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This course introduces the principal algorithms for linear, network, discrete, nonlinear, dynamic optimization and optimal control. Emphasis is on methodology and the underlying mathematical structures. Topics include description of the problem of optimisation and the geometry of Rn, n > 1, convex sets and convex functions, unconstrained optimization: necessary and sufficient conditions for local minima/maxima, constrained optimization: equality and inequality constraints, Lagrange multipliers and the Kuhn-Tucker conditions, computational methods for unconstrained and constrained optimization, steepest descent and Newton's methods, quadratic programming, penalty and barrier methods, sequential quadratic programming (SQP) implementation in MATLAB/OCTAVE.
This course is designed to equip students with the basic techniques for the efficient numerical solution of problems in science and engineering. Topics will include: Curve fitting and function approximation. Approximation formulae for kth derivatives. Composite rules and Romberg integration, Gauss quadrature, numerical method for multiple integrals. Numerical methods for ordinary differential equations. Numerical methods for Eigenvalues, the power method for finding dominant eigennvalues, the inverse power method for finding smallest eigenvalues, the shifted inverse power method, for finding an eigenvalues closest to a given approximate eigenvalue. Piece-wise polynomial interpolation, cubic splines.
This course develops concepts in quantum mechanics such that the behaviour of the physical universe can be understood from a fundamental point of view. It provides a basis for further study of quantum mechanics. Content will include: Historical origin of Quantum Theory: Blackbody radiation, Photoelectric effect, Compton effect, Optical Spectra of atoms. General formalism of Quantum theory: operators, wavefunctions and their physical significance, expectation value, commutation relations, uncertainty principle. The Schroedinger equation, infinite square well, the square well in three dimensions, central potential, step potential. The Harmonic Oscillator, Angular momentum in quantum mechanics. Approximation methods: Stationary Perturbation theory, Variational method, WKB approximation, Theory of Scattering.
This is an introductory mechanics course designed to consolidate the understanding of fundamental concepts in mechanics such as force, energy, momentum etc. more rigorously as needed for further studies in physics, engineering and technology. Topcs covered include kinematics and dynamics of point masses, Newton’s laws, momentum, energy, angular momentum and torque, conservation laws, motion under gravity, central force problem, Virial theorem, Kepler’s laws, Rutherford problem, coupled oscillations, dynamics of rigid bodies, moment of inertia tensor, Euler’s equations, orthogonal transformation and Euler’s angle, Cayley Klein parameters, symmetric top, Lagrangian dynamics, generalized coordinates and forces, Lagrange’s equation, Hamilton’s principle, and variational methods.
This course serves as an introduction to the field of operations research. It will quip students with scientific approaches to decision-making and mathematical modelling techniques required to design, improve and operate complex systems in the best possible way. Topics covered include linear programming, the simplex method, duality and sensitivity analysis, integer programming , nonlinear programming, dynamic programming and network models.
This course is intended to introduce the student to the basic concepts and theorems of functional analysis and its applications. Topics covered include linear spaces, topological spaces, normed linear spaces & Banach Spaces, inner product spaces, Hilbert spaces, linear functional and the Hahn-Banach theorem.
This course is designed to equip students with the basic techniques for the efficient numerical solution of problems in science and engineering. Topics covered include round off errors and floating-point arithmetic, solution of non-linear equations, bracketing, fixed point methods, secant method, Newton's method, zeros of polynomials, Polynomial interpolation, orthogonal polynomial, least squares approximations, approximation by rational function, numerical differentiation, numerical integration, and adaptive quadrature.