost linear systems. Lyapunov’s method. Existence and uniqueness theorems.

**103431 Mathematical Statistics (1) **** **** **** **** ****(3:3-0)**

**Prerequisite: 103332**

Limiting distribution, central limit theorem; Estimation: Point estimation, MLE, MME; Sufficient statistics and its properties; Complete statistics; exponential family; Fisher Information and the Cramér–Rao inequality; Interval estimation; Pivotal quantity method; Statistical test: Uniformly most powerful test.

**103432 Mathematical Statistics (2) **** **** **** **** ****(3:3-0)**

**Prerequisite: 103431**

Estimation, some types of estimators; Sufficient statistics; Minimal sufficient statistics; Completeness; Methods of point estimation and properties of point estimators; Bayesian estimator confidence intervals, testing hypotheses; Neman-Pearson theorem; Randomized tests; Likelihood ratio tests.

**103433 Applied Statistics **** **** **** **** **** ****(3:2-1)**

**Prerequisite: 103431**

This course offers a broad treatment of statistics, concentrating on specific statistical techniques used in science. Topics include: Hypothesis testing and estimation, confidence intervals, chi-square tests goodness of fit Tests, analysis of variance, regression, and correlation.

**103442 Abstract Algebra (2)**** **** **** ****(3:3-0)**

**Prerequisite: 103242**

Ring homomorphisms, polynomial rings. Factorization of polynomials. Reducibility and irreducibility tests. Divisibility in integral domains. Principal ideal domains and unique factorization domains. Algebraic extension of fields. Introduction to Galois Theory.

**103443 Linear Algebra (2)**** **** ****(3:3-0)**

**Prerequisite: 103241**

Abstract treatment of finite dimensional vector spaces. Linear transformations and matrices. Direct sums and factor vector space. Minimal polynomials and Jordan canonical forms. Inner product. Nonnegative and irreducible matrices.

**103445 Matrix Theory**** **** ****(3:3-0)**

**Prerequisite: 103241**

Review of the theory of linear systems. Eigenvalues and eigenvectors. The Jordan canonical form. Bilinear and quadratic forms. Matrix analysis of differential equations. Variational principles and perturbation theory: The Courant minimax theorem, Weyl's inequalities, Gershgorin's theorem, perturbations of the spectrum, vector norms and related matrix norms, the condition number of a matrix.

**103451 Set Theory**** **** **** **** ****(3:3-0)**

**Prerequisite: 103251**

Axioms of set theory: Zermelo-Fraenkel axioms. Equipollence, finite sets, and cardinal numbers. Finite ordinals and denumerable sets. Transfinite induction and ordinal arithmetic. The axiom of choice: Zorn’s lemma and other equivalences.

**103474 Numerical Analysis (2)**** **** **** ****(3:3-0)**

**Prerequisite: 103373**

Introductory numerical integration, Gauss and Romberg integration. Numerical solutions for ordinary differential equations. Initial value problems, Single step and multistep methods. Boundary value problems and finite difference method.

**103470 Math. Software Packages**** **** **** ****(3:3-0)**

**Prerequisite: 9600101**

Training on Mathematica: Build algorithms for problem solving, do numerical and analytical computations and plot specific graphs. Applications on calculus, differential equations, linear algebra, statistics, number theory, programming, calculus of variations, optimal control and graph theory. Writing programs to solve specific problems.

**103472 Applied Mathematics**** **** **** ****(3:3-0)**

**Prerequisite 103201**

Diffusion-type problems. Hyperbolic-type problems. Elliptic-type problems.

**103479 Mathematical Modeling**** **** **** **** ****(3:3-0)**

**Prerequisite: 103241**

Introduction to mathematical classification of Models, constraints and terminology on Models, modeling process, population dynamics models for single species, stability analysis of growth models, Fishing management models, scaling variables, bifurcation analysis of the ODE y’ = f(y, c); Saddle-node, transcritical and Pitchfork bifurcations, models from science and finance, Newton’s law of cooling or heating, Chemical Kinetic reactions, modeling by systems of equations, modeling interacting species; Model building, different types of interactions models.

**103481 Nonlinear Programming**** **** ****(3:3-0)**

**Prerequisite: 103381**

Introduction to nonlinear programming. Necessary and sufficient conditions for unconstrained problems. Minimization of convex functions. A numerical method for solving unconstrained problems. Equality and inequality constrained problems. The Lagrange multipliers theorem. The Kuhn-Tucker condition. A numerical method for solving constrained problems.

**103482 Calculus of Variation**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

The simplest problem of calculus of variation and examples. Necessary conditions for an extremum to include: Euler-Lagrange, Weierstrass, Jacobi and corner conditions. Sufficient conditions for an extremum.

**103483 Optimal Control Theory**** **** ****(3:3-0)**

**Prerequisite: 103222**

Statement of the optimal control problem and examples. The Pontryagin maximum principle. Transversality conditions. Dynamic programming in continuous-time and the Hamilton-Jacobi theory. The linear regulator problem.

**103492 Special Topics in Mathematics **** **** ****(3:3-0)**

**Prerequisite: Fourth Year**

Variable contents. Open for Fourth Year Students interested in studying an advanced topic in mathematics with a departmental faculty member. A student can take this course for credit only once.

**103498 Seminar **** **** **** ****(1:1-0)**

**Prerequisite: Fourth Year**

This course aims at enriching students’ ability of independent readings, writing a report about these readings and presenting a seminar. Each student selects a topic offered by the instructor(s) of the course under his/her (their) guidance.

**103101 Calculus (1)**** **** **** **** ****(3:3-0)**

**Prerequisite: None**

Functions of one variable. Limits and continuity. Differentiation and its applications. The Mean Value Theorem and its applications. Definite integral and the Fundamental Theorem of Calculus. Exponential functions, their derivatives and integrals. Logarithmic functions and their derivatives.

**103102 Calculus (2)**** **** **** ****(3:3-0)**

**Prerequisite: 103101**

Inverse functions. Inverse trigonometric functions and their derivatives. Hyperbolic functions. Techniques of integration. Applications of the definite integral to volumes, areas, arc lengths and surface areas. Improper integrals. L’Hopital’s Rule. Infinite sequences and series. Power series.

**103201 Calculus (3)**** **** **** ****(3:3-0)**

**Prerequisite: 103102**

Conic sections. Parametric equations. Polar coordinates and graphs. Derivatives and integrals of functions in polar coordinates. Vectors and analytic geometry in space. Functions of several variables and partial differentiation. Multiple Integrals.

**103202 Engineering Mathematics (1)**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

Systems of linear equations. Matrices. Determinants. Vector spaces, subspaces, bases and dimension. Invertible matrices. Similar and diagnosable matrices. First order differential equations. Linear differential equations of second and higher order. Homogenous and nonhomogeneous differential equations. Series solutions of differential equations. Laplace transforms. Systems of differential equations.

**103212 Introduction to Real Analysis **** **** ****(3:3-0)**

**Prerequisite: 103201**

Topology of real numbers: Ordering, bounded and connected sets. Sequences: Limits, Cauchy sequences, increasing and decreasing sequences. Functions: Limit of a function, continuity at a point and on an interval, uniform continuity. Differentiation: Rolle's Theorem, Mean Value Theorem.

**103222 Differential Equatio****ns**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

First order and first-degree equations. The homogeneous differential equation with constant coefficients. The methods of undetermined coefficients, reduction of order and variation of parameters. Infinite series solutions. Laplace transforms.

**103231 Principles of Statistics**** **** ****(3:3-0)**

**Prerequisite: 103101**

Introduction to statistics; Descriptive statistics; Measures of centrality and variation; Percentiles; Chebyshev's inequality and empirical rules; Introduction to probability; Probability laws; Counting rules, conditional probability and independence of events; Discrete and continuous random variables; Probability distribution; expected and standard deviation of random variable; Binomial and normal probability distribution; Sampling distributions; Hypothesis testing; Simple linear regression and correlation.

**103241 Linear Algebra (1)**** **** **** ****(3:3-0)**

**Prerequisite: 103102**

Systems of linear equations. Matrices and matrix inverses. Row echelon forms. Determinants and Cramer rule. Vector spaces and subspaces. Basis and orthogonal basis. Linear transformations. Eigenvalues and eigenvectors.

**103242 Abstract Algebra (1) **** **** **** ****(3:3-0)**

**Prerequisite: 103251**

Binary operations. Groups and subgroups. Normal, cyclic, permutation, symmetric, isomorphic groups. Isomorphisms and homomorphisms of groups. Isomorphism theorems. Rings and subrings. Integral domains. Factor rings. Ideals.

**103250 Discrete Mathematics (1)**** **** **** ****(3:3-0)**

**Prerequisite: 103101**

Propositional and Predicate Calculus: Propositions, truth tables, connectives tautologies, contradiction (fallacies) and quantifiers, valid and invalid arguments. Proofs, Sets, functions, Cardinality of sets and countable and uncountable sets. Divisibility, congruence. Mathematical induction. Relations, equivalence relation, partial order relations, equivalence classes, upper and lower bounds, supremum and infimum. Graphs and graph terminology, special types of graphs, connected graphs and graphs isomorphism. Introduction to trees.

**103251 Fundamental of Mathematics**** **** **** **** ****(3:3-0)**

**Prerequisite: 103102**

Logic and proofs; Quantifiers; Rules of inference mathematical proofs, sets: Set operations, extended set operations and indexed families of sets; Relations; Cartesian Products and relations; Equivalence relations; Partitions; Functions onto functions, one-to-one functions; Induced set functions; Cardinality; Equipotent of sets; finite and infinite sets; Countable sets, topology of R, cardinal numbers.

**103253 Discrete Mathematics (2)**** **** **** ****(3:3-0)**

**Prerequisite: 103250**

Prime integers, Composite integer, greatest common divisor, least common multiple, recursive definition. Basics of counting, Pigeonhole Principle. Permutations and Combinations. Binomial coefficients. Generating Functions. Linear recurrence relation. Euler path and circuit, Hamilton path and circuit, planar graph, graph coloring. Rooted trees, spanning trees. Boolean algebra, Boolean function and models.

**103262 Euclidean Geometry **** **** ****(3:3-0)**

**Prerequisite: 103201**

A study of the origin of geometry. The method of axiomatic reasoning. Euclid's and the connection postulates. The postulate of parallel lines. Congruence and similarity in the Euclidean plane. Introduction to ordered and affine geometry.

**103313 Real Analysis (1)**** **** **** **** ****(3:3-0)**

**Prerequisite: 103212**

Real and complex number systems. Metric spaces, connected and compact sets. Sequences and series, tests of convergence, absolute and conditional convergence, power series. Continuity. Differentiation, the Mean Value Theorem. Taylor Theorem. Riemann Integrals: definition, existence and properties.

**103314 Real Analysis (2) **** **** **** **** ****(3:3-0)**

**Prerequisite: 103313**

Sequences and series of functions. Continuity at a point and uniform continuity. Functions of several variables. Inverse and implicit function theorems. Integration of functions of several variables: Fubini’s Theorem. Line and surface integrals. Green's, divergence and Stokes theorems.

**103322 Partial Differential Equations **** **** **** ****(3:3-0)**

**Prerequisite: 103222**

Classification. Some physical models (heat, wave and Laplace equations). Separation of variables. Sturm-Liouville BVP. Fourier series and transforms. Integral transforms. Applications.

**103332 Probability Theory**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

Distributions of random variables; Conditional probability and 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).

**103333 Biostatistics**** **** **** ****(2:2-0)**

**Prerequisite 103101**

Frequency distributions. Measures of centrality and dispersion. Binomial distribution, Poisson distribution, normal distribution. Confidence intervals, testing hypothesis about the mean and the population based on large samples. Pearson and Spearman correlation coefficients. Chi square tests and tests of independence. Vital statistics.

**103344 Number Theory**** **** **** **** **** **** ****(3:3-0)**

**Prerequisite: 103251**

Division. Diophantine equations. Prime numbers. The Fundamental Theorem of Arithmetic. Congruence, linear congruence equations. Fermat's and Wilson's Theorems. Arithmetic functions, Euler's Theorem.

**103365 General Topology**** **** **** ****(3:3-0)**

**Prerequisite: 103251**

Topological spaces, closed sets. Bases and products. Continuous functions. Separation axioms and Hausdorff spaces. Metric spaces. Compact spaces.

**103373 Numerical Analysis (1)**** **** **** ****(3:3-0)**

**Prerequisite: 103102**

Taylor’s Theorem and its applications. Errors. Root findings. Interpolation. Numerical differentiation and integration. Solving systems of linear equations numerically.

**103376 Graph Theory **** **** **** ****(3:3-0)**

**Prerequisite: 103241**

Graphs and subgraphs. Matrices, trees and girth. Eulerian and Hamiltonian problems. Planar and nonplanar graphs. Connected graphs. Matchings, factorization and coverings. Networks.

**103381 Linear Programming **** **** **** ****(3:3-0)**

**Prerequisite: 103241**

Formulation of linear programs and their basic properties. Basic solutions. Graphic solutions. The simplex method. Duality. Sensitivity Analysis. Applications to the transportation model and networks.

**103401 History of Mathematics**** **** **** **** **** ****(3:3-0)**

**Prerequisite: Fourth Year**

Introduction to the history of ancient mathematics: Egyptians, Hindu and Babylonians. Greek math. The school of Pythagoras. A brief biography of Euclid, Archimedes and Ptolemy. Math. In the Arab and Islamic world. Contributions of Arabs in Algebra, geometry and analysis. A brief biography of Al-Khawarismi, Ibn Qurra and Al-Bayrouni. A brief account of the contributions of: Newton, Leibniz, Gauss, Cauchy and Laplace.

**103402 Teaching Mathematics **** **** **** **** ****(3:3-0)**

**Prerequisite: Fourth Year**

The special nature of mathematics, learning and teaching it. Basis for various approaches in mathematics teaching, especially in schools of different levels: Elementary, intermediate and secondary. Preparation and analysis of teaching materials, plans and tests for effective math teaching.

**103413 Theory of Special Functions**** **** **** **** ****(3:3-0)**

**Prerequisite: 103314**

Gamma and Beta functions. Legendre polynomials and functions. Bessel functions. Other special functions (incomplete gamma functions, the error functions, Riemann’s zeta function). Elliptic integrals.

**103414 Complex Analysis**** **** **** ****(3:3-0)**

**Prerequisite: 103313**

Complex numbers. Analytic functions. Limits and continuity. Differentiability. Cauchy- Riemann conditions. Complex integration. Residues and poles. Evaluation of improper integrals. Basic properties of conformal mapping.

**103422 Advanced Differential Equations**** **** **** ****(3:3-0)**

**Prerequisite: 103222**

Linear systems: Homogeneous, nonhomogeneous, constant coefficients and autonomous. Stability. Linear and almost linear systems. Lyapunov’s method. Existence and uniqueness theorems.

**103431 Mathematical Statistics (1) **** **** **** **** ****(3:3-0)**

**Prerequisite: 103332**

Limiting distribution, central limit theorem; Estimation: Point estimation, MLE, MME; Sufficient statistics and its properties; Complete statistics; exponential family; Fisher Information and the Cramér–Rao inequality; Interval estimation; Pivotal quantity method; Statistical test: Uniformly most powerful test.

**103432 Mathematical Statistics (2) **** **** **** **** ****(3:3-0)**

**Prerequisite: 103431**

Estimation, some types of estimators; Sufficient statistics; Minimal sufficient statistics; Completeness; Methods of point estimation and properties of point estimators; Bayesian estimator confidence intervals, testing hypotheses; Neman-Pearson theorem; Randomized tests; Likelihood ratio tests.

**103433 Applied Statistics **** **** **** **** **** ****(3:2-1)**

**Prerequisite: 103431**

This course offers a broad treatment of statistics, concentrating on specific statistical techniques used in science. Topics include: Hypothesis testing and estimation, confidence intervals, chi-square tests goodness of fit Tests, analysis of variance, regression, and correlation.

**103442 Abstract Algebra (2)**** **** **** ****(3:3-0)**

**Prerequisite: 103242**

Ring homomorphisms, polynomial rings. Factorization of polynomials. Reducibility and irreducibility tests. Divisibility in integral domains. Principal ideal domains and unique factorization domains. Algebraic extension of fields. Introduction to Galois Theory.

**103443 Linear Algebra (2)**** **** ****(3:3-0)**

**Prerequisite: 103241**

Abstract treatment of finite dimensional vector spaces. Linear transformations and matrices. Direct sums and factor vector space. Minimal polynomials and Jordan canonical forms. Inner product. Nonnegative and irreducible matrices.

**103445 Matrix Theory**** **** ****(3:3-0)**

**Prerequisite: 103241**

Review of the theory of linear systems. Eigenvalues and eigenvectors. The Jordan canonical form. Bilinear and quadratic forms. Matrix analysis of differential equations. Variational principles and perturbation theory: The Courant minimax theorem, Weyl's inequalities, Gershgorin's theorem, perturbations of the spectrum, vector norms and related matrix norms, the condition number of a matrix.

**103451 Set Theory**** **** **** **** ****(3:3-0)**

**Prerequisite: 103251**

Axioms of set theory: Zermelo-Fraenkel axioms. Equipollence, finite sets, and cardinal numbers. Finite ordinals and denumerable sets. Transfinite induction and ordinal arithmetic. The axiom of choice: Zorn’s lemma and other equivalences.

**103474 Numerical Analysis (2)**** **** **** ****(3:3-0)**

**Prerequisite: 103373**

Introductory numerical integration, Gauss and Romberg integration. Numerical solutions for ordinary differential equations. Initial value problems, Single step and multistep methods. Boundary value problems and finite difference method.

**103470 Math. Software Packages**** **** **** ****(3:3-0)**

**Prerequisite: 9600101**

Training on Mathematica: Build algorithms for problem solving, do numerical and analytical computations and plot specific graphs. Applications on calculus, differential equations, linear algebra, statistics, number theory, programming, calculus of variations, optimal control and graph theory. Writing programs to solve specific problems.

**103472 Applied Mathematics**** **** **** ****(3:3-0)**

**Prerequisite 103201**

Diffusion-type problems. Hyperbolic-type problems. Elliptic-type problems.

**103479 Mathematical Modeling**** **** **** **** ****(3:3-0)**

**Prerequisite: 103241**

Introduction to mathematical classification of Models, constraints and terminology on Models, modeling process, population dynamics models for single species, stability analysis of growth models, Fishing management models, scaling variables, bifurcation analysis of the ODE y’ = f(y, c); Saddle-node, transcritical and Pitchfork bifurcations, models from science and finance, Newton’s law of cooling or heating, Chemical Kinetic reactions, modeling by systems of equations, modeling interacting species; Model building, different types of interactions models.

**103481 Nonlinear Programming (3:3-0)**

**Prerequisite: 103381**

Introduction to nonlinear programming. Necessary and sufficient conditions for unconstrained problems. Minimization of convex functions. A numerical method for solving unconstrained problems. Equality and inequality constrained problems. The Lagrange multipliers theorem. The Kuhn-Tucker condition. A numerical method for solving constrained problems.

**103482 Calculus of Variation**** **** **** ****(3:3-0)**

**Prerequisite: 103201**

The simplest problem of calculus of variation and examples. Necessary conditions for an extremum to include: Euler-Lagrange, Weierstrass, Jacobi and corner conditions. Sufficient conditions for an extremum.

**103483 Optimal Control Theory**** **** ****(3:3-0)**

**Prerequisite: 103222**

Statement of the optimal control problem and examples. The Pontryagin maximum principle. Transversality conditions. Dynamic programming in continuous-time and the Hamilton-Jacobi theory. The linear regulator problem.

**103492 Special Topics in Mathematics **** **** ****(3:3-0)**

**Prerequisite: Fourth Year**

Variable contents. Open for Fourth Year Students interested in studying an advanced topic in mathematics with a departmental faculty member. A student can take this course for credit only once.

**103498 Seminar **** **** **** ****(1:1-0)**

**Prerequisite: Fourth Year**

This course aims at enriching students’ ability of independent readings, writing a report about these readings and presenting a seminar. Each student selects a topic offered by the instructor(s) of the course under his/her (their) guidance.