MAT321β: Functional Analysis (30 lecture hrs + 15 tutorial hrs), Op. (Credit Value 2.5)
Metric spaces: Definition and examples, Open set, Closed set, neighbourhood, Convergence, Cauchy Sequence, Complete Linear, Completion of metric spaces, Banach’s fixed point theorem.
Normed spaces: Linear space, Normed space, Banach space, Finite dimensional normed spaces and sub spaces, Compactness and finite dimensions, Linear operators, Bounded and Continuous linear operators, Linear operators and functional, on finite dimensional spaces, Normed spaces of operators, Dual space, Inner product space, Hilbert spaces.
Fundamental Theorems for Normed and Banach spaces: Zorn’s Lemma, Hann-Banach Theorems, Reflexive spaces, Strong and weak convergence, Numerical integration and weak convergence.
Method of assessment: end of semester examination
MAT322β: Complex Variables (30 lecture hrs + 15 tutorial hrs), Op. (Credit Value 2.5)
Theory of Complex Variables: Complex Functions, Complex differentiability, the Cauchy-Riemann equations, Analytic functions, Cauchy’s Theorem, Cauchy’s Integral Formula, Taylor’s and Laurent’s Theorem, Classification of singularities, Laurent expansions, Contour Integration, The cauchy’s residue Theorem, Integration of rational and trigonometric functions using residue theorem
Method of assessment: end of semester examination
MAT323β: Differential Geometry and Tensor Analysis (30 lecture hrs + 15 tutorial hrs), Op. (Credit Value 2.5)
Differential Geometry: Unit tangent vector, principal normal, binomial vector and curvature of a curve, surfaces, parametric curves, surfaces of revolution, metric, directional ratios and coefficients, Gauss and mean curvature, orthogonal trajectories, families of dual curves, Geodesics.Tensor Analysis: Transformation of coordinates, summation convention, the Kronecker-delta, contravariant and covariant vectors, contravariant, covariant and mixed tensors, symmetric and skew-symmetric tensors, tensor algebra, metric tensor, conjugate metric tensor, Christoffel’s symbols of first and second kind, covariant derivatives.
Method of assessment: end of semester examination
MAT324β: Mathematical Models in Ecology (30 lecture hrs + 15 tutorial hrs), Op. (Credit Value 2.5)
Basic description of mathematical modelling, Introduction of models in Ecology, Analysis of Dynamical systems. Non linear Dynamical systems, Web analysis, population Dynamics. Logistic model, Graphical and Analytical Approaches to Harvesting, Economics of Harvesting. Breeding Season and age structure, Predator-prey system with age structure, Analysis based on competition with aid of logistic equation, Stability and Complexity, The statistical mechanics of population .
Method of assessment: end of semester examination
MAT325β: Introductory Econometrics (30 lecture hrs + 15 tutorial hrs), Op. (Credit Value 2.5)
Matrix algebra: Definition of matrices; rules of matrix algebra; determinants; ranks, inverses and solutions; Cramer’s Rule; quadratic forms; matrix definacy. Applications: solutions of multi-equation models; input-output analysis.
Optimization: Unconstrained optimization in the n-variable case; second order conditions and Hessian matrices. Constrained optimization in the n-variable case; multiple constraint cases and bordered Hessian matrices.
Applications: Maximization and minimization of various economic magnitudes in multi-variable settings. An Introduction to inequality-constrained optimization: profit maximization; non-negativity constraints.
Difference equations: Introduction to dynamics; applications: the cobweb pricing model; macroeconomic trade cycles.
Method of assessment: end of semester examination
MAT326β: Mathematical Foundations of Computer Science (30 lecture hrs + 15 tutorial hrs), (Credit Value 2.5); Op. for Students who do not follow Applied Mathematics or Computer Science
Logic: Syllogisms, propositional logic, propositions, arguments, predicates and quantifiers, logic programming. Number Systems: Number Systems (decimal, Roman etc.), Binary number system, Octal system, Binary arithmetic (including complements methods)
Boolean Algebra and Logic circuits: Boolean Algebra, Switching circuits, logic circuits.
Method of assessment: end of semester examination
IMT321β: Applied Algebra (Algebraic Data Encryption Methods) (30 lecture hrs + 15 tutorial hrs), (Credit Value 2.5) Op. Prerequisites MAT111β, MAT211β, MAT221β
Introduction to the RSA Encryption Scheme: Raising integers to large powers to a given modulus, ’Egyptian exponentiation’, Discussion of primality testing, The Little Fermat and Rabin tests, Implications for the RSA system, Verifying authenticity
Topics in Rings and Fields: GF(p), Polynomials over a ring, The Primitive Element Theorem, Recurrent Sequences, shift registers, The ideal and minimal polynomial of a sequence, Indexing polynomials. Congruence modulo a polynomial, Construction of finite fields, Construction of indexing polynomials, Cyclotomic polynomials, Factorizing polynomials over finite fields
Error detection and correction in telecommunication: ISBN codes, The Hamming metric, The minimum distance of a code, Elementary bounds on the minimum distance of a code, Equivalence of codes, Parity checks, The sphere-packing bound, Reed-Muller codes, Linear Codes, Dual codes, The parity check matrix of a linear code Syndrome decoding, The Hamming codes, Cyclic Codes, Generator polynomials and check polynomials, Construction of binary Hamming codes as cyclic codes, The BCH codes, the Golay code.
Method of assessment:continuous assessment (assignments) and end-of-semesters examination.
IMT322β: Computational Fluid Dynamics (30 lecture hrs + 15 tutorial hrs), Op. (Credit Value 2.5)
Basic Concepts of Fluid Flow: Introduction, Conservation Principles, Dimensionless form of a flow equation Simplified Mathematical Models for fluid flows: Incompressible Flow, Inviscid (Euler), Stokes (Creeping) Flow
Mathematical Classification of Flows: Hyperbolic Flows, Parabolic Flows, Elliptic Flows, Introduction to the Navier- tokes Equation
Introduction to Numerical Methods: Approaches to Fluid Dynamical Problems, What is CFD? , Possibilities and Limitations of Numerical Methods.
Components of a Numerical Solution Process: Mathematical Model, Discretization Method, Numerical Grid, Finite Approximation, Solution process, Convergence Criteria, Properties of Numerical Schemes
Discretization Approaches: Finite Difference Methods, Application of Finite Difference Methods to Different types of Models, Idea of Finite volume and Finite Element Methods with motivating examples.
Method of assessment: end of semester examination
IMT323β: Theory and Applications of Neural Networks (30 lecture hrs + 15 tutorial hrs), Op. (Credit Value 2.5) Prerequisite: IMT2b2β or Level I and Level II of ICT2b13 (CCIT) course
Biological computers and their capabilities over digital computers, problem of classification and recognition, biological neurons, artificial neural networks, Mathematics of single-layer neural networks – the Perceptron, learning and training, learning rate, Perceptron training algorithm, Introducing Mathematica, methods to adjust the learning rate, convergence of solutions, basins of attractions, Baysian inference methods. Types of neural networks (feed-forward, back-propagation etc.) and algorithms for implementation. Monte- Carlo Methods, Hopfield network for optimization problems, e.g., traveling salesman problem, Applications in forecasting problems in finance, meteorology, particle physics.
Method of assessment: continuous assessment (assignments) and end-of-semesters examination.
IMT324β: Statistics with Computer Applications (30 lecture hrs + 15 tutorial hrs), Op. (Credit Value 2.5)
Introduction to Statistical Packages, Data Analysis using a computer package, Descriptive Statistics, Graphical representation of data, Estimation, Hypothesis Testing, Regression, Analysis of Variance, non-parametric methods. Method of assessment: continuous assessment (assignments) and end-of-semesters examination.
AMT321β: Electro Magnetic Theory (30 lecture hrs + 15 tutorial hrs), Op. (Credit Value 2.5)
Electrostatic field equations, electrostatic potential, boundary value problems, magnetostatic field equations, boundary value problems, vector potential, Maxwell’s equations, Lorentz condition and gauge transformations, electromagnetic waves in non-conducting media, Electromagnetic waves in conductors.
Method of assessment: end of semester examination
AMT322β: Theory of Special Relativity (30 lecture hrs + 15 tutorial hrs), Op. for students not following Physics (Credit Value 2.5)
Introduction (Inadequacy of Newtonian mechanics and the need of a new mechanics), The Space-time continuum and separation between events, Events and particles, Space-time, world lines and space-time diagrams, the motion of a material particle, the light-cone, the fundamental quadratic form, space-time as a Riemanian space, proper time and speed of light, Minkowskian coordinates, The Lorentz Transformations, Length contraction, the time dialation, composition of velocities, the velocity 4-vector and acceleration 4-vector, the expanding universe in S.R., The red-shift. Particles and mass, equation of motion, motion under a constant relative force.
Method of assessment: end of semester examination
AMT323β: Mathematical Quantum Mechanics (30 lecture hrs + 15 tutorial hrs), Op. (Credit Value 2.5)
The failure of Newtonian Mechanics to explain phenomena at microscopic level, problem of separation of observable from the observer. Quantum states, representation of quantum states by state (column) vectors, Observables as Hermitian Matrices, mean values and correspondence principle, the angular momentum of a photon, Uncertainty. Equations of motion, quantum particles in one-dimension and three dimension. The Spin of the electron, quantum particle in a spherically symmetric potential. The bound states of the hydrogen atom, The Dirac notation. Fourier transform, Applications to wave-packets, Basic Ideas of Hilbert space theory, theory of linear operators in Hilbert Spaces, Cauchy-Schwarz and Bessel inequalities, Completeness. Special Topics in Quantum Mechanics and applications: The EPR Paradox and Entanglement, Quantum effects in the computer-chip, Introduction to Quantum Computer.
Method of assessment: end of semester examination
AMT324β: Basic Statistics and Data Analysis (30 lecture hrs + 15 tutorial hrs), (Credit Value 2.5) Op. Only for Bio Science Students
Fundamental concepts in probability, Random variables, Mean, variance and expected values, Classification and Description of Sample Data, Sampling Distributions, Estimations, Hypothesis Testing, Regression Analysis, Analysis of Variance and Scientific Applications.
Method of assessment: continuous assessment (assignments) and end-of-semesters examination.