Elementary functions, limits, continuity, limits involving infinity, tangent lines, derivative of elementary functions, differentiation rules, chain rule, implicit differentiation, linear approximation, l'Hospital rule. Graph sketching (extrema, intervals of monotonicity, concavity), optimization. Antiderivatives, definite integrals, Fundamental Theorem of Calculus, integration by substitution, area between curves, improper integrals.

- Apply Calculus To Geometry And To Real World Problems (Such As Graph Sketching, Optimization, Related Rates, Area Computation).
- Compute Limits, Derivatives, Linear Approximations, And Integrals Using Various Techniques.
- Explain Some Important Concepts Of Calculus (Such As Limit, Continuity, Derivative And Integral).
- Justify Some General Results In Single-Variable Calculus From A Theoretical Point Of View.
- Use Technology To Investigate Limits, Graphs, And Integrals.

Integration techniques (by parts, by use of trigonometry, by partial fractions), volume and area of solids of revolution, arc length. Parametric curves: velocity vector, enclosed area, arc length. Curves in polar coordinates: enclosed area, conic sections. Sequences, series, convergence tests, alternating series, absolute convergence, power series, Taylor series, Fourier series.

- MATH105 with a minimum grade D

- Apply Calculus To Real World Problems Such As Area, Volume, And Arc-Length.
- Communicate Solutions Of Calculus Problems Through Peer Group Discussions And Written Assignments.
- Compute Derivatives Of Various Types Of Functions
- Compute Integrals And Taylor Series Using Various Techniques.
- Explain The Concepts Of Sequences, Series, Polar Coordinates And Parametric Equations.
- Justify The Convergence Of Sequences And Series.

Differential calculus of functions of one variable: functions of one variable, techniques of differentiation, derivatives of trigonometric, exponential, and logarithmic functions, chain rule, implicit differentiation, maximum and minimum values, increasing, decreasing and concave functions, inverse trigonometric functions, hyperbolic functions, some engineering applications. Integral calculus of functions of one variable: definite and indefinite integrals, techniques of integration (integration by substitution, integration by trigonometric substitutions, integration by parts, integration by partial fractions), applications of definite integrals in geometry, some engineering applications.

- Analyze The Properties Of Various Types Of Functions And Theirs Graphs
- Apply The Derivative To Model Various Engineering Problems
- Classify Integration Techniques And Integrate Different Types Of Functions
- Use The Integration To Compute Areas Between Curves, Volumes Of Solids, And Volumes Of Revolutions.

Differential calculus of functions of several variables: vectors, vector valued functions, functions of several variables, partial derivatives, chain rule, gradient and directional derivatives, extrema of functions of several variables. Quadratic surfaces. Vector fields and line integrals, double integrals in Cartesian and polar coordinates, triple integrals in Cartesian, cylindrical and spherical coordinates.

- MATH1110 with a minimum grade D

- Apply The Properties Of Vectors, Lines And Planes.
- Compute Partial Derivatives, Rate Of Change, And Extrema Of Functions Of Several Variables.
- Formulate The Concepts Of Vector-Valued Functions, Vector Fields And Line Integrals.
- Manipulate Multiple Integrals To Calculate Areas, Volumes, And Center Of Mass For Different Configurations.
- Use Critical Thinking For Analyzing Engineering Problems.

This course introduces the concepts of differential and integral calculus useful to students in business, economics. Among the topics studied are: curve sketching for some functions relevant to business and economics applications, derivatives and techniques of differentiation, exponential growth, anti-derivatives and methods of integration, definite and indefinite integrals with applications. The course also covers topics on partial derivatives and matrices, in addition to many applications in Business and Economics.

- Apply Mathematical Models And Tools To Various Business And Economics Problems
- Compute Derivatives Of Several Elementary Functions Including Multivariate Functions
- Evaluate Different Types Of Definite And Indefinite Integrals
- Identify Basic Properties Of Several Elementary Functions
- Perform Basic Matrix Operations

Problem solving, fair divisions, Mathematics of Apportionment, Euler circuits, network, scheduling methods, population growth, symmetry, fractal geometry.

- Analyze Appearances Of Patterns And Symmetry In Nature.
- Apply The Concepts And Methods Of Graphing Theory To Solve Problems In Scheduling, Networking And Routing.
- Use Quantitative Methods To Analyze Real-World Problems Involving Social And Management Issues.

Systems of linear equations, matrices and determinants. Vector spaces, inner product spaces. Matrix representations of linear operators. Eigenvalues, eigenvectors, and Cayley-Hamilton Theorem.

- MATU1435

- Communicate Concepts And Results In Linear Algebra To Their Peers.
- Explain The Main Concepts Of Linear Algebra (Matrices, Determinants, Vector Spaces, And Linear Transformations).
- Solve Systems Of Linear Equations And Problems Related To Linear Transformations.
- Use Computational Software To Determine Solutions Of Systems Of Linear Equations, And The Eigenvalues And Eigenvectors Of A Matrix.

Compound and simple propositions, truth table, quantifiers, propositional calculus, methods of proof. Sets and operations on it. Cartesian products, relations, equivalence relation, order relation. Functions. Cardinality.

- Demonstrate ability to integrate knowledge and idea in a coherent manner.
- Justify procedures of abstract proofs by applying Logic.
- Develop counterexamples to claim assertions and to make interpretation of statements.
- Recognize procedures of abstract proofs.

Euclidean space: dot product, cross product, lines, planes, surfaces. Parametric curves in space. Functions of several variables: limits, continuity, partial derivatives, tangent plane, linear approximation, chain rule, gradient, directional derivative, extrema, Lagrange multipliers. Double integrals, applications (area, volume, center of mass), change to polar coordinates. Triple integrals, change to cylindrical and spherical coordinates. Vector fields, line integrals, conservative fields, Green's theorem.

- Apply Calculus To Real World Problems (E.G., Optimization In Several Variables, Area And Volume Computations).
- Demonstrate Knowledge Of Various Curriculum For Students With Special Education. (Plo1,Plo2)
- Discuss Solutions Of Multi-Variable Calculus Problems With Their Peers.
- Explain Important Concepts Of Multi-Dimensional Calculus (Such As: Euclidean Space, Partial Derivatives, Multiple Integrals, Vector Fields).
- Solve Problems Related To Differential And Integral Calculus In Multi-Dimensional Spaces.
- Work In A Group In Calculus Peer-Tutoring Sessions.

Properties of R. Completeness of the line, supremum and infimum, Cantor's nested intervals theorem. Sequences, limits and their properties, monotone sequences, Bolzano-Weierstrass Theorem, Cauchy criterion, properly divergent sequences. Series, absolute and conditional convergence, tests of convergence. Topological properties of R, Metric spaces and general topology.

- MATH205 with a minimum grade D

- Describe The Topological Properties Of Metric Spaces.
- Examine The Properties Of The Real Numbers.
- Solve Problems Related To Convergence Of Sequences And Series Of Real Numbers.

Ordinary differential equations: first order differential equations: separable; homogeneous, linear, Bernoulli, exact-integrating factors. Second order linear differential equations: homogeneous equations with constant coefficients; undetermined coefficients method; variation of parameters method; Euler's Equation; Non-homogeneous equations; higher order linear equations; Solving Homogeneous and Non-Homogeneous Systems of Differential Equations using eigenvalues and eigenvectors. Laplace transforms: basic properties; solving initial value problems using Laplace; solving integral equations; solving systems of differential equations by Laplace transform.

- MATH2220 with a minimum grade D

- Apply Laplace Transform To Solve Initial Value Problems
- Employ Initial Value Problems To Model Engineering Problems
- Solve System Of Differential Equations
- Solve Various Types Of First And Second Order Differential Equations

This course is an introduction to linear algebra. Topics to be covered include solving systems of linear equations, matrices and determinants; Vector spaces, inner product spaces; Eigenvalues, eigenvectors, diagonalization; Least Squares fitting

- MATH1110 with a minimum grade D

- Demonstrate understanding of the main concepts of Linear Algebra (including matrices, determinants, vector spaces).
- Solve a system of homogeneous or non-homogeneous linear equations.
- Find the inverse of a matrix and the solution of a non-singular system using determinants.
- Find the Eigenvalues, Eigenvectors, and diagonalize a matrix.
- Analyze orthogonal and symmetric matrices.
- Work efficiently in a group on a project involving Linear Algebra.

Compound and simple propositions, truth table, quantifiers, propositional calculus, methods of proofs. Sets and operations on sets. Cartesian products, relations, equivalence relation, order relation. Functions, images of sets and cardinality.

- Demonstrate Ability To Integrate Knowledge And Idea In A Coherent Manner.
- Develop Counterexamples To Claim Assertions And To Make Interpretation Of Statements.
- Justify Procedures Of Abstract Proofs By Applying Logic.
- Recognize Procedures Of Abstract Proofs.

Divisibility, Euclidean algorithm, prime numbers, the Fundamental Theorem of Arithmetic, the Sieve of Eratosthenes. Congruences, Diaphontine equations, Chinese Remainder Theorem. Fermat's theorem, Wilson's theorem, Euler's theorem, The Legendre symbol and Quadratic Reciprocity.

- Explain Concepts Of Elementary Number Theory Such As: Divisibility, The Fundamental Theorem Of Arithmetic, And Congruence.
- Show Congruence Identities From Fermat'S Little Theorem And Wilson'S Theorem.
- Solve Linear Diophantine Equations And Linear Congruence Equations.
- Work Efficiently As Part Of A Team To Solve Problems In Number Theory.

Euclid's postulates and plane geometry. Von-Neumann postulates. The parallel postulate. Affine geometry and geometry on the sphere. Projective and hyperbolic geometries. Klein-Beltrami and Poincare models of the plane. Pappus and Desargues theorems. Transformations: automorphisms, motions, similarities, and congruence.

- Communicate Concepts And Results In Geometry To Their Peers.
- Construct Scientifically Geometric Figures With A Ruler And A Compass.
- Establish Geometric Results From Deductive Reasoning.
- Explain Non-Euclidean Geometries, Including Hyperbolic Geometry.
- Obtain Results In Non-Euclidean Geometries From Models In Euclidean Geometry.
- State The Theory And Practice Of Traditional Euclidean Geometry.

First order differential equations: examples, separable equations, homogeneous and exact equations, integrating factor and Bernoulli's equation, linear equations, initial value problems. Higher order differential equations: linear equations, linear independence and Wronskian matrices, existence and uniqueness of solutions. Particular solutions: the method of undetermined coefficients, the method of variation of parameters. Laplace transforms and initial value problems. Series solution of differential equations. System of equations and their matrix form.

- Apply Differential Equations To Model Real Life Problems.
- Classify The Different Types Of Ordinary Differential Equations
- Justify Some Statements In Ordinary Differential Equations (For Instance, About The Existence And Uniqueness Of Solutions).
- Solve First And Second Order Differential Equations Using A Variety Of Techniques

Introduction to mathematical logic, sets, operation on sets, the set of natural numbers, the set of integers, the set of rational numbers, graphical representation of numbers, decimal representation of numbers, other bases, divisibility, solution of arithmetic problems, applications.

- MATU1415

- Analyze Sets And Set Operations.
- Identify Early (Egyptian, Chinese, Babylonian, Roman) And Modern Numeration Systems.
- Manipulate Different Base Number Systems.
- Operate With Natural, Integer, Rational And Real Numbers.
- Recognize Logical Statements And Quantifiers.
- Solve Problems Using Inductive And Deductive Reasoning.

Functions, limits of functions, limits involving infinity, continuity, uniform continuity, Extreme Value Theorem, Intermediate Value Theorem, monotone and inverse functions. Differentiation, Mean Value theorem, L'Hospital's rule, Taylor's theorem. Riemann integral, the Fundamental Theorem of Calculus

- Analyze Statements In Analysis From The Definitions And Theorems.
- Explain The Notions And Results Of Single-Variable Calculus In A Theoretical Way.
- Illustrate Concepts And Results Of Real Analysis To Their Peers.
- Reproduce From The Literature Information For Proving A Result In Analysis.
- Solve Problems In Real Analysis In A Deductive Way.

Vector-valued functions of n variables: limits, continuity, Jacobian matrix, differentiability, general chain rule. Implicit Function Theorem for many variables. Scalar-valued functions of n variables: multidimensional Taylor series, Hessian, optimization, constrained optimization. Multiple integrals: Jacobian, change of variables formula, improper integrals. Parametric surfaces: tangent plane, area, integrals over a surface. Vector Calculus: vector fields, divergence, curl, surface integrals of a vector field, Stokes' and Gauss' theorems.

Complex numbers: properties and representations. Complex functions: limits, continuity, and the derivative. Analytic functions: Cauchy - Riemann equations, harmonic functions, elementary analytic functions. Integration in the complex plane: complex line integrals, Cauchy integral theorem, Morera's theorem, Cauchy integral formula; Maximum principle. Liouville's theorem and the fundamental theorem of algebra.

- Compute Derivatives, Harmonic Conjugates, And Contour Integrals.
- Describe Essential Concepts In Complex Analysis, Such As: Function Of A Complex Variable, Their Limits, Continuity, Derivatives, And Integrals.
- Evaluate Elementary Complex Functions And Multi-Functions.
- Explain Basic Concepts And Results Of Analysis To Their Peers.
- Illustrate Concepts And Results Of Complex Analysis To Their Peers.
- Survey The Literature, Beyond The Textbook.

Error analysis: solutions of non-linear equations in one variable, bisection, fixed point, and false position methods, Newton and secant methods; Solution of a system of linear equations: Gaussian elimination method, Cholesky factorization method. Iterative methods: Interpolation: Lagrange, divided differences, forward, backward, and central methods. Numerical differentiation, two, three and five point formulas. Numerical integration, trapezoidal, Simpson's rules and composite quadrature.

- MATH210 with a minimum grade D

- Analyse The Theoretical And Practical Aspects Of Numerical Techniques.
- Examine The Use Of Numerical Techniques For Solving Problems In Applied Mathematics.
- Solve Linear And Non-Linear Equations Numerically.
- Use Mathematica Software To Solve Numerically Mathematical Problems.

The general Linear Programming Problem. The Simplex method. The revised Simplex method. Computer implementations. Duality. Parametric linear programming. Interior point methods. Applications including: transportation problem, inventory problems, blending problems and game theory.

- MATH210 with a minimum grade D

- Communicate Effectively Solutions Of Linear Programming Problems To Their Peers.
- Discuss The Conditions Of Validity Of Several Linear Programming Methods.
- Explain Various Linear Programming Methods (E.G., Geometric Method, Simplex Method, Duality Method).
- Find Appropriate References For Solving A Linear Programming Problem With Another Method.
- Formulate Problems From Various Fields In The Language Of Linear Programming.
- Solve Linear Programming Problems By Various Methods.

Geometrical figures in plane and space and their properties. Areas and volumes of geometrical figures; unitary and non-unitary linear transformations and their properties. Ratio, proportion, percentage and their practical applications. The geometric problem: construction and solutions methods.

- MATH305 with a minimum grade D

- Compute Perimeters, Areas And Volumes Of Some Two And Three-Dimensional Geometrical Shapes.
- Explain Some Fundamental Ideas Of Algebra (Variables, Expressions, Equations, Inequalities).
- Manipulate Data And Different Units Measures.
- Operate With Ratio, Proportion And Percentage.
- Solve Systems Of Linear Equations.

Groups: examples, subgroups, cyclic subgroups; cosets and Lagrange's theorem; Cyclic groups and permutation groups. Normal subgroups, quotient groups; homomorphisms and isomorphisms; Direct products of groups. Rings: examples, sub rings, ideals, quotient rings, integral domains, Fields. Ring homomorphisms and isomorphisms.

- MATH246 with a minimum grade D

- Classify Algebraic Structures.
- Communicate Concepts And Results Of Abstract Algebra To Their Peers.
- Explain Basic Concepts Related To Various Algebraic Structures (Such As Groups, Rings And Fields).
- Retrieve From The Literature Information For Solving A Problem In Abstract Algebra.
- Solve An Algebraic Problem In A Deductive Way.

Linear Transformations: Isomorphisms of vector spaces, representation by matrices, and change of basis. Eigenvalues and eigenvectors: diagonalization and triangularization of linear operators. Inner product spaces: Orthogonalization and Rieze representation theorem. Self-adjoint operators: the Spectral theorem, Bilinear and quadratic forms.

- MATH140 with a minimum grade D

- Communicate Concepts And Results Of Linear Algebra To Their Peers.
- Demonstrate Diagonalizability Of Linear Operators And Matrices From The Spectral Theorem.
- Explain Concepts Of Linear Algebra Such As: Linear Transformations And Operators, Vector Spaces, And Inner Product Spaces.
- Find The Appropriate References For Solving Other Problems In Linear Algebra.
- Show Ability To Diagonalize, Triangularize, And Orthogonalize Linear Operators.

Definition of a graph. Examples, paths and cycles: Eulerian and Hamiltonian graphs. Application to shortest path and Chinese postman problems, trees, applications, including enumeration of molecules, planar graphs, graphs on other surfaces, dual graphs. Coloring maps, edges, vertices. Digraphs, Markov chains, Hall's marriage theorem and applications. Network flows.

- Apply A Selection Of Algorithms To Solve Practical Problems Involving Discrete Quantities.
- Compute Graph Invariants Such As Chromatic Number, Chromatic Index And Chromatic Polynomial Of Some Examples Of Graphs.
- Explain Basic Graph Theory Concepts To Their Peers.
- Summarize The Basic Concepts In Graph Theory, Including Properties And Characterization Of Bipartite Graphs, Trees, Eulerian And Hamiltonian Graphs And Digraphs.
- Survey In The Literature Beyond The Textbook.

This course introduces students to the principles and practices which are required for secure communication: cryptography and cryptanalysis, including authentication and digital signatures. Mathematical tools and algorithms are used to build and analyze secure cryptographic systems. Basic notions of coding theory will be also covered

- MATH246 with a minimum grade D

- Analyze Error-Correcting Codes.
- Apply Encryption And Signing Techniques.
- Develop Work Attitude In A Team On Related Projects.
- Implement Fundamental Concepts Of Cryptography And Coding Theory.

Definitions and concepts: general and particular solutions. Elimination of arbitrary constants and functions. First order equations (the method of characteristics). Second order equations: classifications (hyperbolic, elliptic, parabolic), the normal form. Boundary value problems: the heat equation, the wave equation, Laplace equation. Methods of solutions: separation of variables, the Fourier and Laplace transforms.

- Apply The Mathematical Software Mathematica To Obtain Analytical Solutions Of Pde’S.
- Classify Different Types Of Partial Differential Equations.
- Solve First Order Linear And Nonlinear Pde’S Using Several Techniques.
- Solve Special Types Of Second Order Pde’S, The Wave, The Heat And The Laplace Equations Using Variety Of Techniques.
- Use Boundary Value Problems To Model Real-Life Problems.

One dimensional discrete dynamical systems. Steady states, stability, periodic points. Chaos. Lyapunov exponents. Symbolic dynamics. 2-dimensional systems. Mandelbort set. Fractals. Applications in ecology population growth, Predator-prey and competition models. Applications in medicine fractal structure of the lung, heart rat variability.

- Analyze The Solution Of Different Types Of Dynamical Systems In Terms Of Steady States, Stability, And Periodic Points.
- Communicate Findings Of A Model In The Language Of The Corresponding Problem.
- Deduce Theoretical Results From Lyapunov Exponents.
- Describe The Dynamic Of Discrete And Continuous Systems.
- Develop Mathematical Models Related To Ecology Population Growth, Predator-Prey And Competition Models, Medicine Fractal Structure Of The Lung And Heart Rat Variability.
- Use Technology To Obtain Relevant Solutions Of Dynamical Systems.

Introduction to the concepts of financial markets and products. Financial derivatives, options, futures and forwards. Pricing, hedging and no arbitrage concepts. The Binomial model. Introduction to stochastic calculus, Stochastic processes, Markov property, martingales. Brownian motion, stochastic integration, stochastic differential equations, Ito's Lemma. Black and Scholes formula, delta hedging. Numerical Methods for finance, Finite Difference Methods, Monte Carlo simulation. Optional topics: Value at Risk, Greeks, Implied volatility, implementation of pricing formulas in VBA for Excel, interest rate models, exotic options, path dependent options, Asian options.

- Collect Material In Financial Mathematics Beyond The Contents Of The Textbook.
- Contribute—As Part Of A Team—To A Project On Modeling And Solving A Financial Derivative Problem.
- Describe The Practical Meanings Of Different Financial Products And Markets And Basic Concepts Of Stochastic Calculus.
- Justify Some Statements In Financial Mathematics By Critical And Deductive Thinking.
- Solve Different Types Of Stochastic Differential Equations.
- Use Binomial Trees Or Stochastic Differential Equations To Model Financial Asset Price Trajectories.

Sequences and series of complex numbers, Power series, Taylor and Laurent expansions, differentiation and integration of power series, application of the Cauchy theorem: Residue theorem, evaluation of improper real integrals, conformal mappings, mapping by elementary functions.

- MATH315 with a minimum grade D

- Apply Techniques Of Complex Analysis To Solve Various Problems.
- Describe The Fundamental Concepts Of Complex Analysis.
- Explain Results In Complex Analysis To Their Peers.
- Survey The Literature Beyond The Textbook.

Approximation theory: Orthogonal and Chebyschev polynomials, rational and trigonometric polynomials, multiple integrals, initial value problems: Taylor's methods, multistep and Runge-Kutta methods, boundary value problems: shooting, finite difference and Rayleigh-Ritz methods.

- Solve problems on some important concepts of Numerical Analysis (such as least squares approximation, double integrals, convergence, and stability).
- Formulate some numerical method for approximating the initial value problems, ordinary and partial boundary value problems.
- Show theorems in numerical analysis using different types of logical proofs.
- Apply Numerical Analysis to real world problems (such as heat equations and wave equations).
- Write Mathematica codes to investigate approximations, integrals, and numerical techniques for solving initial and boundary value problems.
- Work effectively in team on a numerical analysis project.

Rings: introduction to rings properties and subrings; Integral Domain (ID), fields and characteristic of a ring; Ideals and factor rings; ring homomorphisms, polynomial rings and factorization of polynomials; Divisibility in ID and Unique Factorization Domain (UFD). Fields: the Fundamental Theorem of Fields; Splitting Field; Zeroes of Irreducible polynomial; Algebraic extension of Fields; Finite Fields; Introduction to Galois Theory.

- MATH340 with a minimum grade D

- Communicate Concepts And Results Of Abstract Algebra To Their Peers.
- Create Counterexamples To Disprove Algebraic Statements.
- Determine The Zeroes And Splitting Fields Of Polynomials, The Subfields Of (Finite) Fields, And The Galois Group Of Various Field Extensions.
- Discover The Fundamental Concepts Of Abstract Algebra: Rings, Integral Domains, Ring Homomorphisms, Ideals, Factor Rings, And Fields.
- Explore In The Literature, Beyond The Textbook, Topics In Abstract Algebra.

Topological spaces, Bases and sub-bases, subspaces, finite product spaces, continuous maps, homomorphisms, Hausdorff spaces, metric spaces, compactness and connectedness, separation axioms.

- MATH215 with a minimum grade D

- Apply Topological Methods To Solve Other Problems.
- Explain Basic Topological Concepts To Their Peers.
- Justify Proofs Of Several Standard Theorems Related To Compactness, Connectedness And Separation Axioms.
- Summarize The Fundamental Concepts Of Topology, Including: Topological Spaces, Continuity, Metric Spaces, And Hausdorff Spaces.
- Survey Literature Related To A Given Topological Problem.

The modeling process, dimensional analysis, model fitting techniques, discrete models difference equations, logistic equation. Continuous models using derivatives for example: predator-prey, population, harvesting, models. Discussion of stability, phase plane. Applications using Mathematica.

- Analyze Quantitatively The Findings Of A Model.
- Build Team Work For Their Mathematical Modelling Part Of A Project.
- Develop A Mathematical Model For A Given Problem.
- Prepare A Report Linking A Given Problem With Its Model.
- Understand Modeling Process, Dimensional Analysis, Model Fitting Techniques, Discrete Models Difference Equations And Logistic Equation.
- Use Technology To Obtain Relevant Solutions Of A Model.

Introduction and motivation. Problem formulation. Systems models: linear and nonlinear systems. Optimal control problems arising from different fields. Calculus of variation with application to system modeling. Limitation of calculus of variation leading to modern control theory. Time optimal control, attainable state, reachable sets, and Bang-Bang principle. Pontryagin minimum principle and transversality conditions. Linear quadratic control problems. Optimal linear state feedback control. Applications: 3-axis attitude control of communication satellites, road building and fisheries problems, geo-synchronous satellites, speed controls of electric motors.

- Classify Mathematical Representation Of Control Systems.
- Communicate Effectively Solutions Of Optimal Control Problems To Their Peers.
- Determine The Existence And Uniqueness Conditions Of An Optimal Control Using Pontryagin Maximum Principle.
- Show The Stability, Observability And Controllability Of A Linear System.
- Use Logical Approach Of Transfer Functions To Obtain State-Space Realizations From A Linear System And Vice Versa.
- Use Numerical Techniques And Computer Simulation To Find Solutions Of Optimality Conditions.

Students are supervised during their formulation of research proposals. Instructors direct their students in carrying out different tasks leading to the execution of the projects. Students are required to give presentations regarding their achievements, and the written final reports are submitted for evaluation.

- Demonstrate Knowledge Of Mathematics Beyond The Curriculum.
- Explain Mathematical Results In A Deductive Way.
- Find Appropriate References And Read Mathematical Literature Independently.
- Prepare Reports In Mathematics Using Appropriate Tools (E.G. Swp And/Or Latex) And Computational Software.
- Present Mathematical Results Orally In Front Of A Specialized Audience.
- Work On Assigned Tasks Related To Mathematics Individually And As A Part Of A Group.

The Internship training program is coordinated by both the department, academic supervisor and the faculty training committee. The program is continuously monitored and reviewed by a field supervisor staff member at one of the institutions, establishments, or work sites in the United Arab Emirates. (This course is conducted over half a semester (8 weeks) during the third year of study. Offered condensed courses should be taken during the other half of the semester).

- Pre/Co MATH495 with a minimum grade D

- Apply Their Mathematical Knowledge In A Working Environment.
- Determine An Adequate Mathematical Model For A Problem Coming From The Industry.
- Develop Adequate Team Work Attitudes With Supervisors, Co-Workers, And Possibly Customers.
- Interpret A Problem From The Industry In A Logical, Scientific Way.
- Make Up A Well-Structured And Organized Professional Report.
- Solve Mathematical Problems Coming From The Industry.

Sequences of functions, the uniform norm, uniform convergence. Series of functions and tests for uniform convergence. Limits superior and inferior. Lebesgue outer measure and Lebesgue measure, measurable subsets, Borel measurable sets, non-measurable sets. Measurable functions. Integration of non-negative functions, Levi’s monotone convergence theorem, Fatou’s lemma, Integrals of measurable functions. Lebesgue’s dominated convergence theorem. Riemann integral versus Lebesgue integral.

- Compare Riemann Integrals With Lebesgue Integrals
- Construct Lebesgue Outer Measure And Lebesgue Measure
- Describe ?-Algebras, Measurable Sets, And Measurable Functions
- Explain Lebesgue Integrals And Convergence Theorems
- Justify Pointwise And Uniform Convergence Of Sequences And Series Of Functions

Calculus of functions of several variables and of vector fields in Euclidean space. Differentiation and the implicit function theorem. Integration, Fubini’s and Sarad’s theorems. Integration on chains, fields and forms, geometric preliminaries and the fundamental theorem calculus. Integration on manifolds, Stokes’ and Green’s theorem on manifolds, the divergence theorem.

- Apply Tools From Advanced Calculus To The Study Of Manifolds.
- Describe The Relation Between Geometry And Topology Of Spaces
- Generalize Stokes’ Theorem To Manifolds.
- State The Abstract Definition Of Manifolds In The Euclidean Space.

Complex derivative, Cauchy-Riemann equations, conformality, power series and Abel's theorem. Complex integration, exactness and independence of path, Cauchy's theorem for disks, Cauchy's integral formula, higher derivatives, applications. Taylor's finite development, zeroes of analytic functions, classification of isolated singularities, Casorati-Weierstrass theorem. Argument principle, open mapping theorem. Maximum modulus principle, Schwarz' lemma. Chains, cycles, simple connectivity, homology, general form of Cauchy's theorem, periods and residues, the residue method. Compactwise convergence and Weierstrass' theorem, Hurwitz's theorem, Taylor's expansion, Laurent expansion. Mittag-Leffler’s theorem. Infinite products and absolute convergence, Weierstrass’s factorization theorem. Riemann conformal mapping theorem. Montel's theorem. Special functions (gamma, zeta). Introduction to Harmonic analysis.

- Analyze Entire Functions
- Apply Techniques Of Complex Analysis To Solve Boundary Value Problems.
- Describe Techniques Of Power Series Expansions
- Discuss Important Concepts And Results Of Complex Analysis

Introductions to Banach and Hilbert spaces, Bounded operators on Hilbert spaces. Introduction to C*-algebras: definition and examples, projections and unitary groups. Types of C*-algebras, Finite and approximately finite dimensional algebras (AF-algebras), the Bratteli diagrams for AF-algebras. The von Neumann algebras and Factors. Irrational rotation algebras and Cuntz algebras. Dimension groups. Basics of K-theory, classifications of C*-algebras using the K-theory and the unitary groups.

- Analyze Theorems Using Properties Of C*-Algebras.
- Explain Examples Of Banach Algebras.
- Identify Banach Algebras, Hilbert Spaces And C*-Algebras.
- State The Properties Of C*-Algebras And Von Neumann Algebras.

Affine spaces and subspaces, barycentric combinations, independence, frames, intersections of affine subspaces, convex sets, embedding into vector spaces, Euclidean spaces, inner product, orthogonality, duality, adjoint of linear map, linear isometries (orthogonal transformations), applications. Projective spaces and subspaces, projective maps, projective frames, completion of affine spaces, the cross-ratio.

- Compare Between Affine And Euclidean Spaces And Subspaces
- Prove Different Statements In Euclidean And Affine Geometry.
- Solve Problems Related To Euclidean And Affine Geometry.
- State The Main Properties Of Projective Spaces.

Error analysis. Solutions of linear systems: LU factorization and Gaussian elimination, QR factorization, condition numbers and numerical stability, computational cost. Least squares problems: the singular value decomposition (SVD), QR algorithm, numerical stability. Eigenvalue problems: Jordan canonical form and conditioning, Schur factorization, the power method, QR algorithm for eigenvalues. Iterative Methods: construction of Krylov subspace, the conjugate gradient and GMRES methods for linear systems, the Arnoldi and Lanczos method for eigenvalue problems.

- Assess A Method For Its Accuracy, Stability, And Computational Cost.
- Demonstrate Awareness Of The Efficiency Implications In A Computer Implementation Of A Method.
- Model A Real-World Problem As A Problem In Numerical Linear Algebra.
- Select Or Design A Method For Solving A Problem In Numerical Linear Algebra.
- Use Numerical Software Appropriately, I.E., Decide When To Use Certain Methods Depending On Their Limitations.

Theory and implementation of numerical methods for initial and boundary value problems in ordinary differential equations. One-step, linear multi-step, Runge- Kutta, and extrapolation methods; convergence, stability, error estimates, and practical implementation, Study and analysis of shooting, finite difference and projection methods for boundary value problems for ordinary differential equations. Theory and implementation of numerical methods for boundary value problems in partial differential equations (elliptic, parabolic, and hyperbolic).

- Analyze The Error Associated With Each.
- Create Computer Algorithms For Solving Ordinary And Partial Differential Equations.
- Implement Different Numerical Methods For Solving Initial And Boundary Value Problems.
- Implement Different Numerical Methods For Solving Various Types Of Partial Differential Equations Such As The Elliptic, Parabolic And Hyperbolic Types.
- Solve Real-Life Problems Using Different Numerical Methods.

Group theory: definitions, subgroups, permutation groups, cyclic groups, quotient groups, homomorphism, the isomorphism and the correspondence theorems. Ring theory: definitions, rings homomorphism, ideals, quotient rings, fraction fields, polynomial rings, Euclidean domain, and unique factorization domain. Field theory: algebraic field, extensions.

- Classify Different Algebraic Structures.
- Define Various Algebraic Structures.
- Determine Galois Groups Of Polynomials Of Small Degree, Of Finite Fields And Of Cyclotomic Number Fields.
- Develop Proofs Of General Statements Using Abstract Knowledge And Operations.
- Develop Proofs Of General Statements Using Abstract Knowledge.
- Justify Steps Of Abstract Proofs By Invoking Properties Of Algebraic Structures.

Basics of number theory: divisibility, unique factorization, congruence arithmetic, Chinese remainder theorem, integers modulo n, Finite fields, Fermat's little theorem, and Wilson's theorem. Introduction to Algebraic number theory: the Pell equation, the Gaussian integers, Quadratic integers, and the Four square theorem. Quadratic reciprocity and quadratic congruence with composite modules.

- Apply Public-Key Cryptography
- Identify And Use The Properties Of Congruences
- Identify And Use The Properties Of Divisibility
- Recognize The Theory Of Quadratic Residues And Their Relation To Quadratic Congruences
- Solve Some Diophantine Equations

Fundamentals of point set topology: topological spaces, neighborhoods of points, basis, subbases, and weight of spaces. Continuous maps and homeomorphisms, closed and open mappings, quotient mappings. Metric and normal spaces, accountability and separation axioms. Product spaces and quotient spaces. Compactness and connectedness of spaces and properties. Complete metric space and function spaces.

- Examine The Main Topological Properties Of Metric Spaces .
- Justify Elementary Theorems Involving The Concepts Of Topological Spaces, Homeomorphism, Compactness And Connectedness.
- Outline The Fundamental Concepts Of Point-Set Topology.
- Solve Mathematical Questions By Using General Topology.

The theory of initial value and boundary value problems for hyperbolic, parabolic, and elliptic partial differential equations, with emphasis on nonlinear equations. More general types of equations and systems of equations

- Apply Analytical Methods To Prove Existence And Uniqueness Of Solution For Different Types Of Pde’S.
- Classify Second Order Linear Pdes.
- Construct Green’S Functions And Green’S Representation Formula.
- Solve Various Types Of Pde’S Using Different Tools, Such As: The Method Of Characteristics, Separation Of Variables, Fourier And Laplace Transforms.
- Use The Software Mathematica To Solve Various Types Of Pde’S.

Discrete time dynamical systems. Continuous time dynamical systems. Invariant manifolds, homoclinic orbits, local and global bifurcations. Hamiltonian systems, completely integrable systems, KAM theory. Different mechanisms for chaotic dynamics, symbolic dynamics, Applications in physics, biology and economics.

- Use Sarkovskii’S Theorem And Period Doubling Cascades In Chaos Theory.
- Apply Chaotic Dynamical Systems To Science And Engineering Problems.
- Describe The Different Mechanisms Governing Chaotic Systems.
- Determine The Stability And Bifurcations Of Non-Linear Discrete And Continuous Dynamical Systems.
- Explain The Poincaré-Bendixson Theorem And Its Various Implications.
- Explain The Concepts Of Chaos And Fractal

Power series holomorphic functions, representation by integrals, extension of functions holomorhpically to convex domain. Local theory of analytic sets (Weierstrass preparation theorem and consequences). Functions and sets in the projective space P (theorems of Weierstrass and Chow and extensions).

- Analyze Advanced Results About Analytic Functions Of N Variables
- Define The Notion Of Analyticity For Functions Of Several Complex Variables.
- Extend Shapes From The Complex Space To The Projective Complex Space.
- State The Main Properties Of Analytic Functions Of N Variables.
- Use Representation Theorems Of Analytic Functions Of N Variables (Integral Formulae, Taylor’S Development) And Compare With Similar Results For 1 Variable

Metric space topology, continuity, convergence, equicontinuity, compactness, bounded variation, Helly selection theorem, Riemann Stieltjes integral, Lebesque measure, abstract measure space, Lp-spaces, Holder and Minkowski inequalities, Riesz-Fischer theorem.

- Analyze Some Classical Banach Spaces, Including Lp.
- Define The Lebesgue Integral, State The Corresponding Convergence Theorems, And Compare With Riemann Integration Theory.
- Define The Notions Of Measurable Set, Measurable Function, And List Operations Preserving Measurability.
- Survey Main Properties In Abstract Measure Theory.

Banach Spaces, The Banach Fixed Point Theorem; Bounded Linear Operators and functionals; Hilbert Spaces; Representation of functionals on Hilbert Spaces; Compact linear operators in Banach Spaces; Spectral Theory of Bounded Self-Adjoint Linear Operators in Hilbert Spaces.

- Recognize Inner Products, Hilbert Spaces, And State Their Main Properties
- Recognize Normed And Banach Spaces, And State Their Main Properties
- Solve Problems Related To Linear Operators On Hilbert And Banach Spaces.
- Work With Metric Spaces

Numerical methods for partial differential equations, Finite difference methods for elliptic equations, stability and error estimates of finite difference methods. Finite difference methods for heat equations. Preliminaries of finite element methods, Variational formulation, existence and uniqueness, Cea’s theorem, Construction of finite element spaces, Barycentric coordinates, Polynomial approximation theory, Bramble-Hilbert Theorem, transformation formula.

In the first part of the course, introductory research talks will be delivered by faculty members. Ethics issues related to mathematical research will be also discussed. In the second part, each student will give a talk in a research topic of his/her choice.

- Demonstrate ability to integrate in depth specific scientific topics;
- Explain mathematical ideas in a rigorous and ethical way;
- Develop effective team work with peers on projects related to the assigned topics;
- Compile a literature review related to the assigned topics;

Group theory: Sylow theorems, Jordan–Holder theorem, solvable group. Ring theory: unique factorization in polynomial rings and principal ideal domain. Field theory: rules and compass constructions, roots of unity, finite fields, Galois theory, solvability of equations by radicals.

- Justify Steps Of Abstract Proofs By Invoking Properties Of Algebraic Structures.
- Produce Properties Of A Group From The Properties Of Its Subgroups.
- State The Properties Of Basic Algebraic Structures.

Boundary Value Problems, the Mollifier theorem, basic facts about Hilbert space, Fourier-Sobolev spaces, advanced properties of Sobolev spaces, H-space duality, weak formulation of elliptic boundary value problems, spectral properties of elliptic operators, evolution equations, Parabolic and Hyperbolic equations, linear operators, Introduction to simegroups, the Hille-Yosida theorem, the Lumer-Philips theorem, Alternative development of S/G’s, summary of Sg results, analytic semigroups, nonlinear boundary value problems.

- Apply Operator Theory And Group Based Methods To Boundary Value Problems
- Describe The Properties Of Different Spaces Of Functions
- Determine The Equations Governing A Deterministic System

Introduction to the Mathematics’ of financial models. Hedging, pricing by arbitrage. Discrete and continuous stochastic models. Martingales. Brownian motion, stochastic calculus. Black-Scholes model, adaptations to dividend paying equities, currencies and coupon-paying bonds, interest rate market, foreign exchange models.

- Apply Ito Formula For Solving Linear Sdes.
- Apply The Black-Scholes Methodology To Other Financial Models.
- Construct Financial Stochastic Models In Discrete And Continuous Time.
- Derive The Black-Scholes Formulae For Pricing And Hedging Options, Both In Pde And Martingale Approaches.
- Explain Financial Models And Their Related Notions.
- Manipulate Stochastic Processes In Discrete And Continuous Time.

A variety of topics and current research results in Mathematics will be presented by faculty members to students.

- Build Various Mathematical Tools Related To The Topics, And Apply Them To The Investigation Of Other Problems When Appropriate.
- Construct Clear And Accurate Mathematical Proofs In Exploring Further Properties Of The Studied Mathematical Objects.
- Explain The Concepts And Tools Developed In The Topics
- Solve Multi-Steps Problems Using Advanced Mathematics Tools.
- Survey The Main Results Of A Specific Mathematic Topic.

Graduate students will study topics related to their Ph.D. thesis independently. The selection of these topics will be with the consent of advisor.

- Explain The Concepts Of The Studied Topics.
- Formulate Mathematical Proofs In A Clear, Rigorous And Accurate Scientific Method By Exploring The Properties Of The Studied Mathematical Objects.
- Manipulate The Tools Of The Studied Topics.

Normed Spaces; Banach Spaces; Compactness and Finite Dimension; Bounded Linear Operators; Operator Spaces; Inner Product Spaces; Hilbert Spaces; Orthonormal Sets and Sequences; Representation of Functionals on Hilbert Spaces; Self-Adjoint; Unitary and Normal Operators; Zorn's Lemma; The Hahn-Banach Theorem; Adjoint Operator; Reflexive Spaces, The Baire Category Theorem; The Uniform Boundedness Theorem; Strong and Weak Convergence; Numerical Integration and Weak-* Convergence; The Open Mapping Theorem; The Closed Graph Theorem; The Banach Fixed Point Theorem; Spectral Theory of Bounded Linear Operators and Compact Linear Operators in Normed Spaces; Spectral Theory of Bounded Self-Adjoint Linear Operators in Hilbert Spaces.

Outer Measure; The Caratheodory-Hahn Theorem; Measurable Functions; Integration of Measurable Functions; Fatou’s lemma and Convergence Theorems; Abstract Measure Spaces; Product Measures; Fubini’s Theorem; Abstract L_p Spaces; The Completeness of L_p (X,μ); The Riesz Representation Theorem for the Dual of L_p (X,μ).

- Identify abstract measure spaces
- Apply integration of measurable functions.
- Explain the classical Banach spaces; L^p-spaces.
- Explain product measures and Lebesgue measure on R^n.

Banach and Hilbert spaces. Bounded operators on Hilbert spaces, Algebras of operators, The von Neumann algebras, compact and Hilbert-Schmidt operators. Abstract C*-algebras and main examples, The Continuous Functional Calculus Theorem, polar decomposition, Gelfand-Naimark Theorem.

Numerical quadrature, Spectral Methods: Collocation, Tau and Galerkin methods, Elliptic Problems and the Finite Element Method: conservation of heat, behavior of solutions, Two-point boundary value problems and the Laplace and Poisson equations, variational and the Galerkin finite element methods, Convergence, finite difference method, Method of Lines, Numerical stability, stiffness and dissipativity, convergence, Finite difference schemes, consistency, Stability, dissipativity, dispersion, convergence.

Modules, quotient modules, module homomorphisms, direct sums, free modules, tensor products. Vector spaces, matrices, dual spaces, determinants. Modules over principal ideal domains, rational canonical forms, Jordan canonical form. Modules over group rings, Schur lemma, Wedderburn theorem, character theory, orthogonality relations.

Estimates of arithmetic functions, the prime number theorem, Dirichlet series, Dirichlet theorem on primes in arithmetic progressions. Integer partitions, Euler formulas, Jacobi triple product formula. Algebraic numbers, algebraic integers, quadratic fields, units and primes in quadratic fields.

- Apply estimates of arithmetic functions to analytic number theory.
- Describe the Prime Number Theorem.
- Solve problems related to integer partitions.
- Analyze Diophantine equations using Algebraic Number Theory.

Public key cryptosystems (RSA, Rabin, ElGamal), discrete logarithm, Diffie-Hellman key exchange, primality testing, factoring algorithms, multivariate cryptography, other systems, signature schemes, secret sharing, hash functions, identification.

- MATH540 with a minimum grade D

- Employ basic concepts and algorithms of cryptography
- Perform encryption and decryption using public key cryptograph
- Analyze and apply digital signature schemes, secret sharing, hash functions, identification.
- Examine the discrete logarithm problem and illustrate its applications.
- Formulate the mathematical concepts underlying modern cryptography.

Block codes, linear codes, generator and parity check matrices, dual codes, weight and distances, weight enumerators, Hamming codes, Golay codes, Reed-Muller codes, Kerdock codes, bounds on codes, theory of cyclic codes, BCH codes, Reed-Solomon codes, quadratic residue codes, generalized Reed-Muller codes, codes over Z4.

- Employ basic notions of coding theory.
- Analyze linear codes and describe their generator and parity-check matrices.
- Categorize different types of codes
- Determine codes over rings
- Formulate the mathematical concepts underlying coding theory.

Fields, finite fields, field extensions, trace and norm functions, bases, polynomials, primitive polynomials, irreducible polynomials, linearized polynomials, applications of finite fields, linear codes, multivariate cryptography.

Characteristic subgroups, Nilpotent and Solvable Groups, Semidirect and Central products; Automorphisms as Linear Transformation. Representations of Finite Abelian Groups, Complete Reducibility, Clifford’s Theorem, G-Homomorphism and Representation of direct and central products, Character Theory: Frobenius Groups, Coherence and Brauer’s characterization of Characters.

Free module, Projective module, Injective module, Flat modules, Homological dimensions, Noncommutative localization, von Neumann regular rings and generalizations, Frobenius and quasi-Frobenius rings, Morita theory.

Topological spaces, basis, subbases, continuous maps and homeomorphisms, product and quotient topology, metric and normed spaces, connectedness, local connectedness, compactness, local compactness, compactification, countability axioms, normal spaces, Uryshon metrization Theorem, Tietze Extension Theorem, manifolds, complete metric space and function spaces, Metrization Theorem and paracompactness.

- Analyze basic properties of topological spaces.
- Apply various mappings such as continuous mappings, homeomorphisms, closed and open mappings, and quotient mappings.
- Decide whether two given spaces are homeomorphic.
- Construct topological spaces to prove or disprove a given statement.

Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. Homotopy theory, fibration, relations between homotopy and homology, obstruction theory, and topic from spectral sequences, cohomology operations, and characteristic classes.

- Demonstrate understanding of the fundamental concepts of algebraic topology, such as homotopy and homology.
- Demonstrate familiarity with a range of examples illustrating these notions.
- Compute the fundamental group and the homology groups of many examples of topological spaces.
- Demonstrate proficiency in communicating mathematics orally and in writing.

Knots and links, isotopy, Reidemeister moves, numerical invariants, 3-colorings, Braids, Alexander’s Theorem and Markov moves, Jones and bracket polynomials, Tait’s conjectures, Alexander-Conway polynomial, HOMFLY and Kauffmann invariants, Tangle equations and Applications.

- Explain the fundamental notions of knot theory.
- Solve low dimensional topological problems using knot theory.
- Compare between different type of knot invariants.
- Apply knot theoretical techniques to biological problems

n-dimensional Euclidean Space, curves and surfaces, coordinate charts, manifolds, smooth maps, immersion and imbedding, sub-manifolds, partitions of unity, tangent vectors and cotangent vectors, tangent bundles, Riemannian manifolds, tensor and exterior algebra, differential forms, exterior differentiation, de Rham cohomology, Lie groups, quotient spaces.

- State the abstract definition of manifolds in the Euclidean space.
- Apply tools from Advanced Calculus to the study of manifolds.
- Understand the geometry and topology of curves and surfaces.
- Apply methods of abstract algebra to the analysis of manifolds.
- Understand the geometric terms defined on higher-dimensional manifolds.

Method of characteristics, Formation of shocks for the inviscid Burger’s equation, Explicit formulas for solutions of the linear wave equation (d’Alembert, Kirchoff, Poisson) , Cauchy-Kowalevski theorem •, Review of Gronwall’s inequality, Sobolev spaces, Picard iteration, definition of an initial value problem, local well-posedness, global well-posedness, Green’s theorem, Review of Fourier analysis, Vector fields, Energy estimates, Finite speed of propagation, Klainerman-Sobolev inequality (preceded by a review of Sobolev embeddings), Local well-posedness for quasi-linear wave equations and global well-posedness in a subcritical situation, Symmetric hyperbolic systems, Small data global well-posedness for quasi-linear wave equations on R^n , n ≥ 4 Small data global well-posedness for quasi-linear Klein-Gordon equation on R^3, Null forms and small data global well-posedness for quasi-linear wave equations on R^3.

- Analyze Method of characteristics.
- Formation of shocks for the inviscid Burger’s equation
- Find explicit formulas for solutions of the linear wave equation.
- Proof the related theorems Sobolev spaces.
- Anlayize the local well-posedness for quasi-linear wave equations and global well-posedness in a subcritical situation and symmetric hyperbolic systems.
- Analyize the small data global well-posedness for quasi-linear wave equations on R^n , n ≥ 4 .

Integral Equations: Definition of Integral Equations, Kinds of Kernels, Volterra and Fredholm Equations, Method of Successive Approximations, Applications to O.D.E’s, Green’s Functions, Complex Form of Fourier and Laplace Transforms, Singular Integral Equations, Symmetric Kernels, Eigenvalues and Eigenfunctions, Fundamental Properties of Eigenvalues and Eigenvectors, Hilbert-Schmidt Theorem, Rayleigh-Ritz Method for Finding the First Eigenvalue. Calculus of Variations: Maxima and Minima, Euler Equation, Constraints and Lagrange Multipliers, Hamilton’s Principle, Lagrange Equations.

- Approximate the solution of integral equations of different types.
- Explain the applications of calculus of variations.
- Approximate the solution of eigenvalue problems.
- Implement the different approaches in calculus of variation to solve real life problems.
- Analyze the singular integral equations.

Initial Value Problem: Existence and Uniqueness of Solutions; Continuation of Solutions; Continuous and Differential Dependence of Solutions. Linear Systems: Linear Homogeneous And Nonhomogeneous Systems with Constant and Variable Coefficients; Structure of Solutions of Systems with Constant and Periodic Coefficients; Higher Order Linear Differential Equations; Sturmian Theory, Stability: Lyapunov Stability and Instability. Lyapunov Functions; Lyapunov's Second Method; Quasilinear Systems; Linearization; Stability of an Equilibrium and Stable Manifold Theorem for Nonautonomous Differential Equations.

- Describe The Concept Of Stability Of Solutions Of Differential Equations Of Higher Orders And Systems.
- Explain The Concepts Of Equilibrium And Stable Manifolds And Their Role In Stability Of Solutions Of Ode’S.
- Prove The Existence And Uniqueness Of Solutions Of Certain Class Of Ordinary Differential Equations.
- Solve Systems Of Differential Equations Of Different Types.
- Use Available Software To Solve Differential Equations And Systems Of Differential Equations.

Discrete time dynamical systems. Continuous time dynamical systems. Invariant manifolds, homoclinic orbits, local and global bifurcations. Hamiltonian systems, completely integrable systems, KAM theory. Different mechanisms for chaotic dynamics, symbolic dynamics. Applications in physics, biology and economics.

- MATH275 with a minimum grade D

- Locate and analyse the stability of fixed points and periodic orbits of maps and flows
- Identify commonly encountered local and global bifurcations.
- Understand the basic notions of chaotic behaviour for maps and flows.
- Define integrability of Hamiltonian systems, and give a qualitative and semi-quantitative analysis of perturbed integrable dynamics.
- Identify applications of each of the main classes of dynamical systems, stating features of their long time behaviour.

Stochastic process; Brownian motion; Martingales; Ito's integral; Ito's formula; Stochastic differential equations; Geometric Brownian motion; Arbitrage and SDEs; The diffusion equation; Representation theorems; Risk-neutral measures, Change of measure and Girsanov’s theorem; Arbitrage and martingales; The Feynman-Kac connection.

- Manipulate continuous stochastic processes such as Brownian motion.
- Solve different kinds of stochastic differential equations.
- Develop pricing formulas for some financial derivatives using Ito formula and representation and Girsanov theorems.
- Construct PDEs for some financial derivatives prices using Feynman-Kac theorem.

Numerical differentiation (Forward, Backward, central), Measuring the error, Numerical instability, Finite difference methods, Monte-Carlo methods, the Euler-Maruyama and Milstein's higher order methods for Stochastic Differential Equations. Applications to finance such as the simulation of asset prices, Monte Carlo Evaluation of European options, numerical solution for the Black-Scholes PDE.

- Construct numerical simulations for SDEs.
- Evaluate options using the Monte Carlo method.
- Solve options pricing PDEs using the finite difference methods.
- Develop solutions for derivative price using transform techniques.
- Formulate calibration problems to estimate financial asset model parameters.

Graduate students will study topics related to their MSc. thesis independently. The selection of these topics will be with the consent of advisor.

- Acquire an understanding of the studied topics.
- Demonstrate familiarity with the concepts and tools of the studied topics
- Be proficient in expressing clear and accurate mathematical proofs in exploring the properties of the studied Mathematical objects.

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