MS Mathematics
Semester I
This semester contains the advanced courses of Applied Mathematics
Following courses will be offered in semester:
S No Course Name Code Credit Hr
i. Core-I

03
ii. Core-II

03
ii. Core-III
03
S No Course Name Code Credit Hr
Semester II
This semester contains the foundation courses of Nonlinear Dynamics
Following courses will be offered in semester:
i.
Core-IV

03
ii.
Core-V

03
iii.
Elective-I

03
S No Course Name Code Credit Hr
Semester III
This semester contains the foundation courses of Specialized area
Following courses will be offered in semester:
i. Elective-II

03
ii. Elective-III
03
iii. Thesis-I
S No Course Name Code Credit Hr
Semester IV
This semester contains only the research work
Following courses are being offered in semester:
i.
Thesis-II 06

Core Courses

 

Advanced Partial Differential Equations        

Advanced Numerical Techniques     

Nonlinear Dynamics-I   

Numerical Solution to PDEs-I 

Numerical Solution to PDEs-II

Nonlinear Dynamics-II                                     

Numerical Linear Algebra

Newtonian Fluid Mechanics                           

Non-Newtonian Fluid Mechanics                   

Initial and boundary value problems              

Magneto-hydro-dynamics (MHD)

Advanced Number Theory 

Advanced Probability & Statistics               

Introduction to Cryptography

Continuous Optimization

Discrete Optimization

Methods in Optimization

Optimization Modeling with AIMMS

Computer Programming and Softwares for Mathematicians

Stochastic Processes

Electives

Stellar Dynamics                                                                        

General Relativity                                                                                   

Mathematical Modeling & Simulation                               

Modern Control Theory                                                   

Introduction to Cryptography                                                       

Acoustics                                                                                              

Advanced Probability & Statistics                                     

Bayesian Theory                                                             

Computational Fluid Dynamics                                         

Heat Transfer and Mass Transfer                                     

Gravitational Wave Data Analysis                                                 

Operations Research                                                                  

Mathematics of Cryptography                                          

Cryptanalysis                                                                             

Elliptic Curves

Information Theory

  1. Description of Courses

 

Course Title: STELLAR DYNAMICS

Text and Reference Material:

  1. Stellar Dynamics by Leonid P. Ossipkov, Igor I. Nikiforov
  2. Principles of Stellar Dynamics by S. Chandrasekhar
  3. Stellar Dynamics by William Marshall Smart

Contents:

Potential Theory: Spherical systems, potential density pairs, potentials of spheriodal, ellipsoidal and disk systems, Potential of our galaxy, N-body Codes: Direct summation, tree-codes, particle mesh codes, The Orbits of Stars: Orbits in spherical potential, orbits in axisymmetric potential, orbits in triaxial potential, orbits in elliptical galaxies, numerical orbit integration, Equilibria of Collisionless Systems: Boltzmann equation, Jeans theorems, distribution functions, Jeans and virial equation, Kinetic Theory: Relaxation processes, Fokker Planck approximation, the evolution of spherical stellar systems, Dynamical Friction: Chandrasekhar's formula, applications of dynamical friction, decay of black hole orbits, formation and evolution of binary black holes.

Course Title: COMPUTATIONAL FLUID DYNAMICS

Text and Reference Material:

  1. Computational Methods for Fluid Dynamics  by J. H. Fringer, M. Peric
  2. Finite Element Methods by Cuvelier, Segal
  3. Applied CFD Techniques: An introduction based on FEM (John Wiley & Sons, 2001)

Contents:

Classification, implicit & explicit methods, iterative & time/space marching schemes, grids, Discretization process, boundary conditions, aerospace applications, Spectral Element Method, Finite-difference; finite volume methods for solution of Navier- Strokes & Euler equations, Classification of partial differential equations and solution techniques. Truncation errors, stability, conservation and monotonicity, Differencing strategies. Advanced solution algorithms, Grid generation, Construction of complex CFD algorithms, Current applications, Use of CFD codes, CFD Simulation.

Course Title: MATHEMATICAL MODELING AND SIMULATION

Text and Reference Material:

  1. Mathematical Modeling & Simulation by Kai Velten
  2. Mathematical Modeling & Computer Simulation by Daniel P. Maki, Maynard Thompson
  3. A First Course in Mathematical Modeling by Frank Giordano, William P. Fox, Steven Horton

Contents:

Introduction to a Dynamic systems and control, modeling and analysis techniques, the fundamentals and applications of control systems, Modeling and Simulation of Dynamic systems based on Bond graph theory , transfer functions, sensitivity and robust control and digital control. Case studies related to motion control system design, electromechanical system design, Stochastic processes applied to control of various types of systems, Markov chains, Queuing theory, Bifurcations, Perturbation Methods, non-homogeneous Equations, , Training on Lab View software.

Course Title: HEAT TRANSFER AND MASS TRANSFER

Text and Reference Material:

  1. Fundamentals of Heat and Mass Transfer by F. P. Incropera, S. L. Adrienne
  2. Ozonation of water and waste water by C. Gohchalik, J. A. Libra
  3. Mass Transfer: Principles and Applications by D. Basmad Jian
  4. Advances in Heat Transfer by Thomas F. Irvine, Jr. & James P. Hartnett

Contents:

Basic Rules of the Heat Conduction and Heat Conductive Equations, Stable Heat Conduction, Instable Heat Conduction, Possessing the Heat Conduction of Moving Boundaries, Basic Concept of the Heat Radiation, Radiating Heat Transfer of the Solid Surface, Radiation among the Absorption, Radioactivity Medium, Including the Radiating Heat Transformation Convective Heat Transfer in Sealed Cavity of Absorption, Radioactivity Medium. Molecular Diffusion in Liquid, Two-dimension Stable Diffusion, Air Diffusion Coefficient, Conversation Equation, Convection Mass Transfer, Critical Mass Transformation.

Course Title: NON-NEWTONIAN FLUID MECHANICS

Text and Reference Material:

  1. Fluid Mechanics by F. Durst
  2. Dynamics of Polymeric liquids by R. B. Bird
  3. Introduction to Fluid Mechanics & Transport Phenomena by G. Hauke

Contents:

Basic review of fluid properties and basic flow equations, (Navier-Stokes’ equations etc.), laminar flows, Turbulent flows, Compressible and Incompressible flows, Partial differential equations governing the conservation of mass, Momentum and energy of Newtonian fluids are derived. Dimensional analysis used to simplify the governing equations, low Reynolds number flow, strokes flow, high Reynolds number laminar flow, Boundary layer separation phenomena and approximations to the governing equations, laminar stability and transition to turbulent boundary layer conditions.

Course Title: ADVANCED PARTIAL DIFFERENTIAL EQUATIONS

Text and Reference Material:

  1. Introduction to Partial Differential Equations and Boundary Value Problems by R. Dennemyer
  2. Linear Partial Differential Equations for Scientists and Engineers by Tyn Myint-U & Lokenath Debnath
  3. Techniques in Partial Differential Equations by C. R. Chester
  4. Applied Partial Differential Equations with Fourier series and Boundary values Problems by Richard Haberman
  5. Introduction to Partial Differential Equations by Matthew P. Coleman

Contents:

Definition of PDE, Solution of PDEs and principle of superposition, Boundary conditions and their types, Homogeneous PDEs with constant coefficient and separation of variables, Holomorphic functions, Classification of second order linear PDEs, The Heat equation and diffusion equation, Wave equation and vibrating string, Initial and boundary conditions for heat and wave equations, Laplace’s Equation, Solutions of Heat, wave and Laplace’s equations by separation of variables, Fourier transform and properties, Convolution theorem for Fourier transform, Solution of PDEs by Fourier transform, Laplace transform and its properties

Convolution theorem for Laplace transform, Laplace transform of Heaviside unit step and Direct Delta functions, Solutions of partial differential equations by Laplace transform method  

Green’s function and its properties, Method of Green’s function, Nonlinear partial differential equations, Method of characteristics, Solution of nonlinear partial differential equations by method of characteristics

Course Title: MAGNETOHYDRODYNAMICS

Text and Reference Material:

  1. An Introduction to MHD by P. A. Davidson
  2. Magnetohydrodynamics by Cowling, T.G
  3. Magnetohydrodynamics by A.Wesley, Kulikowsky, A.G., and Lyabimov, A.G.
  4. Cosmical Electrodynamics by Alfve’s H., and Falthammar, C.
  5. Plasma Electrodynamics by Akhiezer et.al.
  6. Magnetohydro dynamics by Kendale and Plumpton, C.
  7. Magnetohydrodynamics, Shock Waves by Anderson, J.E.

Contents:

 Equations of electrodynamics, Equations of Fluid Dynamics, Ohm’s law equations of magnetohydrodynamics, Motion of a viscous electrically conducting fluid with linear current flow, steady state motion along a magnetic field, wave motion of an ideal fluid, Effects of molecular structure, Currents in a fully ionized gas, partially ionized gases, interstellar fields, dissipation in hot and cool clouds, Kinematics of MHD: Advection and Diffusion of a Magnetic field, Low-Magnetic Reynold’s number.

Course Title: NEWTONIAN FLUID MECHANICS

Text and Reference Material:

  1. Introduction to Fluid Mechanics by Philip j. Pritchard, Fox and Mc Donald
  2. Fluid Mechanics, Frank M. White
  3. Fluid Mechanics: Statics and Dynamics by John Harris
    1. Fluid Mechanics, Landau Lifshitz
    2. Fluid Dynamics by G. K. Batchelor
  1. Dynamics of Polymeric liquids, R.B. Bird, R.c. Armstrong and O. Hassager

Contents:

Fluids and flows, Viscosity, Newton’s law of viscosity, Classification of fluids, Types of flows, Static equation, Euler’s equation, Conservation laws, Flux, Fourier law of conduction, Fick’s laws, Bernoulli Equation, Navier-Stokes equation and exact solutions Dimensional analysis and Simlitude, Boundary layer approximations and governing equations.

Course Title: INITIAL AND BOUNDARY VALUE PROBLEMS

Text and Reference Material:

  1. Perturbation methods, Nayfeh, A.
  2. Boundary value problems of Mathematical Physics by Stakgold, I.
  3. Methods based on the Wiener-Hopf technique for the solution of Partial Differential Equations by Noble, B.
  4. Analytical Techniques in the Theory of Guided Waves by Mitra, R., and Lee, S.W.

Contents:

Green’s function method with applications to wave-propagation, regular and singular perturbation techniques with applications. Variational methods. A survey of transform techniques; Wiener-Hopf technique with applications to diffraction problems, Asymptotic expansion integrals and properties, Methods of averaging, Convergence of mathematical solutions.

Course Title: NON-NEWTOMIAN FLUID MECHANICS

Text and Reference Material:

  1. Theology and Non-Newtonian Flow by John Harris
  2. Mechanics of Non-Newtonian fluids by W.R. Schowalter
  3. Dynamics of Polymeric liquids by R.B. Birk, R.c. Armstrong and O. Hassager
  4. Principles of Non-Newtonian Fluid mechanics by G Astarita and G. Merrucci

Contents:

Classification of Non-Newtonian Fluids, Rheological formulae (Time-independent fluids, Thixotropic fluids and viscoelastic fluids), Variable viscosity fluids, Cross viscosity fluids, The deformation rate, Viscoelastic equation, Materials with short memories, Time dependent viscosity. The Rivlin-Ericksen fluid, Basic equations of motion in rheological models. The linear viscoelastic liquid, Couette flow, Poiseuille flows. The current semi-infinite field, Axial oscillatory tube flow, Angular oscillatory motion, Periodic transients, Basic equations in boundary layer theory, Orders of magnitude, Truncated solutions for viscoelastic flow, Similarity solutions, Turbulent boundary layers, Stability analysis.

Course Title: GENERAL RELATIVITY

Text and Reference Material:

  1. An Introduction to General Relativity: Spacetime and Geometry by Carroll, S.
  2. General Relativity: An introduction to the theory of the gravitational field by Stephani, H. 
  3. Gravitation by Misner, C. W., Thorne K.S., and Wheeler, J.A.
  4. A First Course in General Relativity by Bernard, S.
  5. An introduction to Einstein’s general relativity by Hartle, J. Gravity
  6. General Relativity by Wald, R.
  7. A Relativist’s Toolkit by Poisson, E.  

Contents:

Flat Spacetime, Vectors and Dual Vectors, Tensors. Special Relativity, Energy and Momentum, Conserved Currents, Stress Energy Tensor, Transformation Law for Tensors, Metric in a Curved Space, Orthonormal and Coordinate Bases; Derivatives; Tensor Densities; Differential Forms and Integration, Gauge/Coordinate Transformations.

Metric in a Curved Space, Orthonormal and Coordinate Bases; Derivatives; Tensor Densities; Differential Forms and Integration, Gauge/Coordinate Transformations. Connection and Curvature, Geodesics, Introduction to Curvature, Geodesic Deviation, Bianchi Identity, Killing Vectors and Symmetries, Einstein’s Equation and Gravitation, Cosmological Constant, Hilbert Action.

Weak Field/Linearized General Relativity, Gauge Invariant Characterization of Gravitational Degrees of Freedom. Spacetime of an Isolated Weakly Gravitating Body, Gravitational Waves, Gravitational Lensing, Cosmology, Friedmann-Robertson-Walker Solution; Distance Measures and Redshift, Schwarzschild Solution, Birkhoff’s Theorem, Metric of a Spherical “Star”, Black Holes, Collapse to Black Hole; Orbits of a Black Hole, Kerr and Reissner-Nordstrom Solutions, Advanced Topics and Current Research in General Relativity.

Course Title: GRAVITAIONAL WAVES AND DATA ANALYSIS

Text and Reference Material:

  1. GW in general: K. S. Thorne. Gravitational radiation. In S. W. Hawking and W. Israel, editors, 300 years of gravitation, chapter 9, pages 330–358. Cambridge, University Press, Cambridge. B. F. Schutz. Gravitational wave astronomy. Classical and Quantum, Gravity, 16(12A):A131–A156. C. Cutler, K. S. Thorne. An overview of gravitational-wave sources. Arxiv preprint gr-qc/0204090, April 2002.
  2. Modeling, parameter estimation &c.: General (mostly frequentist) statistics: A. M. Mood, F. A. Graybill, and D. C. Boes. Introduction to the theory.of statistics. McGraw-Hill, New York. Bayesian methods, computational methods: A. Gelman, J. B. Carlin, H. Stern, D. B. Rubin. Bayesian data analysis.Chapman & Hall / CRC, Boca Raton. P. C. Gregory. Bayesian logical data analysis for the physical sciences.Cambridge University Press, Cambridge. Bayesian time series / Fourier analysis G. L. Bretthorst. Bayesian spectrum analysis and parameter estimation.Lecture Notes in Statistics 48, Springer, Berlin. Gaussian non-white noise modelling, "Whittle" likelihood etc.: L. S. Finn. Detection, measurement, and gravitational radiation, Physical Review D, 46(12):5236–5249.

Contents:

Prior, likelihood, posterior, MAP, ML, Starting from simple examples of single and multiple sinusoid and chirp mass signals, auto-covariance/-correlation, spectrum, white noise, coloured noise, spectrum estimation, Fourier methods, windowing, Marginal likelihood, evidence, Bayes factor,  likelihood ratio test, Neyman-PearsonLemma,generalized likelihood ratio test,   multiple testing, trials factor, "look-elsewhere-effect",Lindley's paradox, detection/false-alarm probabilities sensitivity/specificity), ROC curve,  non-detection limits, Common posterior computations, pseudo random number generation, inverse methodGibbs sampler, Metropolis sampler, Metropolis-Hastings sampler, simulated, annealing,  parallel tempering, nested sampling.

Course Title: MODERN CONTROL THEORY

Text and Reference Material:

i.         Mathematical control theory: deterministic finite dimensional systems (texts in applied

mathematics) (v. 6) by Eduardo d. Sontag

  1. Mathematical Control Theory: An ntroduction (Systems & Control: Foundations & Applications) By Jerzy Zabczyk

Contents:

What Is Mathematical Control Theory?   Proportional-Derivative Control, State-Space and Spectrum Assignment Outputs and Dynamic Feedback Dealing with Nonlinearity, I/O Behaviors,   Discrete-Time Linear Discrete-Time Systems Smooth Discrete-Time Systems Continuous-Time,  Linear Continuous-Time Systems Linearizations Compute Differentials Sampling, Volterra Expansions, Lie Brackets,  Lie Algebras and Flows Accessibility Rank Condition Ad, Distributions, and Frobenius' Theorem Necessity of Accessibility Rank Condition, Constant Linear Feedback ,  Feedback Equivalence Feedback Linearization,  Disturbance Rejection and Invariance,  Stability and Other Asymptotic Notions Unstable and Stable Modes,  Lyapunov and Control-Lyapunov Functions,  Linearization Principle for Stability, Introduction to Nonlinear Stabilization, Observers and Detectability, Dynamic Feedback External Stability for Linear Systems,  Frequency-Domain Considerations,  Parametrization of Stabilizers, Dynamic Programming,  Linear Systems with Quadratic Cost,  Tracking and Kalman Filtering Infinite-Time (Steady-State) Problem Nonlinear Stabilizing Optimal Controls, Review of Smooth Dependence Unconstrained Controls Excursion into the Calculus of Variations Gradient-Based Numerical Methods Constrained Controls: Minimum Principle Notes and Comments Optimality: Minimum-Time for Linear Systems Maximum Principle for Time-Optimality and it applications.

Course Title: INTRODUCTION TO CRYPTOGRAPHY

Text and Reference Material:

  1. Introduction to cryptography with mathematical foundations and computer implementations by Alexander Stanoyevitch
  2. A course in number theory and cryptography by Neal Koblitz

Contents:

Background and overview, One-time encryption using stream ciphers, Semantic security,Block ciphers and pseudorandom functions, Chosen plaintext security and modes of operation, The DES and AES block ciphers,Message integrity. CBC-MAC, HMAC, PMAC, and CW-MAC, Collision resistant hashing, Authenticated encryption. CCM, GCM, TLS, and IPsec. Key derivation functions, Odds and ends: deterministic encryption, non-expanding encryption, and format preserving encryption, Basic key exchange: Diffie-Hellman, RSA and Merkle puzzles, A crash course in computational number theory, Number theoretic hardness assumptions, Public key encryption, Trapdoor permutations and RSA, The ElGamal system and variants, Digital signatures and certificates, Identification protocols, Authenticated key exchange and TLS key exchange, Zero knowledge protocols and proofs of knowledge, Privacy mechanisms: group signatures and credential systems, Private information retrieval and oblivious transfer, Two party computation: Yao's protocol and applications, Elliptic curve cryptography, Quantum computing, Pairing-based cryptography, Lattice-based cryptography, Fully homomorphic encryption

Course Title: ADVANCED NUMBER THEORY

Text and Reference Material:

  1. Elementary Number Theory , Gareth A. Jones and J. Mary Jones
  2. Methods in Number Theory , Melvyn B. Nathanson
  3. Number Theory Ideas and Theory, Fundamental Problems, A.N. Parshin and I.R. Shafarevich

Contents:

Divisors; Bezeout’s identity; LCM, Linear Diophantine equations, Prime numbers and prime-power factorizations; Distribution of primes; Primality-testing and factorization, Modular arithmetic; Linear congruencies; An extension of chineses Remainder Theorem; The arithmetic’s of Zp; Solving congruence’s mod,Units; Euler’s function. The group Un; Primitive roots; The group Un, n is power of odd prime and n is power of 2. Quadratic congruences; The group of quadratic residues; The Legendre symbol, Quadratic reciprocity, Definition and examples; perfect numbers; The Modius Inversion formula., Random integers, Dirichlet series, Euler products, Sums of two Squares; The Gaussian integers; Sums of three Squares; Sums of four Squares, The problem; Pythagorean Theorem; Pythagorean triples; The case n=4; Odd prime exponents.

Course Title: MATHEMATICS OF CRYPTOGRAPHY

Text and Reference Material:

  1. Mathematics of public key cryptography by Steven d. Galbraith.
  2. An introduction to mathematical cryptography by Jeffrey Hoffstein, Jill pipher, j.h. Silverman
  3. Introduction to cryptography with mathematical foundations and computer implementatiuons By Alexander Stanoyevitch

Contents:

Historic background Cryptographic algorithms Types of attacks used to break cryptosystems, Modular arithmetic Greatest common divisors Congruences Chinese Remainder Theorem Primitive roots Finite fields, Substitution ciphers Polyalphabetic ciphers Permutation ciphers, One-way hash functions and properties Secure Hash Algorithm Birthday attacks, Applications to information assurance and cyber security

Course Title: CRYPTANALYSIS

Text and Reference Material:

  1. Cryptanalysis: a study of iphers and their solutionhelen fouché gaines
  2. Modern cryptanalysis: techniques for advanced code breaking by Christopher Swenson
  3. Applied cryptanalysis: breaking ciphers in the real world by mark stamp, Richard M. Low

Contents:

Introduction to cryptanalysis, Monographic substitution systems, Monoalphabetic unilateral substitution systems using standard cipher alphabets, Monoalphabetic unilateral substitution systems using mixed cipher alphabets, Monoalphabetic multilateral substitution systems part three - polygraphic substitution systems, Characteristics of polygraphic substitution systems, Solution of polygraphic substitution systems polyalphabetic substitution systems, Periodic polyalphabetic substitution systems, Solution of periodic polyalphabetic systems, A periodic polyalphabetic ciphers, part five - transposition systems, Types of transposition systems, Solution of numerically-keyed columnar transposition ciphers, Transposition special solutions 
part six - analysis of code systems, Types of code systems, Analysis of syllabary spelling, Frequency distributions of English digraphs, Frequency distributions of English trigraphs, Frequency distributions of English tetragraphs.

Course Title: ADVANCED PROBABILITY & STATISTICS

Text and Reference Material:

Applied Statistics for Engineers and Physical Scientists, Prentice Hall, 3rd edition, 2009, By Johannes Ledolter and Robert V. Hogg Applied Statistics with R

Contents:

Overview of the basic concepts in statistics and probability Tests based on normal distribution, Tests of characteristics of a single distribution; Tests of characteristics of two distributions Tests based on Student's t-distribution, Tests of characteristics of a single distribution; Tests of characteristics of two distributions, Tests of characteristics of two distributions; Certain chi-square tests, Certain chi-square tests; Simple linear regression model, Simple linear regression model, linear correlation; Inferences in the regression model and correlation, More on correlation and Inferences, Adequacy of the fitted model; Multiple linear regression, Multiple linear regression, Multiple linear regression; More on multiple regression, Tests based on F-Distribution, Inferences on variance, Analysis of variance, One-way classification, Two-way classifications, analysis of covariance, Experimental designs, Completely randomized design, Randomized complete block designs.

Course Title: BAYESIAN THEORY

Text and Reference Material:

  1. Aitkin, M. (2010) Statistical Inference: an Integrated Bayesian/Likelihood Approach Chapman & Hall, London
  2. Albert, J. (2007): Bayesian Computation with R, Springer, New York.
  3. Carlin, B.P. and Th.A. Louis (2000): Bayes and Empirical Bayes Methods for Data Analysis, Chapman & Hall, London
  4. Gelman, A., J.B. Carlin, H.S. Stern, D.B. Rubin (2004): Bayesian Data Analysis, Chapman & Hall, London.
  5. Lee, P.M. (2004): Bayesian Statistics: An Introduction, Halsted Press, New York.

Ntzoufras, I. (2009):

  1. Bayesian Modeling using WinBUGS, Wiley, NY

Contents:

Bayes' theorem: discrete case, Likelihood-based functions, Bayes' theorem: continuous case Conjugate examples: Binomial, Normal, Poisson, and Gamma, data, Exchangeability, Sequential Learning, Likelihood Principle, Conditionality Principle, Sufficiency Principle, Stopping Rule Principle Decision-Theoretic Foundations of Statistical Inference, Decision Rules, Loss Functions, Risks, Bayes Estimators Under Standard Loss Functions, Minimax Rules, Admissable Rules, Unbiasedness Subjective priors, Conjugate priors, Noninformative priors, Empirical Bayes priors, Hierarchical priors, Numerical integration, Asymptotic approximations Simple simulation: inverse transform, rejection and mixture method, Stochastic Simulation: rejection and SIR, Metropolis-Hastings algorithm, Practical Implementation Issues, Markov Chain Theory Derivation of the MH Algorithm, Gibbs sampler Adaptive rejection sampling, Slice sampling, Introduction and WinBUGS handouts, Bayesian Linear Regression, Model Checking, Model Comparison via DIC, Analysis of Variance, Generalized Linear Models Hierarchical Models, State-Space Modelling of Time Series, Multivariate Modelling Using Copulas, Bayes factors, Bayesian p-values, Posterior distribution of the deviance.

Course Title: NONLINEAR DYNAMICS-I

Text and Reference Material:

  1. Nonlinear Dynamics & Chaos by Steven Strogatz
  2. Nonlinear Dynamics by M. Lakshmanan and S. Rajasekar
  3. Nonlinear Systems By Hassan K. Khalil

Contents:

An overview to Chaos, Fractals, and Dynamics, Capsule History of Dynamics, The Importance of Being Nonlinear, A Dynamical View of the World, A Geometric Way of Thinking, Fixed Points and Stability, Population Growth, Linear Stability Analysis, Existence and Uniqueness, Impossibility of Oscillations, Potentials, Solving Equations on the Computer, Introduction, Saddle-Node Bifurcation, Transcritical Bifurcation, Laser Threshold, Pitchfork Bifurcation, Overdamped Bead on a Rotating Hoop, Imperfect Bifurcations and Catastrophes, Insect Outbreak, Introduction, Examples and Definitions, Uniform Oscillator, Nonuniform Oscillator, Over damped Pendulum, Fireflies, Superconducting Josephson Junctions, Classification of Linear Systems, Love Affairs, Phase Portraits, Existence, Uniqueness, and Topological Consequences, Fixed Points and Linearization, Rabbit versus Sheep, Conservative Systems, Reversible Systems, Pendulum, Index Theory, Introduction, Examples, Ruling Out Closed Orbits, Poincare-Bendixson Theorem, Lienard Systems, Relaxation Oscillators, Weakly Nonlinear Oscillators

Course Title: NONLINEAR DYNAMICS-II

Text and Reference Material:

  1. Nonlinear Dynamics & Chaos by Steven Strogatz
  2. Nonlinear Dynamics by M. Lakshmanan and S. Rajasekar
    1. Nonlinear Systems By Hassan K. Khalil

Contents:

Saddle-Node, Transcritical and Pitchfork Bifurcations, Hopf Bifurcations, Oscillating Chemical Reactions, Global Bifurcations of Cycles, Hysteresis in the Driven Pendulum and Josephson Junction, Coupled Oscillators and Quasiperiodicity, Poincare Maps, A Chaotic Waterwheel, Simple Properties of the Lorenz Equations, Chaos on a Strange Attractor, Lorenz Map, Exploring Parameter Space, Using Chaos to Send Secret Messages, Introduction, Fixed Points and Cobwebs, Logistic Map: Numerics, Logistic Map: Analysis, Periodic Windows, Liapunov Exponent, Universality and Experiments, Renormalization, Introduction, Countable and Uncountable Sets, Cantor Set, Dimension of Self-Similar Fractals, Box Dimension, Pointwise and Coorelation Dimensions, The Simplest Examples, Henon Map, Rossler System, Chemical Chaos and Attractor Reconstruction, Forced Double-well Oscillator

Course Title: OPERATIONS RESEARCH

Text and Reference Material:

  1. Operations Research: Applications and Algorithms By Wayne L Winston
  2. Operations Research, 7th edition, (USA: Macmillan Publishing Company), 2003 By Taha, Hamdy

Contents:

Introduction to Operations Research (OR): Introduction to Foundation mathematics and statistics, Linear Programming (LP), LP and allocation of resources, LP definition, Linearity requirement, Maximization Then Minimization problems., Graphical LP Minimization solution, Introduction, Simplex method definition, formulating the, Simplex model, Linear Programming – Simplex Method for Maximizing, Simplex maximizing example for similar limitations, Mixed limitations, Example containing mixed constraints, Minimization example for similar limitations, Sensitivity Analysis: Changes in Objective Function, Changes in RHS, The Transportation Model, Basic Assumptions, Solution Methods: Feasible Solution: The Northwest Method, The Lowest Cost Method, Optimal Solution: The Stepping Stone Method, Modified; Distribution (MODI) Method, The Assignment Model:- Basic Assumptions, Solution Methods:-Different Combinations Method, Short-Cut Method (Hungarian Method), MSPT:- The Dijkestra algorithm, and Floyd’s Algorithm {Shortest Route Algorithm}.

Course Title: NUMERICAL LINEAR ALGEBRA

Text and Reference Material:

  1. Numerical Linear Algebra by Lloyd N. Trefethen
  2. Numerical Linear Algebra by Sundrapandian

Contents:

Matrix-Vector Multiplication, Orthogonal Vectors and Matrices, Norms, The Singular Value Decomposition, Projectors, QR factorization, Gram-Schmidt Orthogonaliztion, MATLAB, Householder Triangularization, Least Square Problems, Conditioning and condition numbers, Floating Point Arithmetic, Stability, Stability of Householder Triangularization, Stability of Back substitution, Condition of Least Square Problems, Gaussian Elimination, Pivoting, Stability of Gaussian Elimination, Cholesky Factorization, Eigenvalues Problems, Overview of Eigenvalues Algorithms, Reduction of Hessen berg or Traditional Form Raleigh Quotient, Inverse Iteration, Overview of Iterative methods, The Arnoldi Iteration, How Arnoldi Locates Eigenvalues, GMRES, The Lanczos Iteration.

Course Title: ACOUSTICS

Text and Reference Material:

  1. Kinsler, L.E., Frey,A R Coppens, A.B., Sanders, J.V.Fundamentals of Acoustics, John Wiley & Sons.
    1. Junger, M.C., Feit  D.Sound Structures and their Interaction.
    2. Morse, P.M., Ingard, K.U., Theoretical Acounts, McGraw-Hill Book Company.
    3. Morse, P.M., Vibration and Sound, McGraw-Hill Book Company.

Contents:

Fundamentals of vibrations, Energy of vibration, damped and free oscillations, transient response of an oscillator. Vibrations of strings, Membrances and plates, Forced vibrations, Normal modes, Acoustic waves equation and its solution, Equation of state, Equation of cont, Euler’s equation, Linearized wave equation, Speed of sound in fluid, Energy density, Acoustic intensity, Specific acoustic impedance, Spherical waves, Transmission; Transmission from one fluid to another (Normal incidence) reflection at a surface of solid (normal and oblique incidence). Absorption and attenuation of sound waves in fluids, Pipes Cavities, Wave guides; Underwater acoustics.

Course Title:     ADVANCE NUMERICAL TECHNIQUES

Text and Reference Material:

  1. Numerical Analysis by Richard L. Burden and J. Douglas Faires
  2. Numerical Methods for computer sciences, Engineering and Mathematics by John H. Mathew
  3. Finite Differences And Numerical Analysis by H. C. Saxena

Contents:

Bisection Method, Secant Method, Iteration Method, Regula False Method, Newton Raphson Method, Gauss Elimination Method, Inversion of a matrix using Gauss Elimination Method, Method of triangularization, Crout’s Method, Gauss Jacobi & Gauss Seidel Method, Relaxation Method, QR-decomposition, Solution of Systems of Non-linear Equations, Divided differences

Newton’s divided difference, Lagrange’s Interpolation formula, Gregory Newton forward and backward Interpolation formula, Gauss forward and backward Interpolation formula, Natural splines, Parabolic Runout spline, Cubic Runout splines, Curve fitting with splines, Newton’s forward & backward differences to compute derivatives, Derivatives using Stirlings formula, Trapezoidal rule, Truncation error, Simpson’s rule, Weddle’s rule, Newton-Cote’s formula, Boole’s rule, Optimization, Power Method, Dominant Eigen values & Eigen vectors, Power series approximations, Solutions by Taylor series, Picard’s Method of successive approximations, Euler’s Method, Improved & Modified Euler Method, Runge Kutta Methods, Predictor corrector Methods, Numerical solutions of Elliptic, Parabolic, Hyperbolic type equations, Crank-Nicholson difference method, Relaxation method to solve differential equation

Course Title:     NUMERICAL SOLUTION OF PDE-I

Text and Reference Material:

  1. Numerical Partial Differential Equations: Finite Difference Methods, by Thomas
  2. Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations by Thomas
  3. Numerical Modeling in Science and Engineering by Myron B. Allen., Ismael Herrera and George F. An

Contents:

Classification of PDEs, canonical forms and well-posed problems, behavior of solutions, characteristics. An introduction to finite difference methods, Basics of Finite Difference Approximations,  Derivation of finite difference approximations,  Consistency,  stability for difference approximations, CFL condition,  The Lax Theorem. Matrix and Fourier stability analysis, Parabolic Equations, Explicit and implicit methods for the heat equation, direction splitting and ADI schemes, steady convection-diffusion equations, Hyperbolic Equations, transient convection-diffusion equation, Finite difference methods for the wave equation and high-order methods. Iterative solution of linear algebraic equations, Thomas algorithm for implicit schemes, and Finite difference in higher space dimensions.

Course Title:     NUMERICAL SOLUTION OF PDE-II

Text and Reference Material:

  1. Finite Difference Methods for Ordinary and Partial Differential Equations by LeVeque SIAM, Philadelphia, 2007
  2. Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations by Thomas
  3. Analysis of Fintie Element Method by Strang G., and Fix G
  4. Finite Element Analysis from Concepts to Applications by David S. Burnett

Contents:

First-order nonlinear equations,  quasi-linear and conservation forms,  Characteristics, shock waves and contact discontinuities, Finite volume methods, Godunov methods and Riemann solvers, high resolution schemes,  Dirichlet and Neumann problems,  solvability,  Direct vs. iterative methods of solution, line by line implementation of thomas algorithm, Relaxation and multigrid methods, Multistep schemes, stability of general multistep schemes, Dispersion and dissipation of numerical schemes,  Group velocity and wavepackets in numerical schemes, Numerical solution of systems of hyperbolic PDEs; multilevel schemes; stability and convergence. Introduction to finite element method, finite element method for eliptic and parabolic equations, Recent development in numerical methods

Course Title:     ELLIPTIC CURVES

Text and Reference Material:

  1. Guide to Elliptic Curve Cryptography by Darrel Hankerson, Alfred Menezes, Scott Vanstone
  2. The Arithmetic of Elliptic Curves by Joseph H. Silverman
  3. Elliptic Curves: Number Theory and Cryptography by Lawrence Washington

Contents:

Cryptography basics, Public-key cryptography, Finite Field Arithmetic, Binary field arithmetic, Elliptic Curve Arithmetic, Introduction to Elliptic Curves, Point representation and the group law, Curves with efficiently computable endomorphism, Point multiplication using halving, Cryptographic Protocols, The elliptic curve discrete logarithm problem, Types of Attacks of Elliptic Curves, Domain parameters, Key pairs, Signature schemes, Public-key encryption, Key establishment

Course Title:     STOCHASTIC PROCESSES

Text and Reference Material:

  1. Stochastic Processes by Sheldon M. Ross
  2. Stochastic Processes and Models by David Stirzaker
  3. Stochastic Processes by J. L. Doob

Contents:

Review of probability and random variables, random walk, Stochastic Processes – definition, methods of description, time averaging and ergodicity, continuity, integration and differentiation, autocorrelation, power spectral density, response of linear systems to stochastic inputs, classes of stochastic processes, Shot noise, thermal noise, point processes, Markov processes, Gaussian processes, Mean square error filtering, orthogonality, smoothing, prediction, stochastic gradient algorithm, innovations, Weiner filter, Kalman filter, queuing theory, Poisson arrivals

Continuous Optimization

Introduction to mathematical optimization, duality (Lagrange and saddle point), Optimality Conditions (KKT-theory), Regularity condition (such as Slater’s conditions), Convex optimization, introduction to semi-definite programming.

Discrete Optimization

The course deals with discrete optimization problems and solution techniques. The topic includes: Shortest path problem, Max flow-min cut problem, traveling salesman, matching, integer optimization, methods for integer optimization (cutting plane methods), introduction to complexity.

Methods in Optimization

Simplex methods, Steepest Descent and Conjugate Gradient Methods, Interior point method for convex optimization, Gradient free methods (Nelder–Mead Simplex Algorithm), search methods (bisection search, particle swarm optimization, and genetic algorithm).

Optimization Modelling with AIMMS

This course is given in an interactive environment  where students develop models for optimization problems in the classroom under the supervision of an instructor. The models are then implemented using AIMMS. Topics are: Optimization modeling basics, tricks for optimization modeling, sensitivity analysis.

Computer Programming and Softwares for Mathematicians

The course has two parts:

  1. Structure and object oriented programming
  2. Introductions to the softwares: Mathematica, Maple, Matlab, and R
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