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.
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.
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.
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.
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.
Problem solving, fair divisions, Mathematics of Apportionment, Euler circuits, network, scheduling methods, population growth, symmetry, fractal geometry.
Systems of linear equations, matrices and determinants. Vector spaces, inner product spaces. Matrix representations of linear operators. Eigenvalues, eigenvectors, and Cayley-Hamilton Theorem.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
Topological spaces, Bases and sub-bases, subspaces, finite product spaces, continuous maps, homomorphisms, Hausdorff spaces, metric spaces, compactness and connectedness, separation axioms.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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
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.
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).
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.
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.
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.
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.
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.
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.
A variety of topics and current research results in Mathematics will be presented by faculty members to students.
Graduate students will study topics related to their Ph.D. thesis independently. The selection of these topics will be with the consent of advisor.
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Graduate students will study topics related to their MSc. thesis independently. The selection of these topics will be with the consent of advisor.
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