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بسم هللا الرحمن الرحيم Al Imam Muhammad Bin Saud Islamic University Faculty of Sciences Department of Mathematics Bachelor of Science in Applied Mathematics الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Calculus (1) Course Code MAT Course Num. 101 Course Name Credit Hours Lec Lab Tut 4 3 0 2 Calculus (1) Prerequisites Objectives: - To understand the concept of limits and continuity of a function. To be able to find and interpret the derivatives of functions. To understand the meaning of derivative in terms of a rate of change and local linear approximation and to use derivatives to solve a variety of problems. To understand the meaning of definite integral as a limit of Riemann sum. To be able find the integral of elementary functions. Syllabus: - - - - Limits and Continuity: The Concept of Limit, Computation of Limits, Continuity and its Consequences, The Method of Bisections, Limits Involving Infinity, Asymptotes, Formal Definition of the Limit, Exploring the Definition of Limit Graphically. Differentiation: Tangent Lines and Velocity, The Derivative, Computation of Derivatives: The Power Rule, Higher Order Derivatives, The Product and Quotient Rules, The Chain, Derivatives of Trigonometric Functions and their inverses, Derivatives of Exponential and Logarithmic Functions, Hyperbolic functions and their inverses, Derivatives of hyperbolic functions, Implicit Differentiation, The Mean Value Theorem, Numerical Differentiation. Applications of Differentiation: Linear approximation and Newton’s Method, Indeterminate Forms and L’Hopital’s Rule, Extrema Values, Monotonic Functions and the First Derivative Test, Concavity and the Second Derivative Test, Overview of Curve Sketching, Optimization, Related Rates. Integration: Anti-derivatives, Sums and Sigma Notation, Partitions and Reimann sums, Area, The Definite Integral, The Fundamental Theorem of Calculus, Indefinite Integral and Integration by Substitution, Area between curves. References: - Calculus, Early Transcendental Functions, Robert Smith, Roland Minton, McGrawHill Science Engineering, 2007. Calculus, O. Swokowski, et al, PWS Pub. Co.; 6th edition (1994). Calculus Early Transcendentals, C. Henry Edwards, David E. Penney, Prentice Hall, 2008. 2 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS General Physics (1) Course Code Course Num. PHY 101 Course Name Credit Hours Lec Lab Tut 4 4 2 0 General Physics (1) Prerequisites Objectives: Students will develop an understanding of some of the fundamental laws of nature and their mathematical representation. This will provide them with skills in interpreting natural phenomena in terms of the motion and interaction of masses on macroscopic, microscopic and molecular levels. They will extend their understanding of macroscopic phenomena to include the effects of stationary and moving charges and the role they play in electromagnetic and atomic and nuclear interactions. Syllabus: - - - - - - Vectors: Coordinate systems and frame of reference, vector and scalar quantities, some properties of vectors, components of a vector and unit vectors. Motion in one dimension: Displacement, velocity and acceleration, one dimensional motion with constant acceleration, freely falling objects, kinematics equations derived from calculus. Motion in two dimensions: Displacement, velocity and acceleration vectors , two dimensional motion with constant acceleration, projectile motion, uniform circular motion, tangential and radial acceleration, relative velocity and relative acceleration. Particle dynamics: the concept of force, Newton’s first law, Newton’s second law, Newton’s third law, some applications of Newton’s law, Newton’s second law applied to circular motion, non uniform circular motion. Electric field: properties of electric charges, insulators and conductors, Coulomb's law, electric field created by one charge and group of charges, electric field lines, motion of charged particles in uniform electric field, the oscilloscope. Electric potential: potential difference and electric potential, potential difference in a uniform electric field, electric potential and potential energy due to point charges, potential of a charged conductor. Current and resistance: Electric current, resistance an Ohm’s law, resistance and temperature, electrical energy and power. Direct current circuits: electromotive force, resistors in series and in parallel, kirchhoff’s rules, RC circuits, electrical instruments. Magnetic fields: the magnetic field, magnetic force on a current-carrying conductor, torque on a current loop in a uniform magnetic field, motion of a charged particle in a magnetic field, the hall effect, the quantum hall effect. References: - Physics for Scientists and Engineers (with modern physics) –by Raymond A. Serway, and John W. Jewett – Brooks Cole, 2003. Physics for Scientists and Engineers with modern physics, Randall D. Knight, 3 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS General Chemistry Course Code Course Num. CHE 101 Course Name Credit Hours Lec Lab Tut 4 4 2 0 General Chemistry Prerequisites Objectives: To familiarize the student with the basic principles and concepts of Inorganic Chemistry. Syllabus: - - - - - - - - Atoms, Molecules and Ions: The atomic theory, the structure of the atom, Atomic number, Masse number and Isotopes, the periodic table, Molecules and ions, Chemical formulas, Naming compounds Masse Relationships in chemical reactions: Atomic mass, Avogadro’s number and molar mass, Molecular mass, the mass spectrometer, Experimental determination of empirical formulas, chemical reaction and chemical equations, Amounts of reaction and reactants and products, Limiting reagents Reaction in aqueous solutions: General proprieties of aqueous solutions, Precipitation reactions, Acid-Base reactions, Oxidation-Reduction reactions; Concentration solutions, Gravimetric Analysis, Acid Base Titrations, Redox Titrations. Gases: Substance that exist as Gases, Pressure of a Gas; The Gas Laws, The ideal gas equation, Gas Stoichiometry, Dalton’s Law of Partial Pressures; The kinetic molecular theory of gases. Intermolecular forces and liquids and solids: The kinetic molecular theory of liquids and solids, Intermolecular forces, Proprieties of liquid, Phase changes, Phase Diagrams. Physical proprieties of solutions: Types of solutions, a molecular view of the solutions process, concentration units, the effect of the temperature on the solubility, the effect of pressure on the solubility of gases, colloids Chemical equilibrium: The concept of equilibrium and the equilibrium constant, writing equilibrium constant expression, the relationship between chemical kinetics and chemical equilibrium, what does the equilibrium constant tell us? Factor that affect chemical equilibrium. Chemical Kinetics: The rate of reaction, the rate Law, the relationship between reactant concentration and time, Activation Energy and temperature dependence of rate constants, reaction mechanisms, Catalysis. Molecule bonding: Molecular geometry, dipole Moment, valance bond theory, Hybridization of atomic orbitals, Molecular orbital theory, Molecular orbital configurations, References: - Chemistry, Raymond Chang, Williams College, Mc Graw Hill, Higher Education, 9 th Edition. Chemistry & Chemical Reactivity, by Kotz, John C. & Treichel, Paul, 5th edition 4 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Calculus (2) Course Code MAT Course Num. 102 Course Name Credit Hours Lec Lab Tut 4 3 0 2 Calculus (2) Prerequisites MAT 101 Objectives: - To learn different techniques of integration. To understand the applications of definite integrals to physics and Engineering. To develop the basics of the calculus of infinite series, and their applications. To demonstrate ability to work with polar coordinates and parametric equations. Syllabus: - - - Integration Techniques: Brief Review of Integration by Substitution, Integration by Parts, Integration of Rational Functions Using Partial Fractions, Trigonometric Techniques of Integration, Integrals involving logarithmic, exponential, and hyperbolic functions, Improper Integrals, Numerical Integration. Applications of definite integrals. Applications of Definite Integrals: Volumes By slicing, Volumes using Cylindrical Shells, Arc Length and Surface Area, Application to physics and Engineering. Infinite Series: Sequences of Real Numbers, Convergence and Divergence of Infinite Sequences, Infinite Series, Remarkable Infinite Series (geometric series, p-series, alternating series, telescoping series), Convergence Tests for Positive Series (ratio test, root test, comparison and limit comparison test, integral test), Alternating Series, Absolute and Conditional Convergence, Power Series, Differentiation ad Integration of power series, Taylor and Maclaurin Series, Convergence of Taylor series, Applications of Taylor and Maclaurin Series, Fourier Series, Periodic Functions, Convergence of Fourier Series, Fourier Cosine and Sine Series. Parametric Equations and Polar Coordinates: Plane Curves and Parametric Equations, Calculus and Parametric Equations, Arc Length and Surface in Parametric Equations, Polar Coordinates, Calculus and Polar Coordinates, Conic Sections, Study of Conic Sections in Polar Coordinates. References: - Calculus, Early Transcendental Functions, Robert Smith, Roland Minton, McGrawHill Science Engineering, 2007. Calculus, O. Swokowski, et al, PWS Pub. Co.; 6th edition (1994). Calculus Early Transcendentals, C. Henry Edwards, David E. Penney, Prentice Hall, 2008. 5 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Introduction to Probability & Statistics Course Code Course Num. STA 111 Course Name Credit Hours Lec Lab Tut Introduction to Probability & Statistics 3 3 0 1 Prerequisites MAT 101 Objectives: This course is designed to equip the students with a working knowledge of probability, statistics. The major objective of the course is to help the students to develop an intuition and an interest for random phenomena, and to introduce both theoretical issues and applications that may be useful in real life. By the completion of the course, students will be familiar with ideas of statistical modeling, data analysis and interpretation. They will have learned to use one of the statistical package EXCEL. Syllabus: - - - - Descriptive Statistics: Variables and Data, Types of Variables, Graphs for Categorical Data, Graphs for Quantitative Data, Relative Frequency Histograms, Describing a set of Data with Numerical Measures, Measures of Center, Measure of Variability, On the Practical Significance of the Standard Deviation, Bivariate Data, Graph for Qualitative Variables. Using Technology: Creating, Listing and Describing Data in EXCEL. Counting: Counting Principles, Factorial Notation, Binomial Coefficients, Permutations, Combinations, Tree Diagrams. Basic Probability: Sample Space and Events, Axioms of Probability, Finite Probability Spaces, Infinite Sample Spaces. Conditional Probability and Independence: Conditional Probability, Finite Stochastic Processes and Tree Diagrams, Total Probability and Bayes’ Rule, Independent Events, Independent Repeated Trials. Random Variables: Random Variables, Probability Distributions of Finite Random Variable, Expectation of a Finite Random Variable, Variance and Standard Deviation, Functions of Random Variables, Discrete Random Variables in General, Continuous Random Variables, Cumulative Distribution Function. Using Technology: Generating a Random Sample in EXCEL. Discrete and Continuous Univariate Distributions: Bernoulli Trials, Binomial Distribution, Poisson Distribution, Normal Distribution, Evaluating Normal Probabilities, Normal Approximation of the Binomial Distribution, Geometric Distribution, Exponential Distribution. Using Technology: Binomial Probabilities, Normal Probabilities, and Normal Probability Plots in EXCEL. References: - Probability and Statistics in Engineering, William W. Hines, Douglas C. Montgomery, Connie M. Borror, David M. Goldsman, John Wiley & Sons Inc, 2004. Introduction to Probability and Statistics, William Mendenhall, Robert J. Beaver, Barbara M. Beaver, Duxbury Press, 2006. 6 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Computer Programming (1) Course Code CS Course Num. 140 Course Name Credit Hours Lec Lab Tut 4 4 2 0 Computer Programming (1) Prerequisites Objectives: - Introduce the student to the fundamentals of object-oriented programming, with emphasis on understanding functions, methods, variables, and control structures. Introduce the student to the basic concepts of classes and objects, and how to design object-oriented solutions for problems. Introduce the student to various tools used in programming, including editors, compilers, linkers, and loaders. Syllabus: - Fundamental concepts of object-oriented programming. Data types, control structures, functions, and arrays. Algorithms and problem-solving. An introduction to the concept of abstract data types and their implementation using classes. Analysis of problems using object-oriented concepts. Program correctness and verification. The mechanics of running, testing, and debugging. References: - C++ HOW TO PROGRAM, 5th edition, Deitel & Deitel, Prentice Hall. 7 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Calculus (3) Course Code MAT Course Num. 203 Course Name Credit Hours Lec Lab Tut 4 3 0 2 Calculus (3) Prerequisites MAT 102 Objectives: - To be able to apply derivatives and integrals to problems of arc length, and curvature. To be able to differentiate functions of two and three variables and to find the limits and extrema for multivariable functions; To be able to evaluate double and triple integrals in different systems of coordinates. To be able to define vector fields; find the line and surface integrals; be familiar with conservative fields and path independence, and use Green’s, divergence, and Stokes’s theorems. Syllabus: - - - Vectors and Geometry of Space: Vectors in Space, Dot Product, Cross Product, Lines and Planes in Space, Cylindrical and Spherical Coordinates. Vector-Valued Functions: Vector-Valued Functions of one variable, Calculus of Vector Functions, Motion in Space, Curves and Parameterization, Tangent and Normal Vectors. Functions of several variables and Partial Differentiation: Functions of Several Variables, Limits and Continuity, Partial Derivatives, The Total Derivative, The Gradient and Directional Derivatives, Tangent Plane, Chain Rule, Extrema, Taylor's Series and Approximations for functions of two variables. Multiple Integrals: Double Integrals in Cartesian Coordinates, Areas and Volumes, Double Integrals in Polar Coordinates, Triple Integrals in Cartesian Coordinates, Triple Integrals in Cylindrical and Spherical Coordinates. Vector Calculus: Line and Surface Integrals, Curl and Divergence, Green’s Theorem, Divergence Theorem, Stoke’s Theorem, Some Physical Application of Vector Calculus. References: - Calculus, Early Transcendental Functions, Robert Smith, Roland Minton, McGrawHill Science Engineering, 2007. Calculus, O. Swokowski, et al, PWS Pub. Co.; 6th edition (1994). Calculus Early Transcendentals, C. Henry Edwards, David E. Penney, Prentice Hall, 2008. Calculus, L. Hostetler & Edwards, Houghton Mifflin Publisher, 8th ( 2005). Advanced Engineering Mathematics, E. Kreyszig, John Wiley & Sons , INC 8th ed (1998). 8 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Elements of Sets & Structures Course Code MAT Course Num. 220 Course Name Credit Hours Lec Lab Tut Elements of Sets & Structures 3 2 0 2 Prerequisites MAT 101 Objectives: - To give students the rudiments of mathematical logic and set theory and introduce the important concepts of relations and their types. To let students be familiar to the formal definitions of function and binary operations and study their elementary properties. To expose students to some abstraction by presenting the group concept and studying some of its elementary properties. Syllabus: - - - Elementary logic and set theory: Simple and compound statements, Logical connectives, Truth tables, Basic logic laws, Methods of proofs, Mathematical induction, Operations on sets, Basic laws of set theory, Cartesian product of sets. Relations and functions: Basic definitions on relations, Binary relations and their types, Equivalence relation and set partition, Functions and their types, Bijective function and its inverse. Binary operations: Definitions and basic properties, Identity and inverse elements, Semigroups and monoids. Groups: Definitions and basic properties, Cayley tables, Subgroups, Group order, Order of an element and cyclic groups, Modular groups and symmetric groups. References: - Set Theory and Related Topics, K. Heal & K. Rickard, Cambridge University Press (1997) An Introduction to Mathematical Reasoning: Numbers, Sets and Functions, P. Eccles, Academic Express, 1997. Modern Abstract Algebra, F. Ayres, Schaum's Outline, McGraw-Hill (1965). 9 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Math Software Course Code Course Num. MAT 251 Course Name Credit Hours Lec Lab Tut 2 2 2 0 Math Software Prerequisites MAT 101 Objectives: - - To provide an introduction to the use of some of the high-level mathematical programming language such MATLAB, Maple and Mathematica, as a practical aid in doing mathematics. To provide the student with some basic skills in the use of this software without attempting deep coverage. In the following the phrase “math software “ refers to the specific math language used by the instructor. Syllabus: - - - - - Starting with MATLAB: Introduction to the software, Command window, help and lookfor commands, arithmetic operations, Display Formats, Built-in functions, Variables assignment, Command line editing… Arrays: Creating arrays (vectors, matrices), linspace command, some major matrices, operators, Matrix operations in MATLAB, Array addressing, Adding and deleting elements, Strings… Operators: Operator Precedence, Relational operations, Logical operations, all and any commands, find command, sort command, max and min command… 2D and 3D graphs: Plot and ezplot command, fplot command, multigraphs plots, others plot commands, histograms, formatting a plot, 3D line plot, Mesh and Surface plots, view command… Script files: Creating and saving a file, disp and fprintf commands, loading a file, search path, defining functions, structure of a function file, inline function, feval command, local and global variables… Programming: If-else structure, for and while loops, Break and continue commands, Switch-case statement… Symbolic toolbox: Symbolic object and expressions, algebraic expression manipulation, factorization, simplification, solving equations… References: - MATLAB: An Introduction with Applications, 3rd Edition Amos Gilat, The Ohio State Univ. 2008. MATLAB Primer, K. Sigmon & T. Davis, Champan & Hall, 6th ed., 2002 . Maple V: learning Guide, K. Heal & K. Rickard, Springer Verlag, 1996. Mathematica by example, M. Abell & J. Braselton, Academic Express, 1997. 10 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS General Physics (2) Course Code PHY Course Num. 106 Course Name Credit Hours Lec Lab Tut 4 4 2 0 General Physics (2) Prerequisites PHY 101 Objectives: Students will develop an understanding of some of the fundamental laws of nature and their mathematical representation. Syllabus: - - - Sources of the Magnetic Field: the Biot-Savart’s law, the magnetic force between two parallel conductors, ampere's law, the magnetic field of a solenoid, magnetic flux, Gauss's law in magnetism, displacement current and the generalized Ampere's law. Faraday's law: Faraday's law of induction, motional emf, Lenz's law, induced emfs and electric fields, generators and motors, Eddy currents. Inductance: self-inductance, RL circuits, energy in a magnetic field, mutual inductance, oscillation in an LC circuit, the RLC circuit. Image Formation: reflection, refraction, Dispersion and prisms, images formed by flat mirrors, images formed by spherical mirrors, Images formed by refraction, thin lenses, applications. Interference and Diffraction of Light Waves: conditions for interference, Young’s double –slit experiment, intensity distribution of the double-slit interference pattern, introduction to diffraction patterns, phasor addition of waves, diffraction patterns from narrow slits. References: - Physics for Scientists and Engineers (with modern physics) –by Raymond A. Serway, and John W. Jewett – Brooks Cole – 6th Edition (July 21, 2003) Randall D. Knight, physics for scientists and engineers with modern physics, (December, 2003) 11 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Introduction to Number Theory Course Code MAT Course Num. 222 Course Name Credit Hours Lec Lab Tut 3 3 0 1 Introduction to Number Theory Prerequisites MAT 220 Objectives: - To expose students to the fascinating subject of number theory. To let students gain basic knowledge in number theory which is essential for subsequent courses in mathematics and computer science. To prepare student for abstract mathematics courses like “ Modern Algebra”. Syllabus: - - - Basics: Classical and strong mathematics inductions, well-order principal, binomial theorem. Divisibility and factorizations: Divisibility properties, the division algorithm, representation of a number relative to arbitrary base, the binary digit system, Fundamental theorem of arithmetic, infinitude of prime numbers, greatest common divisors and least common multiple, Euclidean algorithm and Bezout’s identity. Congruences: Congruence and modular arithmetic, Diophantine linear equation, Chinese Remainder Theorem and system of linear Diophantine equations. Wilson’s Theorem, Little Fermat's Theorem, Euler phi function and Euler Theorem . Applications: divisibility tests, round-robin tournaments, pseudo primes, pseudorandom numbers, linear codes, Pythagorean triples and sum of two squares. References: - Elementary Number Theory, K. Rosen, ddison Wesley; 5th ed. (2004). An Introduction to Mathematical Reasoning : Numbers, Sets and Functions, P. Eccles, Academic Express, (1997). Elementary Theory of Numbers, W. Le Veque, Dover Publications (1990). 12 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Linear Algebra Course Code Course Num. MAT 223 Course Name Credit Hours Lec Lab Tut 4 3 0 2 Linear Algebra Prerequisites Objectives: - To introduce students to the subject of linear algebra, this is essential for subsequent courses in mathematics and computer science. To let students be familiar with basics of matrix theory. To let students be familiar with basics of vector spaces and linear transformations. To prepare students for more abstract math courses like “Modern Algebra”. Syllabus: - - - - Matrices and Gauss Elimination: Elementary row operations, Transpose of a matrix, Inverse of a square matrix, Linear equation systems and Gauss eliminations, Determinants and their properties, classical adjoint; Cramer’s rule. Vectors in R2 and R3: Dot product, projections, cross product, mixed product. Vector spaces: Basic definitions, subspaces, linear dependence and independence, bases and dimensions, Rank of a Matrix, Inner product spaces and Gram-Schmidt normalization, orthogonal matrices. Linear transformations: Basic definitions, the matrix of a transform, Kernel and Range of a linear transformation, Matrices of linear transformations, Coordinates and change of basis, homomorphism and isomorphism . Eigenvalues and Eigenvectors: Characteristic polynomial, diagonalization of matrices, Applications involving Powers of matrices. References: - Linear Algebra, Schaum's Outline, S. Lipschutz, M. Lipson, McGraw-Hill 3rd ed. (2000). Linear Algebra, S. Leduc, Cliffs Notes (1996). Linear Algebra: A Modern Introduction, D. Poole, Brooks Cole; 1st ed. (2002). 13 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Introduction to Differential Equations Course Code Course Num. MAT 231 Course Name Credit Hours Lec Lab Tut 3 3 0 1 Introduction to Differential Equations Prerequisites MAT 203 Objectives: - To be familiar with techniques for solving first order. To be familiar with techniques for solving second order equations with constant coefficients. To study The differential operator of order n and its use in solving general linear homogeneous differential equations with constants coefficients . To apply the power series method in searching for a solution of the second order linear differential equations with polynomial coefficients. To learn some techniques of simplifying differential equations by reducing equation order or exchanging variables. To know how to solve some types nonlinear differential equations. To be exposed to matrix calculus and use it in solving linear system of differential equations.. Syllabus: - - First order differential equations: separable equations, exact differential equations, homogeneous differential equations, and solution of general first order linear equations. Second order linear differential equations with constants coefficients: general solution of the homogeneous equation, particular solution of the none-homogeneous equation, the undetermined coefficients and variation of constants methods. Solving linear system of differential equations. The differential operator of order n and its properties: General linear homogeneous differential equations with constants coefficients. Power series solutions of second order linear differential equations with polynomial coefficients. Reducing order of differential equations. Exchanging variables. Bernoulli equations, Riccati equations. References: - A first course in differential equations with applications, Dennis G. Zill, 5th ed, PWS Kent Publishing Company (2000) Differential Equations, F. Ayres, Schaum's Outline, McGraw-Hill (1964). Ordinary Differential Equations, M. Tenenbaum and H. Pallard, Dover Publications (1985). 14 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Mathematical Statistics Course Code Course Num. STA 211 Course Name Credit Hours Lec Lab Tut Mathematical Statistics 4 3 0 2 Prerequisites MAT 102, STA 111 Objectives: - To teach students some important scientific concepts of statistics. To let students be familiar with distributions of random vectors. To expose students to concepts of Expectation and moments. To let students know limit theorems and some of their applications. To teach students techniques of estimations. To let students learn and use some tests of hypothesis. Syllabus: - - - - Basics: Moment generating functions and characteristic functions. Some special distributions: beta, uniform, gamma, Student, Chi-square and Fisher distributions. Geometric, hyper-geometric and multinomial distributions. Discrete and continuous multivariate distributions: Random vectors; multivariate density functions, conditional and marginal density functions. Independent random variables and their sum. Multivariate normal distribution, mean and covariance. Limits Theorems and inequalities: Convergence in law, in Probability and almost sure; Weak and strong law of large numbers; central limit theorems. Chebyshev, Hölder and Minkowski inequalities. Parameter estimations: introduction, maximum likelihood estimators, interval estimates, estimating the difference in means of two normal populations, approximate confidence interval for the mean of a Bernoulli random variable. Hypothesis testing: significance levels, tests concerning the mean of a normal population, testing the equality of means of two normal populations, hypothesis tests concerning the variance of a normal population. References: - Probability and Statistics in Engineering, William W. Hines, Douglas C. Montgomery, Connie M. Borror, David M. Goldsman, John Wiley & Sons Inc, 2004. Introduction to Probability and Statistics, William Mendenhall, Robert J. Beaver, Barbara M. Beaver, Duxbury Press, 2006. Probability and Statistics for Engineering and the Sciences, Jay L. Devore, Duxbury 2004. Introduction to Mathematical Statistics, R. Hogg, et al, Prentice Hall, 2004. 15 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Real Analysis Course Code Course Num. MAT 311 Course Name Credit Hours Lec Lab Tut 4 3 0 2 Real Analysis Prerequisites MAT 203 Objectives: - To give a careful and rigorous treatment of the main ideas of differential calculus which was taught to students in Calculus I and Calculus II . To let students gain experience in dealing with axiomatic thinking and concise proofs of calculus. To expose students to the rudiments of metric and topological spaces. Syllabus: - - Fundamentals: Elementary set theory, Countable and uncountable sets, The real numbers, Sequence of real numbers. Metric spaces: Definition, Open set, closed set, Neighborhood, Convergence and divergence of sequences, Cauchy sequences, Completeness, Completion of metric spaces. Continuity and derivative: Left and right limits, limits, continuity, and uniform continuity. Discontinuity of first and second kind. Variation and fluctuation of a function. Left and right derivatives. Derivatives of functions. Mean value theorem. Riemann’s Integral: The Riemann sum and the Riemann integral. The Fundamental Theorem. First and second mean value theorems. Zero sets and Riemann integrability. The Riemann integral on unbounded set. References: - Elementary Analysis, K. Ross, Springer Verlag 14th ed. (2003) Introduction to Real Analysis, R Bartle, D. Sherbert, Wiley; 3rd ed. (1999). 16 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Numerical Analysis (1) Course Code MAT Course Num. 333 Course Name Credit Hours Lec Lab Tut 4 3 2 1 Numerical Analysis (1) Prerequisites MAT 231, MAT 223 Objectives: This course provides an introduction of computational techniques for finding approximate solutions to difficult mathematical problems. Theory and practice approaches are taught. The error sources and the convergence of the algorithms will be estimated with respect to the various techniques used. The course will involve the use of Matlab or C++ in Lab. Syllabus: - - Introduction to data representation: Numerical Errors; Floating Point Representation; Round-off; Significant Digit; Error Propagation. Root Finding: Bisection Method, Newton’s Method, Secant Method, Fixed Point Iterations. Interpolation and Approximation: Taylor polynomials, Approximation of order n, Polynomial Error, Linear and Quadratic Interpolation, Lagrange Interpolation, Newton Divided Difference Method, Error Evaluation, Spline Interpolation. Numerical Integration and Differentiation: The Trapezoidal and Simpson Rules, Gaussian Quadrature, Numerical Differentiation. Numerical Solution of Linear Systems: Gauss Elimination, LU and Cholesky Decompositions, Iterative Methods: Jacobi and Gauss-Siedel Methods, Error Analysis Numerical solution of differential equations: Euler method, Runge-Kutta methods. Error and convergence analysis. References: - Elementary Numerical Analysis, 3rd Edition, Kendall Atkinson; Weimin Han; (2004). An Introduction to Numerical methods and Analysis, James F. Epperson, Wiley; (2002). Numerical Analysis, R. Burden and J. Faires, 8th ed., Brooks/Cole, 2001. 17 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Mathematical Methods Course Code Course Num. MAT 332 Course Name Credit Hours Lec Lab Tut 4 3 0 2 Mathematical Methods Prerequisites MAT 231 Objectives: - To tech students some important applied mathematical tools. To let students be familiar with the some special function. To let students study Fourier series in more depth and concise way. To let students be familiar with Laplace transforms and some of its applications for solving ordinary differential equations. Syllabus: - - - Series solutions of differential equations: Gamma and beta functions; Stirling’s formula; series solution of differential equations, power series solution of differential equations, Forbeniu's method for solving second order linear differential equations with non-analytic; Euler-Cauchy equations, Bessel’s equations and Bessel’s functions. Fourier Series: Complex form of Fourier series; the Fourier transform; Plancherel’s theorem; orthogonal and orthonormal functions; the mean convergence of Fourier series and Bessel’s inequality; closed and complete set of functions; the piecewise convergence of Fourier series; Dirichlet kernel and piecewise smoothness; differentiation and integration of Fourier series. Laplace Transforms: Basic Definitions and properties; first shifting theorem; partial fractions; differentiation and integration of Laplace transforms, Laplace transform of some particular discontinuous functions, the unit step function; Dirac function, shifting on the t- axes and second shifting theorem, inverse of Laplace transform, solving differential equations, linear system of differential equations. References: - Advanced Engineering Mathematics, E. Kreyszig, John Wiley & Sons, INC 8th ed. - (1998). Methods of Mathematical Physics, R. Courant & D. Hilbert, Wiley-Interscience; 1st ed (1989) Mathematics of Physics and Modern Engineering, I. Sokolnikoff & R. Redheffer, McGraw-Hill College; 2nd ed. (1966). 18 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Introduction to Operations Research Course Code Course Num. MAT 351 Course Name Credit Hours Lec Lab Tut 3 3 0 1 Introduction to Operations Research Prerequisites MAT 223 Objectives: The core subject of this course is Linear Programming (LP)، basically an optimization technique applicable to the solution of problems in which the objective function (expression to be optimized) and constraints are linear. Such techniques find applications across a very wide variety of subjects in engineering، economics، finance and manufacturing. The course should be accompanied by Lab sessions in order to learn students a known software of linear programming (e.g. TORA provided with Taha’s book). Syllabus: - - Introduction to Operations Research. Introduction to Linear programming: Linear programming formulations; Graphical Linear Programming Solution, Graphical Sensitivity analysis. The Simplex Method: Standard Linear Programming; Determination of Basic Feasible Solutions; The Simplex Algorithm. Special Cases of the Simplex: Degeneracy, Alternative optimum, Unbounded solution, Infeasibility. Duality and Sensitivity Analysis: Formulation of the Dual Problem; Relationship between Optimal Primal and Optimal Dual Solutions; Economic interpretation of Duality, Dual Simplex and Sensitivity Analysis. Special linear programming models: The transportation model; The assignment model, Critical Path and PERT. Introduction to Integer Linear Programming: Illustrative applications, Branch and Bound algorithm, Application to the Traveling Salesman Problem. References: - Introduction to Operations Research، by F. Hillier and G. Lieberman، 7th Edition، McGraw Hill، (2001). Operations Research: Applications and Algorithms by Wayne L. Winston, Wadsworth, 3rd Edition (1997). Operations Research: An Introduction by H. Taha، 8th Edition, Prentice Hall, 8th Edition، (2006). 19 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Complex Variables Course Code Course Num. MAT 312 Course Name Credit Hours Lec Lab Tut 4 3 0 2 Complex Variables Prerequisites MAT 311 Objectives: - To let students learn the complex extensions of elementary functions. To let students visualize the geometric meaning of complex functions. To let students learn complex derivatives and the Cauchy-Riemann equations, and the relationship between these concepts. To let students learn complex sequences, series, and power series. To let students learn complex integration, contour integrals, and Cauchy’s theorem Syllabus: - - - - - Basics: Euler formula and exponential form of a complex number, basic topological properties, functions of complex variable, elementary functions. Limits, continuity and uniform continuity. Continuity and differentiability: Limits , continuity and uniform continuity, derivative of a complex function at a point, Cauchy -Riemann equations and differentiability complex functions, derivatives of elementary functions, analytic function at a point, singular points, analytic function and harmonic functions, L’Hopital's rule. Complex integral: line integral and complex integral, complex form of Green’s theorem, Cauchy's and Cauchy-Goursat theorems, complex indefinite integral. Cauchy's integral formula, Argument, Rouche’s, Liouville's, and modulus theorems. Complex sequences and series: Basic definitions, tests of series absolute convergence, power series and uniform convergence, circle of convergence, differentiation and integration of power series, Taylor's series and Laurent's series. type of singular points, Picard’s theorem. Residues: Residues and the residue theorem with applications. Basic concepts of conformal mapping. References: - Complex Variables, M. Spiegel, Schaum's Outline, McGraw-Hill (1968). - Complex Variables and applications, R. Churchill and others, McGraw-Hill 5th ed. (1989). - Complex Variables: Introduction and Applications, M. Ablowitz, et al, 2nd ed. (2003). 20 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Modern Algebra Course Code Course Num. MAT 321 Course Name Credit Hours Lec Lab Tut 4 3 0 2 Modern Algebra Prerequisites MAT 222, MATH 223 Objectives: - To provide the student with a firm foundation in the abstract structure of algebra. To understand and practice the axiomatic methods of thinking. To be familiar with basics of group theory. To be familiar with rudiments of ring theory and integral domains. Syllabus: - Group theory: Definition of a Group, Subgroups, Cyclic Groups, Permutation Groups, Homomorphisms, Cosets Lagrange's theorem, Normal Subgroups, and Factor Groups - Structures of groups: Isomorphism Theorems, Conjugacy, Groups Acting on Sets, The Sylow Theorems. Finite Abelian Groups. Solvable Groups, Simple Groups. - Rings: Basic Definitions. Ring Homomorphisms, Quotient Rings. Ideals. Fields. Euclidean Domains. Principal Ideal Domains. Unique Factorization Domains. - Polynomial Rings: Definitions and Basic Properties. Polynomial Rings over Fields. Irreducibility criteria. References: - - Abstract Algebra, D. Dummit, R. Foote, John Wiley, 3rd edition (2004) Contemporary Abstract Algebra, J. Gallian, Houghton Mifflin Company; 5th Ed. (2001). A first course in Abstract Algebra, J. Fraleigh, Pearson Education, 1st Indian edition (2203). Abstract Algebra: An Introduction, T. Hungerford, Brooks Cole; 2nd ed. (1996). 21 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Combinatorics and Graphs Course Code Course Num. MAT 353 Course Name Credit Hours Lec Lab Tut 4 3 0 2 Combinatorics and Graphs Prerequisites MAT 223 Objectives: This course is divided in two parts: Combinatorics and Graphs. The objectives of the first part are to learn basics of counting, recursion and recurrence relations. The objective of the second part is to introduce students to graph theory, especially Euler and Hamiltonian graphs, planar graphs, coloring graphs, shortest paths, isomorphisms of graphs, spanning trees and network flow. Syllabus: Combinatorics: - Principles of counting: Addition and multiplication principles; inclusion-exclusion principle; Application of inclusion-exclusion; pigeonhole principle and applications, generating functions, multiplication of generating functions. - Recursion: Recurrence definitions. Recurrence relations; generalized binomial theorem. - Linear recurrence equations: Homogeneous and non homogeneous linear recurrences with constant coefficients; binary trees; random walks. Graphs: - Introduction to Graphs: Concepts and definitions, Isomorphisms of graphs, Subgraphs, Matrices of graphs. - Eulerian Graphs: characterization of Eulerian graphs, Chinese Postman problem; - Hamiltonian Graphs: Necessary and sufficient conditions for Hamiltonian graphs, Traveling Salesman problem; - Planar graphs: Euler's Formula, Kuratowski's theorem. - Graph colorings: Vertex colorings. Greedy algorithm. - Shortest path: Shortest path problems, Optimality principle, BFS, DFS, Djkstra's algorithm, - Trees: Spanning trees, minimum spanning trees, Kruskal's algorithm, Prim's algorithm, Greedy algorithm - Networks flow: Definitions, flow augmenting paths, cut sets, maximum flow; FordFullkerson algorithm, Minimum cost capacited flow problem. References: - Applied Combinatorics by Alan Tucker, Wiley; 5th Edition (2006). Discrete Mathematics and Its Applications by K. Rosen, Mc Graw-Hill, 6th Edition (2006). Graph Theory by V. Balakrishnan, Schaum's Outline, McGraw-Hill; 1st Edition. (1997). 22 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Partial Differential Equations Course Code MAT Course Num. 434 Course Name Credit Hours Lec Lab Tut Partial differential Equations 4 3 0 2 Prerequisites MAT 332 Objectives: The aim of this course is to introduce the students to the theory and applications of partial differential equations (PDEs), and to explore various methods of solution. On successful completion of this unit, students should be able to classify PDEs and solve them by using appropriate methods. Syllabus: - Introduction to PDEs: Definition of a PDE; degree, linearity; homogeneous and inhomogeneous equations; First order partial differential equations; The method of characteristics. - Scond-order equation: Classification as Parabolic, Hyperbolic, and Elliptic equations. - Classical PDEs of mathematical physics and Boundary-Value Problems: wave equation, heat equation, Laplace equation. Boundary Conditions; Definition of a Boundary-Value Problem. Dirichlet, Neumann, and Mixed BVP. - Analytic methods for solving PDEs: Separation of variables method; Method of characteristics; Fourier series; Solution of PDEs by Fourier series; Fourier and Laplace transform; Solving PDEs using Fourier and Laplace transform s. References: - Partial Differential Equations of Mathematical Physics R.B. Guenther & J.W. Lee. Prentice Hall/Dover publication Partial Differential Equations Methods and Applications, R. McOwen, Prentice Hall/Pearson Education. 23 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Introd. to Cryptography and Coding Course Code MAT Course Num. 461 Course Name Credit Hours Lec Lab Tut 4 3 0 2 Introd. to Cryptography and Coding Prerequisites MAT 321 Objectives : This course is divided in two parts: Cryptography and Coding Theory. The objective of the first part is to learn students basics of Cryptography through classical ciphers, cryptanalysis and some applications. The objective of the second part is to introduce students to elementary coding theory. Syllabus: Cryptography: - Classical Ciphers: Shift ciphers ،affine ciphers and substitution ciphers. Introduction to stream ciphers ،including linear recursive ciphers. - Introduction to cryptanalysis: the four levels of attack: known ciphertext, known plaintext, chosen plaintext and chosen ciphertext. Exponential ciphers and key distribution. Public-key cryptosystems: RSA, ElGamal and Massey-Omura. Signature schemes with applications. Coding Theory: - Introduction to codes: Error-detection ،error correction and information rate. Linear codes, Perfect codes. Cyclic codes. References: - Introduction to Modern Cryptography by J. Katz and Y. Lindell, Chapman and Hall/CRC, 1st Edition(2007) Elementary Number Theory by K. Rosen ،Addison Wesley; 5th Edition (2004). A First Course in Coding Theory، by R. Hill, Oxford University Press (1997). Coding Theory: A First Course by San Ling and Chaoping Xing, Cambridge University Press (2004). 24 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Numerical Analysis (2) Course Code MAT Course Num. 433 Course Name Credit Hours Lec Lab Tut 4 3 1 1 Numerical Analysis (2) Prerequisites MAT 333, MAT 434 Objectives: - To study basic finite difference methods for partial differential equations. - To understand the concepts of consistency, stability, and convergence. - To learn to solve partial differential equations on the computer. - To introduce finite element method. The course will involve the use of Matlab or C++ in Lab. Syllabus: - Advanced Numerical Linear Algebra: Least Squares Method, Matrix Eigenvalue Problems, Power Method, QR Factorisation. Finite difference techniques: difference equation replacement; implicit and explicit finite difference method. Boundary Value Problems for ODEs: Multistep Methods, Finite Difference Methods for Systems of Differential Equations. Finite Difference Method for PDEs: Numerical Solution of Elliptic PDEs, Numerical Solution of Parabolic PDEs, Numerical Solution of Hyperbolic PDEs, Finite Difference Method for Boundary Value Problems. Introduction to Finite Element Method: application to heat and Laplace equations. References: - Elementary Numerical Analysis, 3rd Edition, Kendall Atkinson; Weimin Han; (2004). Numerical Solution of Partial Differential Equations: An Introduction, K. W. Morton & D. F. Mayers: Cambridge University Press. An Introduction to Numerical methods and Analysis, James F. Epperson, Wiley; (2002). Numerical Analysis, R. Burden and J. Faires, 8th ed., Brooks/Cole, 2001. 25 الوولكة العربية السعودية جاهعة اإلهام هحود بن سعود اإلسالهية كلية العلوم قسن الرياضيات KINGDOM OF SAUDI ARABIA AL-IMAM MUHAMMAD BIN SAUD ISLAMIC UNIVERSITY FACULTY OF SCIENCES DEPARTMENT OF MATHEMATICS Modeling and Simulations Course Code Course Num. MAT 463 Course Name Credit Hours Lec Lab Tut 4 3 0 2 Modeling and Simulations Prerequisites Mat 333 and Mat 334 Objectives: This course provides an introduction to system modeling using both computer simulation and mathematical techniques. Emphasis will be on continuous and discrete-events simulation model development methodologies and implementation techniques. Syllabus: Introduction to Mathematical Modeling Process: Concept; Objectives; Methods and tools Mathematics is the natural modeling language; Definition of mathematical models. Modeling Continuous Systems: Modeling with Differential Equations: Population dynamic; Electrical Circuits; Mechanical Systems; Biological models (Lotka-Volterra systems, Predator-Prey systems). Modeling with Partial Differential Equations: Linear Temperature Diffusion; One-dimensional Hydrodynamic model. Case Studies: Heat diffusion, Wave vibration, Laplace Equation. Modeling Discrete Systems: Modeling with difference equations; Modeling with data; Discrete Velocity Models; Continuous Vs. Discrete Models Simulation: Block-Digrams; State-Space Model; Transfer Functions, State-space Vs. transfer function. Stability and pole locations; Introduction to Matlab\Simulink (Starting Simulink, Basic Elements, Building a System , Running Simulations); Simulation of some models (case study models) and Analysis of Simulation results References: - Kai Veltn, Mathematical Modeling and Simulation: Introduction for Scientists and Engineers, Wiley 2009. Steven T. Karris: Introduction to Simulink® with Engineering Applications, Orchard Publications (2006). 26