Recognize regular, periodic and singular Sturm-Liouville problems.

Obtain self-adjoint form and inner product as well as eigenvalues, eigenfunctions and eigenfunctions expansions.

Obtain Fourier series, Fourier-Bessel and Fourier-Legendre expansions.

Classify linear partial differential equations.

Solve the heat equation, the wave equation and Laplace equation using eigenfunctions expansion, the Laplace transform or Fourier transforms and determine which method is appropriate to which case, in different coordinate systems.

Work with finite dimensional vector spaces, linear transformations, matrix representations.

Solve the algebraic eigenvalue problem and perform jordanization.

** **

Recall theory of integral transforms based upon complex valued integration.

Apply Fourier, Laplace, Mellin and Hankel transforms to boundary value problems involving partial differential or integral equations.

Discuss inversion of Laplace and Fourier transforms using residues and branch line integration.

Solve the Wiener-Hopf problems arising from mixed boundary value problems.

Demonstrate understanding of asymptotic series and asymptotic approximation of integrals.

Use conformal mappings to some physical situations.

** **

Work with sets, functions, images, preimages, and distinguish between finite, countable, and uncountable sets.

Know how the topology on a space is determined by the collection of open sets, by the collection of closed sets, or by a basis of neighborhoods at each point, and know what it means for a function to be continuous.

Apply basic properties of connected spaces, path connected spaces, compact spaces, and locally compact spaces.

Know what it means for a metric space to be complete, and characterize compact metric spaces.

Apply the Urysohn lemma and characterize metrizable spaces.

** **

Compute arc length, tangent, normal and binormal vectors, curvature and torsion of plane and space curves.

Apply Serret-Frenet formulae.

Compute the first and second fundamental forms of surfaces.

Work out isometries, geodesics, Gaussian and mean curvature.

Understand the Gauss Bonnet theorem and its topological implications.

Understand the concepts of manifolds, tangent bundle, cotangent bundle and related computations.

Work with differential forms and their relation to theorems of Gauss and Stokes.

Work with abstract Riemannian manifolds and all the related concepts coming from classical geometry of curves and surfaces.

** **

Know the concept and the basic properties of topological vector spaces.

Recall the definition of locally convex spaces and know standard examples.

Explain the duality between locally convex spaces and convex bornological spaces.

Recall and utilize weak and weak* topologies.

Apply separation theorem for compact convex sets.

Identify extremal points and use Krein-Milman theorem.

Deal with the projective and injective tensor products of topological vector spaces.

Explain the concepts of nuclear spaces and mappings.

Demonstrate conceptual and rigorous understanding as well as computational skill concerning such notions as diagonalizable and triangulable linear operators and invariant subspaces.

Prove and apply results on minimal and characteristic polynomials of linear operators, including Cayley-Hamilton Theorem.

Prove statements on primary and cyclic decompositions of finite-dimensional vector spaces.

Compute the rational and Jordan forms of matrices.

Demonstrate conceptual understanding of inner product spaces and apply the Spectral Theorem for normal and self-adjoint operators.

Prove statements on symmetric and skew-symmetric bilinear forms.

** **

Demonstrate rigorous understanding of exactness of the functors Hom and Tensor, and localization.

Prove and apply basic results on free, projective, and flat modules.

Prove and apply results on modules over principal rings, including decomposition theorems

Prove and apply results on injective modules, including Baer criterion and divisibility.

Prove and apply results on Noetherian modules, associated primes, primary decomposition, and Hilbert basis theorem.

Prove and apply results on Artinian and indecomposable modules, including Krull-Remak-Schmidt theorem and various semisimplicity results.

Prove and apply results on semisimple rings, including structure results.

** **

Demonstrate conceptual understanding of the basics of field extensions, including algebraic, transcendental, and algebraically closed extensions.

Demonstrate rigorous understanding of the maps over simple extensions, splitting fields and multiple roots.

Demonstrate conceptual understanding of separable, normal, and Galois extensions, as well as prove and apply the fundamental theorem of Galois theory, and results on the Galois group of a polynomial and solvability of equations.

Prove and apply results on finite fields and compute Galois groups over Q.

Prove and apply various results on the application of Galois theory, including primitive element theorem, fundamental theorem of algebra, cyclotomic and cyclic extensions.

Prove and apply results on algebraic independence and transcendence bases including Luroth's theorem as well as on separating transcendence bases and transcendental Galois theory.

** **

Demonstrate rigorous understanding of basics of categories and functors, including exactness of Hom and tensor, sum and product, direct and inverse limits, basic adjoint theorems.

Prove and apply advanced results on free, projective, and injective modules.

Prove and apply Watt's theorems a well as advanced results on flatness and localization.

Prove and apply results on homological aspects of rings, including Noetherian, semisimple, von Neumann regular, hereditary, semihereditary, Prüfer, and quasi-Frobenius rings.

Prove and apply results on homology functors, derived functors Ext and Tor, including long exact sequence theorems.

Prove and apply results on homological dimensions, including projective and injective dimensions.

Prove and apply Hilbert syzygy theorem along with applications.

** **

Demonstrate conceptual understanding of the basics of rings and ideals, including nil and Jacobson radicals, extension and contraction of ideals.

Demonstrate rigorous understanding of the basics of modules over commutative rings and tensor product and exactness results, as well as apply Nakayama Lemma.

Demonstrate conceptual understanding of rings and modules of fractions and apply various local-global results.

Prove and apply results on integral dependence and valuation rings, including going-up and going-down theorems, and Hilbert's Nullstellensatz.

Prove and apply results on chain conditions and Noetherian and Artinian rings.

Prove and apply results on discrete valuation rings and Dedekind domains.

Prove and apply results on linear and local topologies and completions via filtrations, graded rings and modules.

Prove and apply results on dimension theory of commutative rings.

** **

Use properties of the Gamma function and Mittag-Leffler functions.

Apply properties of fractional integrals and fractional derivatives.

State the well-posedness for some fractional differential problems.

Describe the appropriate underlying spaces.

Link FDE's to their corresponding Volterra integral equations.

Use transforms to solve linear fractional differential equations.

** **

Recognize Partial Differential Equations and classify them by orders.

Use the Characteristic Method to solve some linear and nonlinear first-order PDE's.

Classify the second-order PDE's by types (elliptic, parabolic, hyperbolic).

Use the characteristic Method to solve some linear second-order PDE's.

Recognize and Solve the wave, heat, and Laplace equations in one and higher dimensions.

Recall and Apply the Maximum principle to establish some property of solutions.

** **

Recall and state the notion of weak derivatives.

Define Sobolev spaces.

Explain and use correctly the basic properties of the Sobolev spaces.

State and apply Sobolev embedding theorems.

Use the lax-Milgram theorem to solve a variety of linear elliptic problems.

Distinguish between a weak and a classical solution.

Demonstrate ability with the elliptic regularity theory and apply it to some PDEs.

State the maximum principle and apply it to establish some property of solutions.

** **

Understand matrix factorization such as LU-factorization, QR-factorization, singular value decomposition and eigenvalue decomposition.

Understand and program algorithms to solve linear system of equations using direct methods like Gaussian Elimination.

Explain the underlying principles of several iterative methods for linear system of equations, such as matrix-splitting, projection, and Krylov subspace methods.

Understand block and sparse matrices and memory requirements.

Understand and program algorithms to solve nonlinear system of equations using iterative methods like conjugate gradient, Newton methods and Continuation methods.

Smooth collected engineering data using linear and nonlinear least squares methods.

Explain the underlying principles of iterative algorithms for computing eigenvalues.

Implement basic optimization algorithms in a computational setting.

Implement, test and validate codes to solve engineering problems numerically.

Develop various finite difference methods for solving initial and boundary value problems.

Propose continuous Galerkin finite element methods for solving initial and boundary value problems.

Explain the stability and the consistency of the computational methods under consideration.

Perform the error analysis of the numerical methods under consideration.

Apply these numerical methods for solving some practical partial differential equations in engineering.

Explore different types of quadrature rules for the numerical integration purposes.

Use advanced programming software such as MATLAB to program the numerical methods under consideration.

** **

Understand the approximation problems particularly in Chebyshev and least squares norms.

Interpolate data using Lagrange and Hermite interpolation.

Understand and apply Chebyshev and least squares polynomial approximation.

Comprehend the properties of orthogonal polynomials with selective applications.

Find Chebyshev and least squares solution of overdetermined systems of linear equations.

Construct cubic and basic B-splines and conduct underlying error analysis.

Have the basic knowledge of Padé and rational approximation with simple applications.

** **

Analyze and program the continuous wavelet transform.

Analyze and program the discrete wavelet transform.

Distinguish the various wavelets in use for the wavelet transform.

Distinguish the advantages of wavelet transforms versus the classical Fourier transform.

Implement wavelet approximation in various applications including signal processing, fractals, ordinary and partial differential equations.

** **

Model and formulate real life problems as linear programs.

Interpret linear programming geometrically through convex polyhedral duality.

Implement the main linear programming algorithms: simplex, revised simplex, dual simplex.

List the advantages and disadvantages of the ellipsoidal algorithm.

Discuss the sensitivity analysis of the main linear programming algorithms.

Relate linear programming to other areas such as game theory, integer programming, parametric programming, etc.

Study selected problems and applications in management, economy, engineering and science.

** **

Derive necessary and sufficient optimality conditions for nonlinear programs.

Derive optimality conditions for nonlinear programs under various constraints qualifications.

Construct the Lagrangian dual problem of a nonlinear program and list some of its basic properties.

List the main algorithms to solve nonlinear programs.

Solve different classes of nonlinear programs using the main computational procedures.

Define the Penalty Functions of certain nonlinear programs.

Use the method of penalty functions to solve nonlinear programs.

Advanced topics not covered in regular courses.

** **

Preparation and delivery of mathematical talks. This is a Pass/Fail course

** **

Characterize stochastic processes

Perform Itô calculus

Analyze and solve some stochastic differential equations.

Apply Stochastic Differential Equations theory to some real-world problems.

** **

Define and discuss the properties of asymptotic sequences and asymptotic series.

Use Watson's Lemma and Laplace Method to obtain asymptotic expansion of some integrals.

Apply methods of stationary phase and steepest descent to obtain asymptotic approximations to integrals.

Review and apply asymptotic methods for differential equations including WKB method.

Recall and apply regular and singular perturbation methods to differential equations and eigenvalue problems.

Apply matched asymptotic method including matching and turning points.

** **

Identify compact operators and their inverses.

Estimate operator norms.

Recognize ill-posedness.

Regularize inverses and estimate discrepancy.

Solve integral equations of first kind by regularization.

Develop stable numerical inversions.

** **

Variable Content. Advanced mathematical topics not covered in regular courses.

** **

Preparation and defense of the MS thesis. This is an NP/NF/IP course.

Recall intuitive interpretations of the Green function.

Solve boundary value problems, involving the computation of Green functions.

Demonstrate understanding of operator and distribution theories, Fredholm alternative.

Calculate the spectrum of an operator.

Review perturbation theory for eigenvalue problems.

** **

Solve integral equations (Volterra and Fredholm equations).

Solve inhomogeneous equations subject to inhomogeneous boundary conditions.

Review spectral theory for second order operators.

Solve some nonlinear equations.

Calculate the continuous spectrum of an operator.

Use variational principles to solve some PDEs.

** **

Understand Tychonoff theorem.

Understand the one-point compactification, the Stone-Cech compactification and their applications.

Understand Paracompactness, Lindelof spaces, Stone's theorem. Metrizability, the Nagata-Smirnov metrization theorem.

Understand the concepts of homotopy and homotopy equivalence of topological spaces;

Understand the fundamental group, homology groups, and covering spaces.

Calculate fundamental groups of the circle, the punctured plane and the n-sphere.

Understand Van Kampen's theorem.

Work with local coordinates, the tangent and cotangent bundles and local coordinates on them.

Work with projective spaces and Grassmannians.

Apply the implicit function theorem to realize subsets defined by differentiable functions as submanifolds of Euclidean space.

Work with differential forms and the theorem of Stokes and be able to relate it to the classical Gauss and Green theorems.

Prove the critical points form a set of measure zero and to derive the weak form of Whitney's embedding theorem and existence of Morse functions as applications.

Demonstrate a working knowledge of Lie groups and Lie algebras via the theorem of Frobenius.

Apply Lie groups to the construction of interesting Riemannian manifolds as homogeneous spaces and work out geodesics and curvature formulas for symmetric spaces.

Define fixed point of operators.

Define nonexpansive operators

Demonstrate ability to use fixed point methods in solving operator equations.

Explain the various kinds of derivatives and integrals of functions on Banach spaces.

Recognize the implicit and inverse function theorems.

Recognize potential operators.

Perform variational settings of operator equations.

Locate extrema of functionals.

Define monotone operators.

Explain monotonicity methods for nonlinear operator equations.

Demonstrate ability to apply abstract operator theory to analyze differential and integral equations and physical problems.

** **

Formulate variational problems and derive the Euler-Lagrange equations for them.

Derive necessary conditions and sufficient conditions for an extremum.

Recognize broken extremal and discuss the additional conditions that must hold at the corner for the extremal to be strong extremum.

Deal with multivariable variational problems with higher order derivatives.

Solve optimal control problems

Use numeric/symbolic software to solve variational and optimal control problems

Give clear self-contained oral presentations and write rigorous reports.

** **

Derive state space realization for linear control system.

Assess the controllability and observability of linear control systems.

Derive controllability subspaces and stabilizability subspaces.

Obtain minimal realizations.

Formulate an optimal control problem.

Solve the linear quadratic regulator problem.

Solve the minimum cost and final value problems in control theory.

Obtain Liapunov functions, Discuss the stability of linear systems.

Assess systems for uniform stability and exponential stability.

** **

Solve Sturm-Liouville boundary values problems (self-adjoint, non-self-adjoint and singular cases).

Analyze asymptotic behavior of the solutions.

Use hypergeometric and special functions in the resolution of differential equations.

Describe bifurcations of periodic and non-periodic solutions.

** **

Classify the first-order systems.

Recall the characteristic Method for systems.

Solve some problems of gas dynamics using the characteristic method.

Demonstrate a clear understanding of dispersive waves and apply to water waves.

Solve some single and double-layer problems using potential theory.

** **

Define exactly Maximal Monotone Operators.

Distinguish between evolution and stationary equations.

Apply Hille-Yosida Theorem to solve the linear wave and heat equations.

Define and distinguish between weak and strong solutions of a PDE.

Define exactly some nonlinear operators and recall their properties.

Use the nonlinear operator theory to solve some nonlinear parabolic equations.

Use the Galerkin Method to solve some nonlinear hyperbolic equations.

** **

Realize integral equations in operator form.

Distinguish modes of convergence of operator approximations.

Analyze rates of convergence of discretization methods.

Calculate approximations of integrals using quadrature methods.

Analyze and program Nystrom's method for solving Fredholm integral equations.

Analyze and program collocation methods, Galerkin methods, method of moments and spline approximation for Volterra integral equations.

Analyze and program iterative methods for linear and nonlinear integral equations.

Analyze the convergence of approximate eigenvalues.

Distinguish the various kinds of operator equations.

Distinguish the theoretical methods for discretizing operator equations.

Analyze stability, consistency and stability of approximation methods.

Discretize operators using projection methods.

Realize iterative methods for solving operator equations.

Analyze the methods of steepest descent, conjugate gradient, averaged successive approximation, splines, finite element and splitting.

Analyze the variational setting of an operator equation and its approximation.

Analyze the methods of vector space, Newton and quasi-Newton for optimization and operator equations.

Recognize the principles and characterization of optimality in dynamical programming.

Analyze the shortest route problem and maximum flow problem.

Recognize the role of the calculus of variation in dynamical programming.

Recognize the role of dynamic programming in networking.

Apply dynamical programming techniques to case study problems.

Variable Content. Advanced topics not covered in regular courses.

** **

Recall the fundamental properties of fluids and fluid flow.

Derive equations of fluid motion from principles of mass and momentum conservation.

Discuss simplified context of Euler flow and steady flow, Bernoulli equation, potential flow.

Review fundamental concepts of boundary layer, drag and lift, vorticity, and circulation.

Explain fundamentals of turbulence.

Use statistically homogeneous turbulence and spectral methods.

Recognize comparisons between laminar and turbulent properties of, wakes, jets, flow past a cylinder, drag and lift.

Describe advanced methods of solution of problems in fluid mechanical systems.

Use the concepts and methods mentioned above to solve advanced problems in fluid mechanical problems.

Set up a numerical problem, and the use of CFD software, to solve fluid mechanical systems.

Formulate the problem at hand.

Discuss the problem at hand and state the theoretical tools and arguments to solve it.

Accommodate slight variations in the statement of the problem.

Use numeric/symbolic software, when necessary, to solve the problem.

Give clear self-contained oral presentations and write rigorous reports.

**Learning Outcomes:** Content-dependent.

Selected topics from the following: groups, rings, modules, and general algebraic systems.

**Learning Outcomes:** Content-dependent.

** **

**Learning Outcomes: **Content-dependent.