# Faculty of Science , Mansoura University

## Mathematics Courses Content

Math Courses

 First Level Math.111 Fundamental Math (1) Algebra and Geometry, 2 h/W Descriptions: Algebra: Mathematical induction and Partial fractions. Binomial theorem and its applications, Solution of cubic equations, Solution of 4th degree equations, Sets, subsets, set operations and inductively definition of sets, equivalence relations, equivalence classes, partitions and partial order, maps, composition of maps, kinds of maps and inverse functions.Geometry: 1- coordinate plane: Rectangular coordinates and polar coordinate – distance – change of axes - Straight line in plane and the common equation of two lines - Circle - The conic section: Parabola – Ellipse –Hyperbola - The general equation of the second degree in two variables Tutorial: 2h/W Math.112 Fundamental Mathematics (2) Differentiation and integration, 2h/W Descriptions: A. Review and Preparation for Calculus. B. Limits and Their Properties. C. Differentiation. D. Applications of Differentiation. E. Integration. F. Logarithmic, Exponential and Other Functions. G. Applications of Integration. H. Integration Techniques. Tutorial: 2h/W Math.121Mechancs (1), 2h/W Descriptions: 1- Vector Algebra. 2- Reduction of forces. Equivalence of sets of Forces. 3- Equilibrium of Frames, Smooth hinges. 4- The center of mass. 5- Line integral. The work and energy. 6 – The virtual work principle. Tutorial: 2h/W Math.122 Mechance (2), 2h/W Descriptions: Motion of a particle in a straight line – Motion in a resisting medium – Vertical motion under the earth's attraction - Simple harmonic motion and its applications. Inertial frames. Motion in moving frames. Projectiles.Motion of a particle on a circle.Impluse, impulsive forces and impact of elastic bodies.Motion of bodies of variable mass.Rockets.Mechanical systems.D’Alembert’s principle.Constraints.The general equation of Dynamics.Conservative systems.Lagrange’s theorem on stability. Tutorial: 2h/W Second Level Math 211 Real Analysis, (Math.), 2h/W Descriptions: 1- Sets and functions: operations on sets – real-valued functions – equivalence and countability – real numbers – least upper bounds. 2- Sequences of real numbers: convergent, bounded and monotone sequences –lim sup and lim inf. 3- Series of real numbers: convergence and divergence – alternating series – series with nonnegative terms – absolute and conditional convergence. 4- Sequences and series of functions: power series – uniform convergence – Weiestrass's approximation Theorem. 5- Fourier expansions. Tutorial: 2h/W Math 212 Abstract Algebra (1), 2h/W Descriptions: Maps, kinds of maps, operations, groupoids and all essential kinds of groupoids Groups, subgroups and its properties Cyclic groups. Symmetric groups and permutation groups.Normal subgroups and factor groups.Homomorphism theorems of groups and Automorphisms group. P-Groups and Sylow Theorems.Rings and fields. Tutorial: 2h/W Math 214 Ordinary Differential Equations, 2h/W Descriptions: Definitions. First-order differential equations: linear, separable, exact and homogenous, Second-order differential equations. Reduction of order, constant coefficients, second-order linear equations: ordinary points and regular singular points. Euler's equation.Series solutions of second-order linear differential equations. Power series, solutions about an ordinary point. Solutions about a regular singular point. Equal roots of indicial equation and roots differing by an integer. Tutorial: 2h/W Math 215 Linear Algebra (1), 2h/W Descriptions: What is a field and examples of the well-known fields. Matrices defined over a field, operations on matrices, Echelon form. Algebra of square matrices, inverted matrix and system of linear equations.What is a vector space, subspaces, intersection and addition of subspaces. Linear combination, dependently and independently set of vectors, Basis and Dimension of a vector space.Linear transformations and linear operators and its proprties.Transformation from basis to another basis. Eigenvalues and eigenvectors. Similar matrices and diagonalization for square matrices. Applications. Tutorial: 2h/W Math 216 Calculus of Several Variables, 2h/W Descriptions: This course develops further the basic topics of the differentiation and integration of functions of several variables. This course consists of three main parts. Part 1 : Differential calculus of functions of several variables - Limits and continuity - Partial derivatives - Directional derivatives and the gradient -Normal lines and tangent planes - Extreme - Lagrange multipliers. Part 2: Multiple Integrals- Double integrals in different spaces and their application- Triple integrals in different spaces and their applications - Transformation of coordinates - Change of variables in multiple. Part 3: Topics in vector Calculus- Line integrals and applications - Green's Theorem - Independent of path of line integrals in the plane and dimensional spaces - Surface integrals - The divergence Theorem - Stock's Theorem. Tutorial: 2h/W Math 217 Introduction to Logic, 2h/W Descriptions: The Propositional Logic 1 - Propositional Calculus and proofs - Predicate Logic and Quantifiers - Divisibility Theory of Integers - The Theory of Congruences - Primes and Their Distributions - Finite Continued Fractions. Tutorial: 2h/W Math. 218 Solid Analytic Geometry, 2h/W Descriptions: Cartesian and parametric equation for plane in space. Cartesian and parametric equations for Line in space. Cartesian and parametric sphere equations for sphere. Cartesian and parametric circles equations for circles in space. Relations between Lines, planes and spheres in space. Paraboloid and ellipsoid, surface in space. Tangent plane of surfaces in space. The general equation of the 2nd degree in 3 variables. General theory of quadratic surfaces. Tutorial: 2h/W Math. 221 Mech. (3), 2h/W Descriptions: 1- Vector Analysis. 2- Moments of inertia 3- Equilibrium of Beams and chains. 4- Bending and shear forces. 5- Hydrostatics. 6- Electro-statics (Attraction and Potential) Tutorial: 2h/W Math. 223 Mechanics (4), 2h/W Descriptions: Plane motion of a particle. Kinematics. Intrinsic coordinates. Constrained motion. Stability of equilibrium and stationary motion – Damped simple harmonic motion – forced vibrations. Central orbits. Elements of celestial Mechanics. Orbital motion of planets and Satellites. Motion of a particle in three dimensions. Motion on a smooth surface. Motion on a rotating earth. Plane motion of a rigid body. Tutorial: 2h/W Math. 224 Mathematical Biology, 2h/W Descriptions: The Theory of Linear Difference Equations Applied to population Growth. - Nonlinear Difference Equations; Steady states and fixed points. Stability. Period doubling bifurcations. Chaos. Applications of Nonlinear Difference Equations to Population Biology; Host-parasitiod systems - Continuous Processes and Ordinary Differential Equations; An Introduction to Continuous Models; Phase-Plane Methods and Qualitative Solutions; Structural stability and instability. Lyapunov functions. Applications of Continuous Models to Population Dynamics, Prey-predator models. Limit Cycles, Oscillations, and Excitable Systems.Epidemic models. Tutorial: 2h/W Math. 231 Introduction to Statistics and Probability, 2h/W Descriptions: 1-Descriptive Methods for Qualitative Data. 2-Descriptive Methods for Quantitative Data. 3-Coefficient of Correlation Pearson’s and Spearman’s rank correlation coefficients. 4-Simple Linear Regression Model. 5-Introduction to probability, Sample space, Events, Operations with Events,Counting Sample Points, Probability of an Event, Some Probability Laws, Conditional Probability, Bayes Rule. 6-Introduction to the random variables: classification of random variables , probability mass and density function , distribution function, expectation and variance of random variables. 7- The moment generating function , the properties of the m.g.f. 8-Some probability distributions. Binomial, Poisson, Geometric distribution, Normal distribution, Standard normal distribution. Tutorial: 2h/W Math 241 Computer Science (2), 2h/W Descriptions: Programming: coverage of all the C++ programming language, with special emphasis on pointers and their usage. C++ Libraries and Tools: Compilation, linking and archiving; dividing programs into reusable units (libraries), C++ programming methods for data hiding and opaque typing; professional Issues: programming style; debugging techniques, safety-first programming methods Practical: 2h/W Math 242 Computer Algebra, 2h/W Descriptions: 1- Introduction to symbolic mathematics systems in general and Maple in particular. 2- Effective use of Maple. 3- Numbers and Functions. 4- Manipulating Algebraic Expressions. 5- Solutions of Equations. 6- Programming in Maple. 7- Graphics. 8- Other applications based on student interests. Practical: 2h/W Math 201 Pure Mathematics, 2h/W Descriptions: Functions of more than one variable - Continuity – Partial differentiation and it's applications (Implicit function, Taylor expansion, differentiation under the sign of integration, Maximum and minimum values) – Multiple integrals – Linear integrals - Conditional maxima and minima (Lagrange Multiplier). Differential Equations: Separable – Homogeneous – Equations tends to homogenous and separable – Exact – Integrating factor – Bernoulli's equations – Applications – Linear differential equation with order two and three. Tutorial: 1h/W Math.202 Introduction to Statistics and Probability, 2h/W Descriptions: Descriptive Methods for Qualitative Data - Descriptive Methods for Quantitative Data- Coefficient of Correlation- Pearson’s correlation coefficients- Spearman’s rank correlation coefficients - Simple Linear Regression Model- Introduction to probability- Sample space – Events- Operations with Events- Counting Sample Points- Probability of an Event- Some Probability Laws - Conditional Probability - Bayes Rule - Introduction to the random variables -Some probability distributions - Binomial distribution – Poisson distribution - Normal distribution - Standard normal distribution. Tutorial: 1h/W Math 203 Linear Algebra and Geometry, 2h/W Descriptions: System of linear equations – Matrices – vector Space –Eigen values and eigen vectors of a square matrix. Coordinates in R3 - Straight linear – Plane. Tutorial: 1h/W Math 204 Differential Equations, 2h/W Descriptions: First order differential equations: Separable - Homogenous – Equations tends to homogenous and separable - Exact – Integration factor – Bernoulli's equations. Applications – Linear differential equations. Second order differential equations. Systems of differential equations. Tutorial: 1h/W Math 205 Advanced Calculus, 2h/W Descriptions: Function of more than one variable: Limits – Continuity – Partial differentiation, and its applications. Multiple integrals – Linear integrals. Tutorial: 1h/W Math 206 Pure Math., 2/W Descriptions: Definitions. First-order differential equations: linear, separable, exact and homogenous, Second-order differential equations, reduction of order, constant coefficients; Second-order linear equations: ordinary points and regular singular points. Euler's equation. Introduction to systems of first-order equations. Solutions of two linear first-order equations. Introduction to Partial differential equations – order – homogenous and non homogenous – degree-linear and nonlinear- Heat equation, Wave equation and Laplace's equation in both one and higher dimensions. Separation of Variables and solutions of boundary value problems. Tutorial: 2h/W Math 207 Biomathematics, 2h/W Descriptions: Single species population dynamics: Models in discrete and continuous time: basic reproductive number; compensatory and depensatory competition; transcritical, tangent and period doubling bifurcations, chaos. Harvesting: maximum sustainable yield; yield effort curves. Population dynamics of interacting species: host-parasitoid interactions: Nicholson-Bailey model; Jury conditions and Naimark-Sacker bifurcations. Predator-prey models: Lotka-Volterra model; phase plane analysis; Routh-Hurwitz conditions and Hopf bifurcations; Poincare-Bendixon theorem, Dulac condition; Lyapunov functions; Volterra's principle. Competition: Gauses principle of competitive exclusion. Infectious diseases: SIS epidemic models: basic reproductive number; threshold theorem. SIR epidemic and endemic models: threshold criterion; size of the epidemic; eradication and control. Vector-borne diseases and sexually transmitted diseases. Tutorial: 1h/W Math 208 Difference Equations, 2h/W Descriptions: Sequences, Difference Operators, Solutions of linear homogeneous equations, Solutions of linear nonhomogeneous equations Fundamentals of Linear Difference systems fixed points and Stability theory, nonlinear difference equations, The z-Transform. Tutorial: 1h/W Third Level Math 310 Linear Algebra 2, 2h/W Descriptions: Inner product vector spaces. Symmetric, Hermitian, Unittary and Orthogonal matrices. Eigenvalues and Eigenvectors, Diagonlization of Matrices, Characteristic and minimum polynomials. Triangular Form, LU factorization, Invariant subspaces and primary decomposition. Nilpotent Operators and Jordan Canonical form, Rational Canonical form. Applications. Pre-requisite: Math.111, Tutorial: 2h/W Math 311 Measure Theory, 2h/W Descriptions: 1) Basic definitions. 2) Riemann integration. 3)Measure function, Lesbegue measure. 4) Measurable functions. 5) Lesbegue integration. 6) Measure dynamics. Pre-requisite: Math.112, Tutorial: 2h/W Math. 312 Complex Analysis (1), 2h/W Descriptions: 1 – The complex number system. 2 – Functions of complex variable. 3 – Complex differentiation and the Cauchy – Riemann Equation. 4 – Complex integration and Cauchy Theorem. 5 – Cauchy's integral formulas and related Theorems. 6 –Series and Residues and poles Evaluation of integrals. 7 – Taylor and Laurent's series. Pre-requisite: Math.112, Practical: 2h/W Math 313 Numerical Analysis (1), 2h/W Descriptions: IDEAS OF APPROXIMATION. Round-off and truncation errors, . ROOT FINDING, nonlinear equations (Bisection method, fixed-point, Newton’s method, secant method), APPROXIMATION METHODS Interpolation, Lagrange & Hermite polynomials, Splines. Dived difference formula, Newton interpolation polynomial, NUMERICAL INTEGRATION AND DIFFERENTIATION. Trapezoidal rule, composite trapezoidal rule, Simpson’s rule, Romberg integration. Gaussian quadrature. Euler-Maclaurin. ODEs: INITIAL VALUE PROBLEMS Enler’s method, Taylor’s metod, Runge Kutta methods, multi-step methods. Math 315 Abstract Algebra 2, 2h/W Descriptions: What is a ring All essential kinds of rings. Integral Domain and its properties. Unites, primes and irreducible elements. Subrings, ideals, prime and maximal ideals. Factor rings and homomorphism theorems. Fields, subfields and prime subfields, Extention of an integral domain to a field. Pre-requisite: Math.111, Tutorial: 1h/W Math 316 Topology (1), 2h/W Descriptions: 1- Topological Spaces( Open sets – Neighborhoods- Closed sets – Closure of sets- Interior of sets – Exterior and boundary of sets – limit points – Dense and nowhere dense sets – Comparison of topologies – Subspaces.) 2- Bases and Subbases( Base for a topology – Subbase – topologies generated by classes of sets – Local bases.) 3-Continuity( Continuous functions – Continuity at a point – Open and closed functions – Homomorphisms – topological properties – Topologies induced by functions.) 4-Separation Axioms( T0 – spaces, T1 - spaces, T2 - space, Regular and normal spaces, T3 – spaces, T4 – spaces) 5-Metric spaces(Metrics – Topology for a metric space – Distance between sets – Diameter of a set – Properties of metric topologies – Equivalent metrics.) Pre-requisite: Math.111, Tutorial: 1h/W Math 317 Abstract Algebra 3, 2h/W Descriptions: Euclidean domain and its properties, Polynomials over a ring and over a field, Prime and irreducible polynomials, Gauss theorem and Eisenstein's criterion, Field extensions, Splitting fields, Finite fields and its properties, Classification of extensions. Pre-requisite: Math.111, Tutorial: 2h/W Math. 318 Integral Equations, 2h/W Descriptions: 1-Volterra Integral equations: ( Relationship between linear differential equations and Volterra integral equation- Resolvent Kernel - Method of successive approximations - Convolution type and solution by Laplaces' transformation - Volterra integral equation of the first kind - Apel's integral equation and Euler integral) 2- Fredholm Integral Equations:( The method of Fredholm's determinant- Iterated Kernels- Degenerate Kernels- Homogeneous integral equations - Nonhomogenous symmetric equations) Pre-requisite: Math.112, Tutorial: 1h/W Math 319 Number Theory, 2h/W Descriptions: Divisibility. Congruence. Quadratic Reciprocity and Quadratic forms. Some Diophantine equations. Continued Fractions. Some Number- Theoretic Functions. Primes and Multiplicative Number Theory. Algebraic Numbers. Pre-requisite: Math.111, Tutorial: 2h/W Math 321 Electrodynamics, 2h/W Descriptions: Electrostatics, Magnetostatics, Electrodynamics, Basics of special relativity, Lorentz transformations, Relativistic kinematics and dynamics, 4-d geometry, The Electrostatic Scalar Potential, The Magnetostatic Vector Potential, The Electrodynamic Potentials- Covariant Classical Electrodynamics The Wave Equations, Plane Waves. Pre-requisite: Math.122, Tutorial: 2h/W Math. 322 Theory of Elasticity (1), 2h/W Descriptions: Stress theory. Strain theory. The relation between stresses and strains – Hook's law. Basic equations of elasticity theory and fundamental problems. Semi – inverse method and applications. Plane stress and strain problems. Torsion in beams. Pre-requisite: Math.122, Tutorial: 1h/W Math. 324 Theory of Functions and Special Functions, 2h/W Descriptions: 1-Series solution of differential equations. 2- Gamma and Beta Functions. 3- Legender polynomials and functions. 4- Bessel functions. 5- Hermite polynomials. 6-Laguerre polynomials. 7- Hypergeometmic functions. 8- Shebyshev polynomials. 9- Elliptic integrals. Jacobi’s elliptic functions. Pre-requisite: Math.112, Tutorial: 1h/W Math. 301 Biostatistics, 2h/W Descriptions:Descriptive Methods for Qualitative Data, Descriptive Methods for Quantitative Data, Coefficient of Correlation, Pearson’s correlation coefficients, Spearman’s rank correlation coefficients, Simple Linear Regression Model, Introduction to probability, Introduction to the random variables, Some probability distributions, Introduction to the sampling theory, Introduction to the interval estimation for one population, Introduction to the test of hypotheses for one population, The test of hypothesis about the population variance. Pre-requisite: Math.112, Tutorial: 2h/W Math. 302 Applied Statistics, 2h/W Descriptions: Descriptive Methods for Qualitative Data, Descriptive Methods for Quantitative Data, Coefficient of Correlation, Pearson’s correlation coefficients, Spearman’s rank correlation coefficients, Simple Linear Regression Model, Introduction to probability, Introduction to the random variables, Some probability distributions, Binomial and Poisson distribution, Normal and Standard normal distribution, Introduction to interval estimation, Introduction to Tests of hypothesis Pre-requisite: Math.112, Tutorial: 2h/W Math. 303 Numerical Analysis, 2h/W Descriptions: 1) Introduction, Computer Arithmetic and Errors. 2) Calculus of Finite Differences. 3) Difference Equations. 4) Linear Systems of Equations. 5) Interpolation. 6) Applications - Computer Subroutine Packages. Including MAPLE and MATLAB). Tutorial: 2h/W Math 341 Structured Programming, 2h/W Descriptions: The course teaches one of the structure languages in a way that emphasizes algorithms design using a structured, modular, and object-oriented approach. Then it studies recursion, introduces the abstract data type of lists, and shows how one can implement them in the used language using fundamental data structures. Introduction material on software engineering and program development. Topics include both the common heritage of the programming language (e.g. syntax, primitive types, equality, relational, and logical operators, arithmetic operations iteration, conditional expressions, functions, arrays, pointers, proper documentation techniques, testing and debugging, error handling and dynamic memory allocation, methods and the conventional standard libraries) as well as the introduction to object-oriented and unique aspects of programming with the used language. Completion of this class will prepare the student for advanced programming language. Practical : 2h/W Math.342 Database Management Systems, 2h/W Descriptions: Basic principles of database management systems (DBMS) and of DBMS application development. DBMS objectives, systems architecture, database models with emphasis on Entity-Relationship and Relational models, data definition and manipulation languages, the Structured Query Language (SQL), database design, application development tools, access methods interfaces, security, concurrency control and recovery, query processors and optimizers. Practical : 1h/W Math. 343 Systems of Computer Algebra, (2h/W Descriptions: General Introduction. Brief history of mathematical computing. Mathematical software packages, programming languages. Programming in MATLAB. Essentials of MATLAB; vectors and matrices, colon notation, numeric output, graphics, control structures and logical tests. MATLAB functions. Symbolic and high precision computations. Essentials of programming Matlab using m-file scripts and functions. Graphic visualizations. Practical : 2h/W Math. 344 Artificial Intelligence and Expert Systems, 2h/W Descriptions: Studying the foundations of Artificial Intelligence in today's environment and instilling an understanding of representations and external constraints with the idea of enabling a student to think creatively. There are many cognitive tasks that people can do easily and almost unconsciously but that have proven extremely difficult to program on a computer. Artificial intelligence is the problem of developing computer systems that can carry out these tasks. The course can focus on some central areas in AI such as: representation and reasoning, learning, AI languages such as Prolog and Lisp, expert systems, machine learning, robots, and natural language processing. , Practical : 2h/W Math. 345 Operating Systems, 2h/W Descriptions: Principals of operating systems. The operating systems as a control program and as a resource allocator. The concept of a process and a concurrency problems. Synchronization, mutual exclusion, deadlock, Additional topics include memory management, file systems process scheduling threads and protection. Practical : 2h/W Math. 346 Computer Network, 2h/W Descriptions: An introduction to the structure and components of computer networks, functions and services. Also, this course discuses current protocols, technologies, network layers, security, wireless, mobile network, performance issues, and multimedia networking. Practical : 2h/W Fourth Level Math 415 Lie Algebra, (Math.), 2h/W Descriptions: 1 – Basic concepts. 2 – Ideal and homeomorphisms. 3 – Solvable and nilpotent Lie algebras. 4 – Semi simple Lie algebras. 5 – Representation theory (module-Structure). 6 – Representation of sl (2) 7 – Root system. 8 – Manifold theory. 9 – Lie groups. 10 – Connection between Lie group and Lie algebra. Pre-requisite: Math.212, Tutorial: 1h/W Math.416 Differential Geometry, 2h/W Descriptions:  The theory of curves [(i) plane curves. (ii) space curves : arc – Length, curvature, torsion and Frenet's formulas]  The theory of surfaces in space: [(i) regular surface – tangent vectors and tangent plane at a point. (ii) Curves on a surface, first and second fundamental forms. (iii) Curvatures of surface. (v) Gauss equation and Christoffel symbols. (iv) Principal curvatures and lines of curvatures. Pre-requisite: Math.214, Tutorial: 1h/W Math. 417 Complex Analysis (2), 2h/W Descriptions: 1 – Conformal mapping. 2 – Analytic continuation and Riemann surfaces. 3 – The Schwarz –Christoffel Transformation. 4 – Integral Formulas of Poisson Type. 5 – Harmonic Functions. Pre-requisite: Math.211, Tutorial: 1h/W Math 418 Lattice Theory, 2h/W Descriptions: Partially ordered, totally ordered and inductively ordered sets. Lattice and complete lattice. Sublattices and direct product of lattices. Lattices and ordered homomorphisms and the other kinds of homomorphisms. Distributive and Modular lattices. Complemented lattices. Boolean algebra. Applications in switching and logic circuits. Ideals and congruence relations. Pre-requisite: Math.212, Tutorial: 1h/W Math 419 Topology (2), (Math.) , 2h/W Descriptions: Methods of generating topologies. Compact spaces Connected spaces - Sequences in topological spaces- Convergence in topological spaces: by filter and nets - Fuzzy topological spaces. Pre-requisite: Math.211, Tutorial: 1h/W Math.421 Operations Research, 2h/W Descriptions: Linear Programming. Simplex method. Duality and sensitivity analysis. Transportation and assignment problems. Network models. Dynamic programming. Pre-requisite: Math.215, or Math.111 for Phys. -comp. Sci. Tutorial: 1h/W Math 422 General Relativity, 2h/W Descriptions: 1) Equivalence and covariance principles. 2) Tensor analysis. 3) Gravity as a metric phenomenon. 4) Einstein equation in vacuum. 5) Schwarzschild solution and black holes. 6) Perihilion shift and bending of light rays in gravitational fields. 7) Einstein equation in matter and cosmological models. Pre-requisite: Math.223, Tutorial: 1h/W Math 423 Quantum Mechanics, 2h/W Descriptions: Problems with Classical mech. (Black body radiation, photoelectric effects, stability of Hydrogen atom). Old Mechanics and its problems. Uncertainty principle, Particle-Wave duality and Schrödinger equation. Some Math. Results and Ehrenfest th. (Contact with Classical Mech.). Solving Schrödinger equation for some constant potential. Schrödinger equation for the harmonic Oscillator. Schrödinger equation for the Hydrogen atom. Time independent perturbation theory. Pre-requisite: Math.223, Tutorial: 1h/W Math. 424 Advanced Mechanics, 2h/W Descriptions: Dynamics of rigid bodies: ( Euler’s case of free body motion. Explicit solution. Poinsot’s geometric interpretation of the motion. Study of stability of uniform rotation. Lagrange’s case of motion of a symmetric body. Separation of variables. Qualitative analysis of motion. Stability of standing gyroscope. Conditional integrable cases. Gyrostat. Integrable cases. Gyroscopic stabilization. Motion of a body in the field of a far Newtonian centre. Motion of a body not fixed from a point. Motion of a top on a horizontal plane. Attitude dynamics of Earth satellite (case of circular orbit). Analytical Dynamics(Hamilton’s principle. Poincare-Cartan invariant. Canonical transformations. Hamilton-Jacobi equation. Separation of variables. Integrable and nonintegrable systems. Action-angle variables. Regular and chaotic motions). Pre-requisite: Math.223, Tutorial: 1h/W Math. 425 Hydrodynamics, 2h/W Descriptions: 1 – Fundamental principles. 2 – Some general theorems. 3 – Potential flows. 4 – Two-dimensional fluid motion. 5 – The viscous fluid. Pre-requisite: Math.223, Tutorial: 1h/W Math 426 Modeling and Simulations, 2h/W Descriptions: Continuous time population models (Malthus, Logistic, Their equilibrium and stability) - Discrete time population models (Logistic, Ricker Their equilibrium and stability) Multi-species population models (Predator-prey, Competition, epidemics, fractals Equilibrium and stability). Space Shuttle motion. Tacoma bridge accident. Multiobjective optimization (crop spray, fishing fleets, radiotherapy etc…) Then student projects are done. Some applications of control theory. Pre-requisite: Math.215, Tutorial: 1h/W Math. 427 Theory of Elasticity (2), 2h/W Descriptions: Mechanical and thermo dynamical foundations - Uncoupled thermoelastic theory - Thermal stress analysis for elastic systems and some basic problems - Thermal stress analysis for viscoelasti and plastic systems - Mechanics of anisotropic elastic bodies - Theories of orthotropic plates and shells (Bending, Buckling and vibration). Pre-requisite: Math.223, Tutorial: 1h/W Math 428 Boundary Value Problems, 2h/W Descriptions:Laplace’s equation: in rectangular, polar, and cylindrical coordinate systems, Solution in the plane. Use of complex analytic functions and conformal mapping. Boundary conditions (Dirichlet and Neumann). Fourier method of separation of variables. Well-posed and ill-posed problems. Two and three dimensional examples from potential theory, electrostatics, hydrodynamics in different coordinate systems. The Heat Equation: Heat flow in a rod. Separation of variables. Axisymmetric flow of ground water. The Wave Equation: Solution of vibrating string problem, D’Alembert’s solution. Pre-requisite: Math.214, Tutorial: 1h/W Math 429 Partial Differential Equations, 2h/W Descriptions: Introduction to Partial differential equations – order –homogenous and non homogenous – degree-linear and nonlinear- Heat equation, Wave equation and Laplace's equation in both one and higher dimensions. Separation of Variables, boundary value problems, Fourier Series. Sturm-Liouville eigenvalue problems, eigenfunction expansions. Green's function methods. Fourier and Laplace Transform techniques. Applications. Pre-requisite: Math.214, Tutorial: 2h/W Math. 431 Statistics Theory (2), 2h/W Descriptions: hypotheses testing: Applications( tests of hypotheses of two samples - tests concerning meas- tests concerning difference between meas- tests concerning difference between proportions- Goodness of fit- The wilcoxon two sample rank test.- tests based on Runs). 2- Hypotheses Testing Theory:(Statistical hypotheses. Testing a Statistical hypotheses- Losses and risks- Neyman pearson Lemma- The Power function of a test- Likelihood ratio test). 3- Non-Parametric Test(The sign test - The signed rank test- Rank sum test: the U test, the H test). Pre-requisite: Math.231, Tutorial: 1h/W Math 432 Stochastic Processes (1), 2h/W Descriptions:Introduction in Stochastic processes - Poisson Processes and Their Generalizations - Markov Chain - Birth-Death Processes. Pre-requisite: Math.231, Tutorial: 1h/W Math. 434 Analysis of Variance, 2h/W Descriptions: 1- One–way ANOVA, balanced and non-balanced designs. 2- Two –way ANOVA, with fixed effect and interaction. Pre-requisite: Math.231, Tutorial: 1h/W Math. 435 Time Series and Forecasting, 2h/W Descriptions: 1- Time series models- 2- AR,MA,ARMA and their properties - 3- Parameter estimation - 4- Model identification - 5- Forecasting -6- The Box –jencins approach to forecasting -7 –Second time series models. Pre-requisite: Math.231, Tutorial: 1h/W Math. 436 Reliability Theory, 2h/W Descriptions: 1. Introduction to reliability theory- 2. Network modeling and evaluation of simple systems - Network modeling concepts- Series and Parallel systems- Series-parallel systems- Partially redundant systems- Standby redundant systems. 3. Network modeling and evolution of complex systems- Modeling and evolution concepts- Conditional probability approach- Cut set method -Tie set method- Event trees. 4. Probability distributions in reliability evaluation- General reliability functions- Evaluation of the reliability functions- Shape of reliability functions- The Poisson distribution- The normal distribution- The exponential distribution- The weibull distribution-The gamma distribution. 5- System reliability evaluation using probability distributions. Pre-requisite: Math.231, Tutorial: 1h/W Math 441 Data Structure, 2h/W Descriptions: Basic elements of data structure. Topics covered in the course includes: data abstraction, a survey of linear data structures (array representations and applications, stack and queues, array application and representation, implementation of recursion using stack, linked list, ), nonlinear data structures (tree, graph and priority queues), file organization and access methods, stack and queues, memory management, a discussion of more advanced internal and external sort a search algorithms, and the trade-offs involved in the use of different data organizations. The emphasis will be on algorithm analysis and trade-offs study. Practical : 1h/W Math. 442 Neural Networks, 2h/W Descriptions: Introduction - Simple neural nets for pattern classification - Back propagation neural net and radial basis functions - Pattern association - Neural nets based on competition - Other neural networks - Applications of neural networks. Practical : 1h/W Math 443 Computer Graphics, 2h/W Descriptions: Overview of Computer Graphics - Basic principles - Graphics Systems and Primitives – Library / system support for graphics - Point plotting - Straight line drawing - Curved line drawing - Two-Dimensional Graphics - Mathematical background and Coordinate System – illumiuation, shoding, renderg and texturing - Transformations (Translation, Scaling, Rotation) – Animation -Approaches (segments vs. direct) - Filling (Painting) – Windowing – Clipping. Advanced software tools. Practical : 1h/W Math. 444 Design and Analysis of Algorithms, 2h/W Descriptions: Building existing skills in mathematical analysis of algorithm complexity, including lower bounds, worst-case and average-case behavior. General techniques in algorithm design (such as divide and conquer, greedy and dynamic programming approaches) in the context of problem domain like graph, sorting, searching and optimization problems. Introduction the topic of NP-complete problems. Pre-requisite: Math.241,Practical : 2h/W Math. 445 Image Processing, 2h/W Descriptions: 1- fundamentals (What is digital image processing-Digital image representation-Image types). 2- Intensity transformatins function( Histogram processing- Spatial filtering). 3- Image restoration( A model of the image degradation- Noise models - Direct inverse filtring). 4- Color image processing(Color image representation-Converting to other color spaces- Color transformatins). 5- Image compression( Coding redundancy- Interpixel redundancy- JPEG compression). 6- Image segmentation( Point, line and edge detection – Thresholding - Region-Based segmentation). 7- Wavelets( The Fast wavelet Transform- Wavelet decomposition structures- Wavelet in image processing). Pre-requisite: Math.241, Practical : 1h/W Math. 446 Special issues in Computer science (2) 2h/W Pre-requisite: Math.241, Practical : 1h/W Math 401 Complex Analysis, 2h/W Descriptions: 1–The complex number system- 2–Functions of complex variable. 3–Complex differentiation and the Cauchy– Riemann Equation. 4– Complex integration and Cauchy Theorem- 5– Cauchy's integral formulas and related Theorems. 6 – Taylor and Laurent's series. Pre-requisite: Math.206, Tutorial: 2h/W Math. 402 Linear Programming, 2h/w Introduction of Linear programming problems- Geometry solutions of LP problems- The simplex method- Algebraic manipulations - Dual problem- The simplex method in matrix notations-Game theory and its application in economic. Pre-requisite: Math.206, Tutorial: 2h/W Math.400 Project of Research and Report, 1h/W for two semesters The project of research and report, to develop students to use their scientific knowledge, their ability to plan and execute an extended experimental or theoretical investigation and use all their communication skills to describe their results. To provide an understanding of some techniques of research, including the presentation of results. They should have produced an impressive report on their project and discuss its content with confidence.

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