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MTH 501 Mathematical Analysis Boolean algebra. mapping. Denumerable sets. Real numbers: Axioms. Ordering. Tietze Urysohn extension theorem. Metric spaces: fundamentals. Continuous mapping. Limits. Cauchy sequences. Compactness. Space of continuous functions: Derivatives. Formal rules of derivation. Sets and functions. Axioms for the real numbers. Suprema and infima and the axioms of bounds. Real sequence. Real series. Hilbert spaces. Differential calculus. Integration.
MTH 502 Linear Algebra Linear vector spaces. Basis. Linear dependence and independence. Linear transformation. Eigenvalues and eigenvectors. Positive definite matrices. Linear programming.
MTH 503 Numerical Analysis Solutions of algebraic equations. Interpolation. Numerical integration and differentiation. Numerical solution of ordinary differential equations. Approximation theory. Error analysis. Computer applications.
MTH 504 Theory of Probability Concepts of probability, conditional probability and independence. Random variables and distributions. Central limit theorem. Laws of large numbers. Introduction to stochastic processes theory.
MTH 505 Functional Analysis Normed spaces. Banach spaces. Seperability. Arzela-Ascoli theorem. Ston-Weierstrass theorem. Hahn Banach theorem. Dual spaces. Riesz representation theorem. Hilbert spaces.
MTH 506 Differential Equations Existence and uniqueness. Linear systems. Analytic systems. Autonomous systems. Stability theory. Sturm-Liouville theory. Introduction to partial differential equations.
MTH 601 Ordinary Differential Equations Qualitative theory of ordinary differential equations. Existence and uniqueness of solutions. Stability theory. Periodic solutions. Limit cycles. Applications to theory of oscillations.
MTH 602 Methods of Mathematical Physics Eigenfunctions/eigenvalues problems. Diffusion/wave problems and exposition of characteristics and Green’s function. Integral transforms. Variational. Perturbation and distribu¬tion theoretic methods for solving differential. Difference and integral equations.
MTH 603 Partial Differential Equations Diffusion equations for nonuniform media. Solution of the nonhomogeneous equation subjected to time dependent boundary conditions both in Cartesian as well as other coordinate systems. The Laplace and Poisson equations. Applications of complex variables technique. Green’s function for time independent and time dependent problems. The method of characteristics for the wave equation and propagation of discontinuities. Introduction to the application of integral transforms.
MTH 604 Integral Equations Methods of solution. Singular types and their methods of solution. Alternative integral equation reformulation of boundary value problem. Affiliated variational principles.
MTH 605 Perturbation and Asymptotic Methods Exposition of various methods. Regular perturbation theory. Singular perturbation theory. Initial and boundary layers. Method of multiple scales. Ray theory. Two times methods. Application problems.
MTH 606 Numerical Solution of Partial Differential Equations Using the Finite Difference Method Fundamentals of the finite difference method. Implementation to the diffusion, wave and Poisson equations. Discussion of stability, consistency and convergence of the methods presented. The method of characteristics for hyperbolic equations. Examples.
MTH 607 The Finite Element Method Variational methods. Rayleigh Ritz and Galerkin formulation. The Finite Element idealization. Applications. Higher order elements and isoparametric elements. Higher degrees of freedom and curved sided elements. Discussion of convergence and error estimation. Computational aspects.
MTH 608 The Boundary Element Method The Green’s theorem and Green’s identities. Boundary integral formulation of differential equations. Boundary elements and applications to steady and time dependent problems. Computa¬tional aspects of the method.
MTH 609 Mathematical Modeling This course is directed towards the development of necessary modeling concepts and skills.
MTH 610 Operations Research Classical optimization. Optimization of functions of a single variable and multiple variables. Constructed optimization. Inequality constraints. Search techniques. Unconstrained problems. One and multiple dimensional problems (simultaneous and sequential methods). Various types of mathematical programming.
MTH 611 Optimization Theory and Techniques Unconstrained optimization. Nonlinear programming. Nondifferential optimization. Applications
MTH 612 Process Control Applications of Z-transform samples data systems. Dead time compensation. Self tuning regulator. Industrial applications.
MTH 613 Group Theory Fundamentals. Jordan Holder theorem. Sylow theorem. Structure of finite dimensional algebras. Application to representation of finite groups. The classical linear groups.
MTH 614 Algebraic Topology Fundamental group and covering spaces. Simplicial and singular homology theory with applications. Cohomology theory. Duality theory. Homotopy theory. Applications. Relation between homotopy and homology.
MTH 615 Topological Groups Fundamentals. Haar measures. General theory of topological transformation groups. The existence of slices and application. The Smith theory of transformation groups.
MTH 616 Foundation of Statistics Sampling distributions. Point and interval estimations. Properties of estimators. Testing statistical hypothesis. Types of errors. Linear and multiple regression.
MTH 617 Introductory Probability and Applications Sample spaces and random variables. Special distributions. Bivariate random variables. Joint distributions. Marginal distributions. Independence. Applications.
MTH 618 Advanced Probability Theory Mathematical foundations of probability theory and stochastic processes: probability space. Measures. Expectation. Independence and conditioning. Convergence types. Laws of large numbers. Central limit theorem. Brownian motion. Martingales. Renewal processes.
MTH 619 Time Series and Applications Fundamental of time series analysis. Stationarity. Autocovariance. Autocorrelation and spectral analysis. Integral representation of a stationary time series. Linear filtering. Transfer functions. Estimation of a spectrum.
MTH 620 Regression Analysis and Design of Experiments Linear regression. Test of lack of fit. Multiple regression using matrix approach. Nonlinear regression. Completely randomized designs. Randomized complete blocks. Latin square. Factorial and incomplete blocks.
MTH 621 Theory of Reliability and Life Testing Life models. Relevant distribution (Poisson and Weilbull distributions). Reliability and hazard functions. Making in life testing. Design of experiments in life testing.
MTH 622 Stochastic Processes A systematic account of several principal areas in stochastic processes; branching processes. Markov chains (discrete and continuous parameter). Poisson processes. Gaussian processes. Brownian motion.
MTH 623 Stochastic Analysis and Stochastic Differential Equations Univariate and multivariate Brownian motion theory. Classi¬fication of diffusion models with killing. Stochastic diff¬erential equations and stochastic integrals. Examples and applications.
MTH 624 Ergodic Theory The fundamentals of ergodic theory. Ergodicity. Strong mixing. Weak mixing. Ergodic theorems. Recurrence. Entropy. Information theory. Topological entropy.
MTH 625 Differential Geometry Representation of planes. Representation of space curves. Curvature and osculating plane. Torsion. Involuite and evolute. Representation of surfaces and characteristics. Mapping of different types.
MTH 626 Computational Geometry for Design Curves and surfaces design. Composite curves and splines. Composite surfaces. Cross-sectional design. Computing methods for surfaces design.