First order and first degree equations. The homogeneous differential equations with constant coefficients. The methods of undetermined coefficients, reduction of order, and variation of parameters. The Cauchy-Euler equation. Series solutions. Systems of linear differential equations. Applications.

Pre-Requisites: MATH102

Special functions. Bessel's functions and Legendre polynomials. Vector analysis including vector fields, divergence, curl, line and surface integrals, Green's, Gauss' and Stokes' theorems. Systems of differential equations. Sturm-Liouville theory. Fourier series and transforms. Introduction to partial differential equations and boundary value problems.

Minkowski space. Tensor analysis on differentiable manifolds. The Einstein field equations. Exact solutions; the Schwarzschild and Reissner-Nordstrom solutions. The three classical tests of general relativity. Energy momentum tensor for perfect fluids and the electromagnetic field. The interior Schwarzschild solution. Black holes and analytic extensions. Robertson-Walker and other cosmological models of the universe. Distance measurements in cosmology.

Selected topics from linear algebra, multivariate calculus, and optimization for Data Science with an emphasis on the implementation using numerical and symbolic software, toolboxes, and libraries for data science like NumPy, SciPy, Pandas, SymPy. Topics include data transformation using linear algebra, vector spaces, linear transformations, matrix representations, matrix decompositions (eigenvectors, LU, QR, SVD, Cholesky); multivariate calculus for continuous, convex, and non-convex optimization methods; Basic neural network design

An introduction to Cartesian tensors. Stress and strain tensors. Conservation of mass, energy and momentum. Hooke's law and constitutive equations. Isotropic solids and some exact solutions of elasticity. Elastodynamic equations. Elastic waves in an unbounded medium. Plane waves in a half space. Reflection and refraction at an interface. Surface waves.

All aspects of the data science pipeline using the software, toolboxes, and libraries like NumPy, SciPy, Pandas, Matplotlib, and Seaborn: data acquisition, cleaning, handling missing data, EDA, visualization, feature engineering, modeling, model evaluation, bias-variance tradeoff, sampling, training, testing, experimenting with a classical model

Laplace transforms including the convolution theorem, error and gamma functions. The method of Frobenius for series solutions to differential equations. Fourier series, Fourier-Bessel series and boundary value problems, Sturm-Liouville theory. Partial differential equations: separation of variables and Laplace transforms and Fourier integrals methods. The heat equation. Laplace equation, and wave equation. Eigenvalue problems for matrices, diagonalization. Prerequisite: Graduate Standing. (Not open to Mathematics majors. Cannot be taken for credit with MATH 333.)

Integral transforms: Fourier, Laplace, Hankel and Mellin transforms and their applications. Singular integral equations. Wiener-Hopf techniques. Applications of conformal mapping. Introduction to asymptotic expansion.

Basic set theory (countable and uncountable sets, Cartesian products). Topological spaces (basis for a topology, product topology, functions, homeomorphisms, standard examples). Connected spaces, path connectedness. Compact spaces, compactness in metrizable spaces. Countability axioms, first countable and second countable spaces. Separation axioms, Urysohn's Lemma, Urysohn's metrization theory. Complete metric spaces.

The fundamental group. The fundamental group of the circle. Van Kampen’s Theorem. Covering spaces and unique lifting property. Homology: definition, invariance under homotopies. Homology of a pair. The relationship between the fundamental group and the first homology group. Basic exact sequences in homology: Mayer-Vietoris and excision. Homology of CWcomplexes. Homology of product spaces.

A basic introduction to graph theory for advanced students in computer science, Mathematics, and related fields. Connectivity, matching, factorization and covering of graphs, embeddings, edge and vertex coloring. Line graphs. Reconstruction of graphs. Networks

Curves in Euclidean spaces: arclength, tangent, normal and binormal vectors, curvature and torsion. Frenet formulas. Isoperimetric inequality. Differential geometry and local theory of surfaces, the first and second fundamental forms. Local isometries. Geodesics. Gaussian and mean curvature of surfaces. The Gauss- Bonnet theorem. Manifolds and differential forms. Introduction to Riemannian geometry.

Rudiment of linear algebra, discrete and continuous differential geometry of curves and surfaces, geometry processing on meshes, convex and non-convex optimization

Lebesgue measure and outer measure. Measurable functions. The Lebesgue integral. Lebesgue Convergence Theorem. Differentiation and integration. 𝐿􀯣 spaces. Riesz Representation Theorem. Introduction to Banach and Hilbert spaces. Product spaces, Fubini’s Theorem.

Analytic functions. Cauchy's theorem and consequences. Singularities and expansion theorems. Maximum modulus principle. Residue theorem and its application. Compactness and convergence in space of analytic and meromorphic functions. Elementary conformal mappings.

Normed linear spaces, Banach spaces, Hilbert spaces, Banach Algebras (definitions, examples, geometric properties), bounded linear operators, convex sets, linear functionals, duality, reflexive spaces, weak topology and weak convergence, Banach fixed point theorem, Hahn-Banach theorem, uniform boundedness principle, open mapping theorem, closed graph theorem, representation of functionals on Hilbert spaces (Riesz Representation Theorem).

Algebra of bounded operators, self-adjoint operators in Hilbert Spaces, Normal operators, compact operators, projections, spectral theory of linear operators in normed spaces and Hilbert spaces, spectral mapping theorem, Banach-Alaoglu theorem.

Topological vector spaces, locally convex spaces, Krein-Milman theorem, duality in locally convex spaces, separation theorem for compact convex sets, topological tensor products, nuclear mappings and spaces.

A quick review of basic properties of topological, metric, Banach and Hilbert spaces. Introduction of Hausdorff metric and iterated function system. Fixed point theorems and their applications. Introduction to infinite dimension calculus Frechet and Gateaux derivatives, Bochner integral. Introduction to weak and w*-topologies. Algorithmic optimization including complementarity problems and variational inequalities.

Fourier series on the circle group (Fourier coefficients, Fourier series of squaresummable functions, absolutely convergent Fourier series, Fourier coefficients of linear functionals), The convergence of Fourier series, Fourier transforms on L1(R), Fourier transforms on Lp(R), The Payley-Wienner theorems. Fourier analysis on locally compact groups (locally compact groups, the Haar measure, characteristic and the dual group, Fourier transforms, almost periodic functions and the Bohr compactification).

Polynomial time algorithms and intractable problems; relationship between the classes P, NP, and NP-complete; Cook's theorem and the basic NP-complete problems. Techniques for proving NP-completeness; NP-hardness. Hierarchy of complexity classes.

Basic properties of vector spaces and linear transformations, algebra of polynomials, characteristic values and diagonalizable operators, invariant subspaces and triangulable operators. The primary decomposition theorem, cyclic decompositions and the generalized Cayley-Hamilton theorem. Rational and Jordan forms, inner product spaces, The spectral theorem, bilinear forms, symmetric and skew symmetric bilinear forms.

Basic definitions of rings and modules, homomorphisms, sums and products, exactness, Hom and tensor, adjoint isomorphism, free, projective and injective modules. Chain conditions, primary decomposition, Noetherian rings and modules, Artinian rings, structure theorem.

Field extensions, the fundamental theorem. Splitting fields and algebraic closure, finite fields, separability, cyclic, cyclotomic, and radical extensions. Structure of fields: transcendence bases.

Review of free, projective, and injective modules, direct limits. Watt's theorems. Flat modules. Localization. Noetherian, semisimple, Von Neumann regular, hereditary, and semi-hereditary rings. Homology, homology functors, derived functors. Ext. and Tor. homological dimensions, Hilbert syzygy theorem.

Pre-Requisites: MATH551

Classical ring structure theorems, functors between module categories, equivalence and duality for module categories. Decomposition properties of injective and projective modules. Specific Artinian rings.

Basics of rings and ideals. Rings of fractions, integral dependence, valuation rings, discrete valuation rings, Dedekind domains, fractional ideals. Topologies and completions, filtrations, graded rings and modules. Dimension theory.

Pre-Requisites: MATH551

Affine and projective algebraic varieties. Zariski topology. Morphisms of algebraic varieties, Veronese and Segre maps, coordinate rings. Hilbert’s Nullstellensatz. Hilbert polynomial, dimension and degree. Quasi-projective varieties and their morphisms. Regular functions. Tangent space, smoothness. Birational maps and blow ups. Resolution of singularities.

Basics concepts from linear algebra and numerical analysis. Direct methods for large, sparse linear systems, Cholesky and LU factorizations. Regularization of ill-conditioned least squares problems. SVD and QR factorizations. Sensitivity and conditioning of linear systems and least square problems. Stationary and non-stationary iterative methods, multigrid methods. Matrix theory including spectral decompositions, and eigenvalue perturbation theory. Eigenvalue and QR algorithm, and computations of SVD. Applications

Categories, functors. Abstract algebraic categories, algebraic theories. Filtered colimits. Algebraic functors. Birkhoff’s Variety Theorem. Concrete algebraic categories, equational categories. Monads, Abelian categories. Categories of modules over rings.

Concepts from linear algebra and numerical analysis. Direct methods for large, sparse linear systems. Cholesky, LU, QR and SVD factorizations. Sensitivity and conditioning of linear systems and least square problems. Iterative methods. Numerical methods for eigenvalues. Computations of SVD. Computer software and applications. Prerequisite: Graduate Standing. (Cannot be taken for credit with MATH 471.)

Simple linear regression. Testing of intercept and slope. Multiple linear regression. Estimation parameters and testing of regression coefficients. Prediction and correlation analysis. Analysis of variance technique. Completely randomized and randomized block designs. Latin square design. Incomplete block design. Factorial design, 2k factorial design and blocking and confounding in 2k factorial designt. Note: (Students cannot receive credit for both MATH 560 and STAT 430 or SE 535)

Examples of simple time series. Stationary time series and autocorrelation. Autoregressive moving average processes. Modeling and forecasting with ARMA processes. Maximum likelihood and least squares estimator. Nonstationary time series. Not to be taken for credit with STAT 460

Basic principles of option pricing, binomial model, the Black-Scholes model, arbitrage, complete and incomplete markets, trading strategies, European options, American options. Topics include Risk-neutral Valuation, options on stock Indices, currencies, futures, the Greek letters, Interest Rate Derivatives, Black-Scholes PDE and formula

Pre-Requisites: MATH564

Foundations of probability theory. Measure-theoretic approach to definitions of probability space, random variables and distribution functions. Modes of convergence and relations between the various modes. Independence, Kolmogorovtype inequalities. Tail events and the Kolomogorov 0-1 law. Borel-Cantelli lemma. Convergence of random series and laws of large numbers. Convergence in distribution. Characteristic functions. The central limit theorem. Weak convergence of probability measures. Conditional expectations and martingales.

Differential equations, Geometric Brownian motion, financial examples., Feynman-Kac formula, Description: Stochastic processes, Gaussian processes, Brownian motion, Ito stochastic integral, the Ito lemma, the geometric Brownian motion and its application to finance.Introduction to stochastic GirsanovFormula and application to Black Scholes PDE and formula.

Existence, uniqueness and continuity of solutions. Linear systems, solution space, linear systems with constant and periodic coefficients. Phase space, classification of critical points, Poincare-Bendixson theory. Stability theory of linear and almost linear systems. Stability of periodic solutions. Laypunov's direct method and applications.

Special functions (Gamma, Mittage-Leffler, and wright), Riemann fractional integral Riemann-Liouville and Caputo fractional derivatives, composition rules, embeddings, equivalence with integral equations, well posedness for Cauchy type problems, successive approximation method, Laplace and Mellin transform methods

First order linear and nonlinear equations. Classification of Second order equations. The wave equation, heat equation and Laplace's equation. Green's functions, conformal mapping. Separation of variables, Sturm-Liouville theory. Maximum principles and regularity theorems.

Sobolev spaces. Mollifiers. Dual spaces. Poincaré’s inequality. Lax-Milgram Theorem. Linear elliptic problems. Weak formulation. Weak derivatives. Weak solutions. Existence uniqueness and regularity. Maximum principle.

Basis and Linear Independence, Linear Operators and Matrices, Eigenvectors and Eigenvalues, Inner Products, Divisibility, Euclidean Algorithm, Congruence, Definition of Groups, Subgroups, Lagrange Theorem, Euler Phi Function, Euler’s Theorem, Cyclic Groups, Groups Isomorphism, Group Homomorphism, Permutation Groups, Finite fields, Quadratic Residues, Reciprocity, Discrete Probability Theory.

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). Finite difference and finite element methods: convergence, stability, and error estimates. Projection methods and fundamentals of variational methods. Ritz-Galerkin and weighted residual methods.

Survey of practical techniques of numerical analysis for engineering and science graduate students. Topics include computational and theoretical aspects of direct and iterative methods for linear systems, iterative solutions of nonlinear systems (successive approximations, relaxation, conjugate gradient, and quasi- Newton methods), sparse materials, least-squares problems (both linear and nonlinear), eigenvalue problems, and optimization problems. Problems include case studies in various disciplines. Note: (Not Open for Mathematics Majors).

Concepts of consistency, stability, and convergence of numerical schemes. Initial and boundary value problems for ordinary differential equations. Various finite difference and finite element methods and their applications to fundamental partial differential equations in engineering and applied sciences. Case studies selected from computational fluid mechanics, solid mechanics, structural analysis, and plasma dynamics. Note: (Not Open to Mathematics Majors)

Best approximation in normed linear spaces: basic concepts. Lagrange and Hermite interpolation. Approximate solution of over-determined system of linear equations. Linear approximation of continuous functions in Chebyshev and least squares norms. Rational approximation. Piecewise polynomial approximation. Cubic and B-splines.

This course introduces implementable numerical methods for solving initial value problems, stability and convergence. One-step, multistep, and Runge-Kutta methods. Shooting and bisection methods. Finite difference methods and applications to equilibrium and non-equilibrium models including steady-state, heat, and wave problems.

Why and how industrial mathematics? The description of air bag sensor. How to judge the quality of a non-woven fabric? Damage estimation in a machine (fatigue life time). Mathematics to solve the above mentioned problems.

This course introduces finite element, finite difference, and finite volume methods. Applications of these methods to steady-state, diffusion and wave models. Stability and convergence. Homogenization, upscale and multiscale methods. Implementations and computer labs. Prerequisite: MATH 576, MATH 557 or Consent of the instructor

The continuous wavelet transform, the discrete wavelet transform, advantages of using wavelet transforms over the classical Fourier transform. Applications of wavelets in solution of differential and partial differential equations. Iterated function system and deterministic fractals.

Basic properties of convex sets and convex functions. Analytical and geometrical aspects of convexity and duality. Convexity connections with optimization. Lagrangian and Fenchel duality. Non-smooth convex optimization and subdifferential calculus.

Theory of linear programming and its duality. Simplex method, revised simplex method, dual simplex method. Interior point method. Sensitivity analysis. Integer and mixed integer programming (cutting plane, and branch and bound methods). Implementation and current solvers. Applications selected from transportation problems, assignment problems, game theory, goal programming, multiobjective programming, linear fractional programming, regression, classification and finance. Prerequisite: Graduate Standing. (Cannot be taken for credit with ISE 503.)

An advanced introduction to theory of nonlinear programming, with emphasis on convex programs. First and second order optimality conditions, constraint qualifications, Lagrangian convexity and duality. Penalty function methods. Theory and algorithms of main computational methods of nonlinear programming. Representative applications of nonlinear programming in Economics, Operations Research and Mathematics.

Central concepts of geometric modeling, basic shape representations (parametric and implicit curves and surfaces, meshes, point clouds), freeform curve and surface design in spline representation, subdivision surfaces, surface quality assessment, geometry processing on meshes.

Physically-based simulation methods for modeling shape and motion (rigid bodies, deformable objects, fluids), interactive dynamic animations (representation, dynamics, collisions detection), data driven animation methods.

This course introduces students to fundamental concepts in linear and nonlinear inverse problems. Emphasis is placed on describing how to integrate various information sources from measured data and prior knowledge about the inverted model. Subjects studied will include topics and tools such as: Regression, Least squares, Maximum likelihood estimation, Rank deficiency, Ill-conditioning, Generalized and Truncated SVD solutions, regularizations (Tikohonov, spectral filtering), proximal and primal-dual iterative schemes, Nonlinear inverse (gradient-based and global optimization methods), OCCAM method. Computer lab sessions will be organized to combine classroom learning with hands-on applications.

Concepts of statistical designs and linear models. Basic designs: Completely randomized design. Randomized block design. Latin square designs (computer aided selection) models: Fixed, random and mixed models, estimation of parameter using Gauss-Markov theorem. Expectation of mean squares with and without use of matrix theory. Incomplete block designs. Factorial experiment, 2p confounding, fractional replicate and orthogonal designs. 3p confounding, fractional replicate and orthogonal designs. P q N confounding; fractional replicate and orthogonal designs. Tagouchi method as applied to design of experiments for engineering, industrial and agricultural data analysis. Extensive use of computer packages and computer aided designs.

Least square method and properties. Simple and multiple linear regression with matrix approach. Development of liner models. Residual analysis. Polynomial models. Use of dummy variables in multiple linear regression. Analysis of variance approach. Selection of `best' regression equation. Concepts of mathematical model building. Non-linear regression and estimation. Extensive use of computer packages.

Advanced topics not covered in regular courses.

Research and expository survey journals in mathematical sciences, review journals, citation journals, journal abbreviations and literature citations. Classification of mathematical subjects. Library search: books, bound journals, current periodicals, microfilms. Searching for publications on a specific subject or by a certain author. Structure and organization of a research paper in mathematics. Methods of dissemination of mathematical results: abstracts, conferences, research papers, books and monographs. Major mathematical societies and publishers and their publication programs. The course will consist of one lecture a week and «workshop» sessions at the KFUPM Library supervised by the instructor.

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Prerequisite: Graduate Standing

Prerequisite: Graduate Standing

Graduate students are required to attend departmental seminars delivered by faculty members, visiting scholars, and fellow graduate students. Additionally, each student must present at least one seminar on a current research topic. Among other things, this course is designed to give students an overview of research in Mathematics and a familiarity with research methodology and journal publications in their discipline. This is a Pass/Fail course.

Probability spaces, characteristics functions, stochastic processes, martingales, Markov chains, Brownian motion, Ito calculus, ito formulas, stochastic differential equations, applications of stochastic differential equations. MATH 531 or instructor’s consent

Kinematics and dynamics. Potential flow. Navier-Stokes equations. Some exact solutions. Laminar boundary layers. Stokes and Oseen flows. Sound waves. Topics in gas dynamics. Surface waves. Flow in porous media. Darcy's law and equation of diffusivity.

Asymptotic sequences and series. Asymptotic expansions of integrals. Solutions of differential equations at regular and irregular singular points. Nonlinear differential equations. Perturbation methods. Regular and singular perturbations. Matched asymptotic expansions and boundary layer theory. Multiple scales. WKB theory.

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

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

Mathematical and numerical analysis of linear inverse and/or illposed problems for partial differential, integral and operator equations. Tikhonov regularization. Constraints and a priori bounds. Methodologies for achieving «optimal» compromise between accuracy and stability. Applications to practical problems in remote sensing, profile inversion, geophysics, inverse scattering and tomography. Prerequisite: Graduate Standing for math majors and MATH 513 for non-math majors.

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

Pre-Requisites: MATH599*

Co-Requisites: MATH 599

Review of normed and product spaces. Theory of distributions, weak solution. Complete orthonormal sets and generalized Fourier expansions. Green's functions and boundary-value problems, modified Green's functions. Operator theory, invertibility, adjoint operators, solvability conditions. Fredholm alternative. Spectrum of an operator. Extremal principles for eigenvalues and perturbation of eigenvalue problems. Applications.

Integral equations; Fredholm integral equation, spectrum of a self-adjoint compact operator, inhomogeneous equation. Variational principles and related approximation methods. Spectral theory of second-order differential operator, Weyl's classification of singular problems. Continuous spectrum. Applications. Introduction to nonlinear problems. Perturbation theory. Techniques for nonlinear problems.

A graduate student will arrange with a faculty member to conduct an industrial research project related to the Data Science field. Subsequently the students shall acquire skills and gain experiences in developing and running actual industry-based project. This project culminates in the writing of a technical report, and an oral technical presentation in front of a board of professors and industry experts.

The Tychonoff theorem, one-point compactification, the Stone-Cech compactification. Paracompactness, Lindelof spaces, Stone's theorem. Metrizability, the Nagata-Smirnov metrization theorem. Homotopy paths, fundamental group, simply-connected spaces, retracts and deformation retracts; the fundamental groups of the circle, the punctured plane and the n-sphere; Van Kampen's theorem.

Calculus on manifolds. Differentiable manifolds, mappings, and embeddings. Implicit functions theorem, exterior differential forms, and affine connections. Tangent bundles. Stoke's theorem. Critical points. Sard's theorem. Whitney's embedding theorem. Introduction to Lie groups and Lie algebras. Applications.

Topics to be chosen from Measure and Integration, Measurable Selections, Locally Convex Spaces, Topological Groups, Harmonic Analysis, Banach Algebras.

Harmonic functions. The Riemann mapping theorem. Conformal mappings for multi-connected domains. Elliptic functions and Picard's theorem. Analytic continuation. Entire functions. Range of an analytic function. Topics in univalent functions and geometric function theory.

Algebra of bounded operators, self-adjoint operators in Hilbert Spaces, Normal operators, compact operators. Projections. Spectral theory of linear operators in normed spaces and Hilbert spaces. Spectral Mapping Theorem. Banach-Alaoglu Theorem.

Fixed points methods. Nonexpansive mappings. Differential and integral calculus in Banach spaces. Implicit and inverse function theorems. Potential operators and variational methods for linear and nonlinear operator equations. Extrema of functionals. Monotone operators and monotonicity methods for nonlinear operator equations. Applications to differential and integral equations and physical problems.

Gateaux and Fre'chet differentials. Classical calculus of variations. Necessary conditions. Sufficient conditions for extrema. Jacobi and Legendre conditions. Natural boundary conditions. Broken extrema, Erdmann-Weierstrass condition. Multiple integral problems. Constrained extrema. Hamilton principle with applications to mechanics and theory of small oscillations. Problems of optimal control. Direct methods including the Galerkin and the Ritz-Kantorovich methods. Variational methods for eigenvalue problems.

Selected topics from the following: Variational inequalities, weak lower semicontinuity and extremal problems in abstract spaces, theory of optimal control, stochastic control, distributed parameter systems, optimization problems over infinite horizons, algorithmic and penalty methods in optimization.

Review of systems of linear differential equations to include existence and uniqueness, contraction mappings, fixed points, transition matrix, matrix exponentials, the Laplace transform and stability. Linear control systems. Controllability, observability and duality. Weighting patterns and minimal realizations. Feedback. Linear regulator problem and matrix Riccati equations. Fixed-end point problems. Minimum cost and final-value problems in control theory. Stability of linear systems. Uniform stability. Exponential stability.

Enumerative analysis, generating functions. Sorting and searching. Theory of codes. Block design. Computational combinatorics. Methods of transforming combinatorial ideas into efficient algorithms. Algorithms on graphs, network flow.

Lattices: basic properties, distributive and modular lattices, complete lattices, equivalence relations and algebraic lattices; Algebras: definition and examples, isomorphisms, subalgebras congruences and quotient algebras, homomorphism theorems, direct products, subdirect products, simple algebras, class operators and varieties, terms and term algebras, free algebras, Birkhoff's theorem, equational logic, Boolean algebras: Boolean algebras and Boolean rings, filters and ideals, Stone duality, connections with model theory: First-order languages and structures, reduced products and ultraproducts.

Advanced theory of solvable and nilpotent groups. General free groups. Krull- Schmidt theorem. Extensions. The general linear group. Group rings and group algebras. Representation theory of groups.

Selected topics from: prime spectra and dimension theory; class groups; ideal systems and star operations; multiplicative ideal theory; generator Property; homological aspects of commutative rings; pullbacks of commutative rings.

Selected topics from: Groups, rings, modules, and general algebraic systems.

Contents vary. Concepts and methods in algebra which have wide applications in mathematics as well as in computer science, systems theory, information theory, physical sciences, and other areas. Topics may be chosen from fields of advanced matrix theory; algebraic coding theory; group theory; Gröbner bases; or other topics of computational and applied algebra.

Theory of point estimation, Properties of estimators. Unbiased estimation and lower bounds for the variance of an estimator. Methods of moments and maximum likelihood. Bayes' and minimax estimation. Minimal sufficient statistics. Neymann-Pearson theory of testing of hypotheses. Unbiased and invariant tests. Confidence estimation. Confidence intervals (shortest length, unbiased and Bayes'). The general linear hypothesis and regression. Analysis of variance. Nonparametric statistical inference.

Measurable functions and integration. Radon-Nikodym theorem. Probability space. Random vectors and their distributions. Independent and conditional probabilities. Expectation. Strong laws of large numbers. The weak compactness theorem. Basic concepts of martingales. Invariance principles. The Law of the Iterated Logarithm. Stable distributions and infinitely divisible distributions.

Self-adjoint boundary-value problems, Sturm-Liouville theory. Oscillation and comparison theorems. Asymptotic behavior of solutions. Singular Sturm-Liouville problems and non self-adjoint problems. Hypergeometric functions and related special functions. Bifurcation phenomena.

Classification of first order systems. Hyperbolic systems, method of characteristics. Applications to gas dynamics. Dispersive waves; application to water waves. Potential theory, single and double layers, existence theory for Dirichlet and Neumann problems.

Maximum Monotone Operators. Bounded and unbounded operators. Pseudo monotone operators. Self-adjoint. Evolution Equations in Hilbert and Banach spaces. Hille-Yosida Theorem, application to linear heat and wave equations. Nonlinear Evolution equations. The Galerkin Method.

Review of the Fredholm and Hilbert-Schmidt theories for Fredholm integral equations of the second kind. Kernels with weak and logarithmic singularities. Singular integral equations of the first and second kind (Abel, Carleman, and Wiener- Hopf equations). Nonlinear integral equations (Volterra and Hammerstein equations). Application of the Schauder fixed point theorem. Nonlinear eigenvalue problems and integral equation methods for nonlinear boundary-value problems. Nonlinear singular integral equations. Applications to engineering and physics (the nonlinear oscillator, the airfoil equation, nonlinear integral equations arising the radiation transfer, hydrodynamics, water waves, heat conduction, elasticity, and communication theory).

Pre-Requisites: MATH535 Or MATH535

Numerical methods and approximate solutions of Fredholm integral equations of the second kind (both linear and nonlinear). Approximation of integral operators and quadrature methods. Nystrom method. Method of degenerate kernels. Collectively compact operator approximations. Numerical methods for Volterra integral equations. Methods of collocation, Galerkin, moments, and spline approximations for integral equations. Iterative methods for linear and nonlinear integral equations. Eigenvalue problems.

Theoretical topics in numerical analysis based on functional analysis methods. Operator approximation theory. Iterative and projection methods for linear and nonlinear operator equations. Methods of steepest descent, conjugate gradient, averaged successive approximations, and splittings. Stability and convergence. Abstract variational methods and theoretical aspects of spline and finite element analysis. Minimization of functionals. Vector space methods of optimization. Newton and quasi-Newton methods for operator equations and minimization.

Development of the dynamic programming algorithm. Optimality principle and characterizations of optimal policies based on dynamic programming. Shortest route problems and maximum flow problems. Adaptive process. One-dimensional allocation processes. Reduction of dimensionality. Additional topics include imperfect state information models, the relation of dynamic programming to the calculus of variations, and network programming. Computational experience will be acquired by working on individual projects of applying dynamic programming to case study problems.

Contents vary. Topics selected from: Nonconvex optimization, geometric programming, Lagrangian algorithms, sensitivity analysis, large-scale programming, nonsmooth optimization problems and optimality conditions in infinite-dimensional spaces, combinatorial optimization, computation of fixed points, complementarity problems, multiple-criteria optimization, and semi-infinite programming.

Variable Content. Advanced topics not covered in regular courses. Prerequisite: Admission to Ph.D. Program.

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Prerequisite: Admission to Ph.D. Program

Prerequisite: Admission to Ph.D. Program

Ph.D. students are required to attend Departmental seminars delivered by faculty, visiting scholars and graduate students. Additionally, each Ph.D. student should present at least one seminar on a timely research topic. Ph.D. students should pass the comprehensive examination as part of this course. This course is a prerequisite to registering the Ph.D. Pre-dissertation MATH 711. The course is graded on Pass or Fail basis. IC grade is awarded if the Ph.D. Comprehensive exam is not yet passed.

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

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

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