COURSE STRUCTURE
Pre-requisite for the Course : Mathematics Courses of the B.A./B.Sc. first two years/Engineering Mathematics Course of B.E. as taught in Indian Universities.
Part – III (Duration : One Year)
MIM 301 Any One of the following :
(a) Probability and Measure
(b) Functional Analysis
MIM 302 Software Engineering
MIM 303 Compiler Techniques and Computer Networks
MIM 304 Operation Research
MIM 305 Data Base Systems and Artificial Intelligence
MIM 306 Practicals related to Papers MIM 303 (Computer Networks), MIM 304 and MIM 305
MIM 307 One Project work related to the Courses in all the three years as given above or Real Industrial
Problems/ or One of the following Papers :
(i) Performance modeling of communication networks
(ii) Modeling, Simulation and Monte Carlo Methods.
(iii) Computational Biology.
(iv) Mathematics of Finance and Insurance.
(v) Computational Fluid Dynamics
(vi) Chemo metrics and Quality Control in Industry
(vii) Industrial Processes
Practicals related to the above papers (vi) and (vii)
(viii) Wavelets
(ix) Biomechanics
(x) Fuzzy Sets and Their Applications
(xi) Any other topic according to the availability of Subject expert.
Additional Recommended References
Avner Friedman, Mathematics in Industrial Problems (All Volumes of the series), Springer-Verlag, New York Inc.
DETAILS OF SYLLABUS
(a)
Probability and Measure
Class of sets. Measures and Probability Spaces.
Binomial Random Variables-Poisson Theorem. Interchangeable Events. Bernoulli. Borel theorems. Central Limit
Theorem for Binomial Random Variables. Large Deviations.
Independence-Borel-Cantelli Lemma. Kolomogorov’s Zero-One Law. Convergence in Probability. Almost Sure Convergence. Their Equivalence for Sum of Independent Random variables.
Integration-Definition. Monotone Convergence. Dominated Convergence. Indefinite Integrals.
Mean Convergence. Jensen, Holder and Schwartz Inequalities.
Sums of Independent Random Variables-Three Series Theorem. Laws of Large Numbers. Stopping Times. Elementary Renewal Theorem. Optimal Stopping.
Measure Extensions-Lebesgue - Stieltjes Measure. Fubini’s Theorem. Radon-Nikodym Theorem. Kolomogorov Consistency Theorem.
Conditional Expectation. Conditional Independence. Introduction to Martingales.
Distribution and Characteristic Functions.
Central Limit Theorems.
Limit Theorems for Independent Random Variables-Laws of Large Numbers. Law of the Iterated Logarithm. Dominated Ergodic Theorems. Maxima of Random Walks.
Recommended Text
1. Y.S.Chow and H. Teicher, Probability Theory–Independence, Interchangeability, Martingales. Springer /
Narosa,1979.
References :
1. K.L.Chung, Elementary Probability Theory with Stochastic Processes, Springer International Student Edition,
Narosa, 1975.
(b) FUNCTIONAL ANALYSIS
Unit-1 Normed linear space: definition and examples, continuous linear transformations, spaces BL(X,Y), BL(X) and BL(X,X), lP & LP (for 09p9µ) Branch spaces, Riesz-Fischer lemma, Minkowski’s and Holder’s inequalities, the Hahn-Banach theorem and its applications, open mapping theorem.
Unit-2 Bounded linear maps on banach spaces, uniform bounded principle, closed graph theorem, the natural imbedding of N in N**, dual spaces of lP & LP (for l<p<µ) bounded inverse theorem, two norm theorem, operators, conjugate of an operator, transpose of an operator.
Unit-3 Inner product space and its examples, Hilbert space, orthogonal complements, orthonormal set, Gram-Schmidt orthonormalization, Bessel’s inequality, projection theorem, Riesz representation theorem. Conjugate space H*, ad joint of an operator, self-ad joint operators, normal and unitary operators, projections, determinants and spectral theorem.
Banach algebra: definition and examples, regular and singular elements of Banach algebra.
Unit-4 Topological divisors of zeros, the spectrum, spectrum of self adjoint, normal and unitary operators, the formula for the spectral radius, the radical and semi-simplicity, the Gelfand mapping, applications of formula of the spectral radius, involutions in Banach algebra, the Gelfand-Neumark theorem.
Note : The course is roughly covered by the following books:
(1) G.F. Simmons : Introduction to Topology and Modern Analyses
(2) B.V. Limaye : Functional Analysis, 2nd Edition, New Age
International Limited, Publishers, 2nd Edition.
Reference Books :-
(1) S.K. Berberain : Lectures in Functional Analysis and Operator Theory, Spring Verlag.
(2) Goff man and George Padre : First Course in Functional Analysis, Prentice Hall of India.
(3) Martin Schechter : Principles of Functional Analysis (Student
Edition) Academic Press, New York.
Introduction-Evolution & Scope of Software.
Software Product & Process.
Software Process Models-Linear Sequential Model. Prototyping Model. Evolutionary Model.
Requirements Analysis-Specifications. Algebraic Axioms. Regular Expressions. Decision Tables.
Event Tables. Transition Table.
Software Design-Architectural & Detailed Design-Abstraction, Information hiding, modularity, Concurrency,
Coupling, cohesion, Data Flow Diagrams, Structure Charts.
Object Oriented Design-Object-Oriented Design Concepts & Methods. Class and Object Definition. Refining Operation. Program Components and Interfaces. A notation for OOD.
Implementation Issues-Structured Coding. Recursion. Documentation Guidelines.
Software Testing Methods & Strategies-Structural, Functional, unit, integration, validation, system
testing.
Software Metrics-Quality Metrics and Metrics for analysis, design, coding, testing & maintenance.
Software Reliability & Software Quality Assurance.
References :
1. Roger S. Pressman-Software Engineering: A practitioner’s approach, fifth Edition, McGraw-Hill Ian
2. Sommerville-Sofware Engineering: Fifth Edition, Addison Wesley.
Compiler Techniques
Overview of the Compiling Process. Some Typical Compiler Structures.
Lexical Analysis (Scanner)-Regular Expression. Finite Automata. Specification and Recognition to Tokens. Simple Approaches to Lexical Analyzer Design.
Syntax Analysis (Parser)-Syntax trees, ambiguity. Context tree grammar & derivation of parse trees. Basic parsing techniques, derivation Top-Down and Bottom-Up Parsing. Operator-precedence Parsing. LR Parsers. Syntax Directed Definition. Translation schemes. L-attributed & S-attributed Definition.
Symbol Table Organization-Data Structures for Symbol Table (ST). Design of a ST. ST for Block Structured Language.
Run-Time Memory Allocation-Storage Allocation Strategies. Static Dynamic & Heap Memory Allocation. Memory Allocation in Block Structured Languages, in recursion. Memory allocation in Fortan.
Compilation Process and Code Optimization-Compilation of Expressions. Control Structures and I/O Statements.
Error Detection and Recovery. Issues in Optimization. Optimizing Transformations. Local and Global
Optimization. Loop optimization.
Computer Networks
Elements of data communication-Concepts and terminology, analog and digital data transmission, signals, attenuation, delay distortion, noise, channel capacity, transmission media, data encoding, asynchronous and synchronous transmission, multiplexing.
Networking-Communication network, comparison of circuit switching, message switching and packet switching, radio and satellite networks, local network topology, star topology, ring topology, bus/tree topology, medium access control protocols. LAN protocol performance.
Network Architecture and Distributed Processing-OSI reference model, layered and hierarchical approaches, network interface, principles or inter-networking.
TCP/IP reference model. Internet protocols and standards. Network services, virtual terminal protocol, and
file transfer protocol, electronic mail, Distributed processing-Definition, logical aspect of application design, case studies.
References
:
1. J.Tremblay & P.G. Sorenson, The Theory & Practice of Compiler Writing,
McGraw Hill.
2. D.M. Dhamdhere, Compiler Construction-Principles & Practice, Macmillan & Co.
3. W.A. Barrett el al., Compiler Construction Theory & Practice, Galgotia.
4. A.V. Aho, R.Sethi & J.D.Ullman, Compilers-Principles, Techniques & Tools, Addison-Wesley
5. A.S. Tanenbaum, Computer Networks, Prentice Hall of India.
6. W. Stallings, Data and Computer Communication, McMillan & Co.
7. J.Martin, Computer Network and Distributed Data Processing, Prentice Hall.
8. W.Stallings, Local Networks, McMillan & Co.
Unit-1 Introduction : Nature and scope of operations research(OR) and its impact.
Linear Programming : Linear Programming (LP) Model, assumptions of LP, graphical linear programming solution, slack, surplus and unrestricted variables. Principles of Simplex Method , Algebra of simplex method, simplex algorithm, artificial starting solution, M-method and two-phase method, special cases the simplex method application-degeneracy, unbounded, and infeasible solutions.
Unit-2 Duality and Sensitivity Analysis : The essence of duality theory, primal-dual relationships, the role of duality theory in sensitivity analysis, special types of L. P. problems: Transportation Problem : Simplex method for transportation
problem. The transshipment problem, The Assignment Problem : Network analysis including PERT-CPM The terminology of network, the shortest path problem, the minimum spanning tree problem, maximum flow problem,
minimum cost flow problem, project planning and control with PERT-CPM.
Unit-3 Dynamic Programming : Characteristic of dynamic programming, deterministic dynamic programming, forward and backward recursion technique, selected dynamic programming of applications, Game Theory : Decision making under risk expected value criterion, decision under uncertainty, simple game and games with mixed strategies, optimized solutions of two-person zero – sum games, solving games by L. P. Integer Programming: solution algorithm.
Branch and bound technique, branch and bound Algorithm for binary (zero – one) linear programming, bound and scan algorithm for mixed integer linear programming, cutting plane algorithm.
Unit-4
Non Linear Programming : Classical optimization theory, unconstrained problems with necessary and sufficient conditions constrained problems with equality and inequality constraints, Kuhn Tucker conditions constrained algorithms-separable, quadratic, convex, geometric programming.
Basic Probability : Sample space, events algebra of events, measure theoretic approach to probabilities, probability space, Definition of a discrete and continuous random variable on a probability-space, conditional probability and independent events, expectation of a random variable and moments, distribution function of a random variable, special study of binomial, Poisson, geometric, normal exponential and grammar distributions and their properties.
Morkov Chains: Stochastic process and its classification, Morkov chains, Chapman-Kolmogorov’s equation, classification of states of Morkov chains, Longterm properties of Morkov chains, absorption states and simple applications, continuous time Morkov chains.
Queuing Theory : Basic structure of queuing models, examples of queuing systems drawn from real life situations, role of exponential distribution. Birth and death process and queuing models based on birth and death process.
Note : The syllabus is roughly covered by the chapter 1,2,3,4,5,6,7 and 10,11,12,13,14,15,16 of the book entitled “Operations Research” by Fredrick, S. Hillier and Gerudal J. Hiberman. (Fifth edition) McGrwaw-Hill Publishing Co.
Reference Books :
1. “Introduction to Operation Research” by H.A. Taha, PHI, 1995.
2. “Principles of Operation Research” by H.M. Wanger, PHI, 1996.
3. “Linear Programming” by G. Hadley, Narosa, 1990.
4. “Optimization Methods in O.R. and System Analysis” by Mittal & Mollen, New Age, 1996.
5. “Non Linear and dynamic programming” by Sharma, S.D. Kedarnath, 1993
Database Systems
Data abstraction, database system architecture. Schemas and Sub schemas, data independence. Physical Data Organization. Hashed, Index file, B-tree.
Data Models-Data modeling using entity relationship.
Hierarchical and Network Model-DBTG proposals, data manipulation language.
Relational Model-Relational Algebra and Calculus. Storage organization for relations. Functional. Multivalued and Project-Join Dependencies. Decomposition.
Normal Forms-First. Second. Third. BCNF. Fourth and P J normal forms. Relational Query language. Query processing. Query optimization-General strategies of optimization. Optimizing Algebraic Expression. Systematic Query optimization using cost estimate. Security in DBMS to gain integrity mechanism of roll back and recovery validation and data translation of database operation and management. Case study of Oracle or Ingress.
Artificial Intelligence
Definition and Introduction to AI & AI Techniques. Problem Definition-Problem Definition as a State space search. Production System. Control Strategies. Problem characteristics.
Problem-Solving Methods-Forward & Backward Reasoning. Matching Indexing. Search Techniques. Depth-First and Breadth-First Search Techniques. Adding Heuristics. Hill-Climbing Search Techniques. Problem Reduction.
Constraint Satisfaction. Game Playing.
Knowledge Representation-Knowledge Representation in Predicate and Propositional Logic. Resolution in Predicate and Propositional Logic. Deduction and Theorem Proving. Question Answering. Structured Representation of knowledge. Declarative Representation. Semantic Networks. Conceptual Dependencies. Frames and Scripts.
Procedural Representation.
Implementing A.I. System-A.I. Language & their Important Characteristics. Overview of LISP and PROLOG.
Computer A.I. Applications. Lisp Machines and Parallel Machines.
Introduction to Expert Systems.
References :
1. J.D. Ullman, Principles of Database System, Galgotia
2. C.J. Date, An Introduction to Database System, Vols. I & II, Addison-Wesley.
3. P.H. Winston, Artificial Intelligence, Addison-Wesley.
4. N.J. Nilsson, Principles of Artificial Intelligence , Springer-Verlag.
5. E-Rich, Artificial Intelligence, McGraw Hill.
6. N.C. Rowe, Artificial through PROLOG, Prentice-Hall of India.
7. A. Bonnet, Artificial Intelligence : Promise and Performance, Prentice Hall.
(i) Performance modeling of computer communication networks.
Essential of stochastic process-Markov chains. Markov process. Reversibility. Renewal theory.
Queuing Systems-(case studies involving queuing theory).
Simple queuing systems(Poisson process, M/M/I queue, extended models such as Erlang B and C, Transcient effects, M/G/I queue)
Network of Queues (Product form solutions, recursive non-product form solutions, negative customer queuing models).
Numerical solutions of Models (Convolution and mean value analysis algorithms with worked out examples, PANACEA, Norton’s equivalent theorem for queuing networks, simulation).
Stochastic petri networks (The model, SPN’s with and without product form solution).
Discrete time queuing systems (arrival process, Geom./Geom/m and Geom/Geom/1 discrete queues, three industrial case studies involving discrete time queuing system).
Network traffic models (Continuous and discrete time models of recent interest, burstiness, self-similar traffic, solution techniques).
Performance analysis of multiple access protocols. Performance issues in mobile communication.
References
1. Thomas Robertazzi, Computer Networks and Systems: Queuing theory and Performance Evaluation,
Springer-Verlag, 2000.
2. B.R.Haverkort, Performance of computer communication systems (A model based approach), Wiley, 1998.
(ii) Modeling, Simulation and Monte Carlo Methods
System awareness-System concepts. System-Continuous/discrete, stochastic / deterministic, open / closed. System methodology – modeling. Advantages and disadvantages of simulation, simulation terminology.
Generation of random numbers and their applications-Pseudo Random numbers. Linear congruential method.
Inverse-transform method. Generation of non-standard uniform numbers. Generation of Normal random numbers (including Box-Muller approach). Binomially, exponentially. Poisson distributed random numbers. Rejection method.
Composite Method. Monte Carlo (MC) integration. Hit or Miss MC method. Error analysis. Sample mean MC method. Efficiency of MC method. Integration in the presence of noise. Variance reduction
techniques-importance sampling. Antithetic Variates.
Monte Carlo Solution of Differential Equations-Gambler’s ruin. Solution of simple Differential Equations. Solutions of the Fokker - Planck Equation. Dirilchlet Problem. Solution of General Elliptic Differential equation.
Discrete System Simulation and Model Validation-Simulation terminology. Time management methods. Object generation. Events and event Synchronization. Queue Management and List Processing. Collection and recording simulation data. Evaluation of the Simulation Model. Validation Description. Sampling Methods.
Design of Simulation experiment and output analysis-Validation of simulator. Completely randomized design.
Randomized complete block design. Factorial design. Estimation of model parameters-Optimization of response surface. Heuristic search. Random search. Analysis of simulation results. Estimation of confidence limits. Variance reduction.
Language for discrete system simulation-Language characteristics. Use of multipurpose language. Example :
Simulation of queuing system. Simulation languages-GPSS, Special-purpose languages-SIMSCRIPT II.5, GASP IV.
References
J.R. Thompson, Simulation-A Modeler’s Approach, Wiley, 2000. 2. Udo W Pooch and James A Wall, Discrete event simulation (A practical Approach), CRC Press, 1993.
R.Y. Rubinstein, Simulation and the Monte Carlo Method, John Wiley & Sons, 1981.
A.M. Law and W.D. Kelton, Simulation modeling and analysis, McGraw Hill Intl. Ed. (second Edition) 1991.
J.J. Schreiber, An introduction to simulation using GPSS/H (Both 5.25 and 3.5 inch disk included), John Wiley,1991.
(iii) Computational Biology
Basic concepts of molecular biology. DNA and Proteins. The Central Dogma. Gene and Genome Sequences.
Restriction Maps-Graphs, Interval graphs. Measuring Fragment sizes.
Algorithms for double digest problem (DDP)-Algorithms and complexity, approaches to DDP Integer Programming.
Partition problems. Traveling Salesman Problem (TSP) simulated annealing.
Sequence Assembly-Sequencing strategies. Assembly in practices fragment overlap statistics, fragment alignment, sequence accuracy.
Sequence comparisons Methods-Local and global alignment. Dynamic programming method. Multiple sequence alignment.
Probability and Statistics for sequence alignment and sequence patterns-Hidden Markov models for biological sequences.
References
1. M.S. Waterman, Introduction to Computational Biology, Chapman & Hall (1995).
2. A. Baxevanis and B Ouellette, Bioinformatics, A Practical Guide to the analysis of Genes and Proteins, Wiley
Interscience (1998).
(iv) Mathematics of Finance and Insurance.
Prerequisite knowledge of Application of Mathematics to Finance as taught in an optional paper at Graduate level – UGC curriculum, 2002
Financial Derivatives-An Introduction; Types of Financial Derivatives-Forwards and Futures; Options and its kinds; and SWAPS.
The Arbitrage Theorem and Introduction to Portfolio Section and Capital Market Theory : Static and Continuous-Time Model.
Pricing by Arbitrage-A Single-Period option Pricing Model; Multi-Period Pricing Model-Cox-Ross-Rubin stein Model; Bounds on Option Prices.
The Ito’s Lemma and the Ito’s Integral.
The Dynamics of Derivative Prices-Stochastic Differential Equations (SDEs)-Major Models of SDEs: Linear Constant Coefficient SDEs; Geometric SDEs; Square Root Process; Mean Reverting Process and Omstein – Uhlenbeck Process.
Martingale Measures and Risk-Neutral Probabilities: Pricing of Binomial Options with equivalent Martingale Measures.
The Black-Scholes Option Pricing Model-using no arbitrage approach, limiting case of Binomial Option Pricing and risk-Neutral Probabilities.
The American Option Pricing-Extended Trading Strategies; Analysis of American Put Option; early Exercise premium and relation to free boundary problems.
Concepts from Insurance-Introduction; The Claim Number Process; The Claim Size Process; Solvability of the Portfolio; Reinsurance and Ruin Problem.
Premium and Ordering of Risks-Premium Calculation Principles and Ordering Distributions.
Distribution of Aggregate Claim Amount-Individual and Collective Model; Compound Distributions; Claim Number of Distributions; Recursive Computation Methods; Lundberg Bounds and Approximation.
Risk Processes-Time-Dependent Risk Models; Poisson Arrival Processes; Ruin Probabilities and Bounds Asymptotics and Approximation by Compound Distributions.
Time Dependent Risk Model-Ruin Problems and Computations of Rui Functions; Dual Queuing Model; Risk Models in Continuous Time and Numerical Evaluation of Ruin Functions.
References
John C. Hull, Options, Futures, and Other Derivatives, Prentice-Hall of India Private Limited.
Sheldon M. Ross, An Introduction to Mathematical Finance, Cambridge University Press.
Salih N. Neftci, An Introduction to the Mathematics of Financial Derivatives, Academic Press, Inc.
Robert J. Elliott and P. Ekkehard Kopp, Mathematics of Financial Markets, Springer-Verlag, New York Inc.
Robert C. Merton, Continuous-Time Finance, Basil Blackwell Inc.
C.D. Daykin, T. Pentikainen and M. Pesonen, Practical Risk Theory for Actuaries, Chapman & Hill.
Tomasz Rolski, Hanspter Schmidil, Volker Schmidt and Jozef Tewugels, Stochastic Processes for Insurance and Finance, John Wiley & Sons Limited.
(v) Computational Fluid Dynamics
Prerequisites-Numerical Analysis. Computer Programming. Partial Differential Equations.
Basic equations of Fluid Dynamics.
Analytic Aspects of PDE.
Finite Volume and Finite Difference Methods on Nonuniform Grids.
Stationary Convection-Diffusion Equation (Finite Volume Discretization, Schemes of Positive Type, Upwind Discretization).
Nonstationary Convection-Diffusion Equation-Stability.
Discrete Maximum Principle.
Incompressible Navier-Stockes(NS) Equations-Boundary Conditions. Spatial and Temporal Discretizations on
Collocated and on Staggered Grids.
Iterative Methods-Stationary Methods. Krylov Subspace Methods. Multigrid Methods. Fast Poison Solvers. Iterative Methods for Incompressible NS Equations.
Shallow water Equations-One and Two Dimensional Cases.
Scalar Conservation Laws-Godunov’s Order Barrier Theorem, Linear Schemes.
Euler Equation in One Space Dimension-Analytic Aspects, Approximate Riemann Solver. Ocher Scheme. Flux Splitting Schemes. Stability. James-Schmidt-Turkel Scheme. Higher Order Schemes.
Discretization in General Domains-Boundary Fitted Grids. Equations of Motion in General Coordinates.
Numerical Solution of Euler Equation in General Coordinates.
Numerical Solution of NS Equations in General Domains.
Unified Methods for Computing Compressible and Incompressible Flow.
Recommended Tex
1. Wesseling, P., Principles of Computational Fluid Dynamics, Spriger-Velag, 2000
References
1. Wendt. J.F., Anderson, J.D., Degrez, G. and Dick, E., Computational Fluid Dynamics: An Introduction, An
introduction, Springer-Verlag, 1996.
2. Anderson, J.D. Computational Fluid Dynamics: The Basics with Applications, McGraw Hill, 1995.
Note : This is a rapidly emerging area and books are being published at a very fast rate. Visit the website
www.cfd-online.com. for an up-to-date list.
(vi) Chemo metrics and Quality Control in Industry
Introduction. Validation of analytical data. Various methods of validation of data. Statistical parameters (accuracy and
precision). Confidence interval. Significant numbers. Error evaluations. Preparation of quality control charts. Various tests of significance. Material testing by comparison of mean with true values. Comparison of two means and other relevant significant tests; determination of significance of methods of analysis, equipment, and materials using various tests under quality control and forensic programmes.
Regression analysis (least-squares methods) standard addition calibration method, normal error curve (Gaussian distribution) and its mathematical equation general treatment of equilibria (acid-base, complexations, redox) in Nater, calculation of pH and knowledge of good lab practice (GLP) and other requirements of generate data for qualitative and quantitative production from industry.
References
1. G.D. Christain, Analytical Chemistry 5th edition (1994), John Wiley & Sons, New York.
2. M.A. Sharat and D.L. IIIuran, Chemometrics, John Wiley, New York.
3. R. Canlcutt and R.Roddy, Statistics for Analytical Chemists, Chapman and Hall, New York.
(vii) Industrial Processes
Process Control. Automation in microanalysis. Autoanalysers. Chemical microscopy. Good manufacturing process (GMP). Development of standard technical procedure (STP). Mathematical and statistical methods for process optimization applicable in manufacturing industries. Modeling and parameter estimation. Resolution for the purpose of separation & purification of industrial materials. Factor analysis and its application. Signal processing. Pattern recognition. Structural property relationship.
Knowledge of elementary electronics in industry. Simple integrated circuit. Power supply. Transformer. Operational amplifiers. Detectors. Transducers. Rectifiers, signal to noise ratio. Electronic components used in instrumentation.
Principles of analytical separations. Resolution (fundamental equation). Distillation (fractional and molecular). Planar Chromatography and high performance liquid chromatography and methods of crystallization emphasizing mathematical background involved. Pollution modeling and pollutant distributions in industrial wastes and effluents.
References
1. H.A. Strobel, Chemical Instrumentation: a systematic approach, 2nd Edition (1973), Addition Wesley,
Reading, Mass.
2. E.W. Berg, Chemical Methods of Separations, 1st Edition (1963), McGraw Hill, New York.
Practical Related to the above papers (vi) & (vii)
Determination of accuracy. Precision. Mean deviation. Standard deviation. Coefficient of variation. Normal error Curve and Least square fitting of certain sets of data obtained from industrial quality control laboratory. Use of analytical instruments (HPLC, HPTLC, flourimeter, polarized microscope etc.) and organization of results in terms of optimization, resolution, modeling, Signal processing and pattern recognitions. Validation of methods. Data and computer modeling. Comparison of two sets of results in term of significance by students’ t-test and F-test.
Preparations of materials. Solutions and buffers for bench work in an industry. Tests for purity of industrial chemicals.
(viii) Wavelets
Preliminaries.
Different ways of constructing wavelets-Orthonormal bases generated by a single function; the Balian-Low theorem.
Smooth projections on L2(R). Local sine and cosine bases and the construction of some wavelets. The unitary folding operators and the smooth projections. Multiresolution analysis and construction of wavelets. Construction of compactly supported wavelets and estimates for its smoothness. Band limited wavelets. Orthonormality. Completeness. Characterization of Lemarie-Meyer wavelets and some other characterizations. Franklin wavelets and Spline wavelets on the real line. Orthonormal bases of piecewise linear continuous functions for L2(T). Orthonormal bases of periodic splines. Periodization of wavelets defined on the real line.
Characterizations in the theory of wavelets – the basic equations and some of its applications. Characterizations of MRA wavelets, low-pass filters and scaling functions. Non existence of smooth wavelets in H2 (R).
Frames – the reconstruction formula and the Balian-Low theorem for frames. Frames from translations and dilations.
Smooth frames for H2R).
Discrete transforms and algorithms – the discrete and the fast Fourier transforms. The discrete and the fast cosine transforms. The discrete version of the local sine and cosine bases.
Recommended Text
1. Eugenio Hernandez and Guido Weiss. A First Course on Wavelets, CRC Press, New York, 1996.
References
1. C.K. Chui, An Introduction to Wavelets, Academic Press, 1992.
2. I.Daubechies, Ten Lectures on Wavelets, CBS-NSF Regional Conferences in Applied Mathematics, 61,
SIAM, 1992.
3. Y. Mayer, Wavelets, algorithms and applications (translated by R.D. Rayan, SIAM, 1993).
4. M.V. Wickerhauser, Adapted wavelet analysis from theory to software, Wellesley, MA, A.K. Peters, 1994.
(ix) Biomechanics
Prerequisite: Fluid Mechanics
Newton’s equations of motion. Mathematical modeling. Continuum approach. Segmental Movement and
Vibrations.
External Flow: Fluid Dynamic Forces Acting on Moving Bodies.
Flying and Swimming.
Blood Flow in Heart, Lung, Arteries, and Veins.
Micro-and Macro circulation.
Respiratory Gas Flow.
The Laws of Thermodynamics. Molecular Diffusion, Mechanisms in Membrances, and Multiphasic Structure.
Mass Transport in Capillaries, Tissues, Interstitial Space, Lymphatics, Indicator Dilution Method, and Peristalsis.
Description of Internal Deformation and Forces.
Stress, Strain, and Stability of Organs.
Strength, Trauma, and Tolerance.
Biomechanical Aspects of Growth. Engineering of Blood Vessels. Tissue Engineering of Skin.
Recommended Text
1. Y.C. Fung, Biomechanics, Springer-Verlag, New York Inc., 1990.
(x) Fuzzy Sets and Their Applications
Fuzzy Sets – Basic definitions. a-level sets. Convex fuzzy sets, Basic operations on fuzzy sets. Types of fuzzy sets.
Cartesian products. Algebraic products. Bounded sum and difference. t – conorms.
The Extension Principle – The Zadeh’s extension principle. Image and inverse image of fuzzy sets. Fuzzy numbers.
Elements of fuzzy arithmetic.
Fuzzy Relations and Fuzzy Graphs – Fuzzy relations on fuzzy sets. Composition of fuzzy relations. Min-Max composition and its properties. Fuzzy equivalence I relations. Fuzzy compatibility relations. Fuzzy relation equations.
Fuzzy graphs. Similarity relation.
Possibility Theory – Fuzzy measures. Evidence theory. Necessity measure. Possibility measure. Possibility distribution.
Possibility theory and Ifulzzy sets. Possibility theory versus probability theory.
Fuzzy Logic – An overview of classical logic, Multivalued Ilogics. Fuzzy propositions. Fuzzy quantifiers. Linguistic variables and hedges. Inference from conditional fuzzy propositions, the compositional rule of inference.
Approximate Reasoning – An overview of fuzzy expert system. Fuzzy implications and their selection. Multiconditional approximate reasoning. The role of fuzzy relation equation.
An Introduction to Fuzzy Control – Fuzzy controllers. Fuzzy rule base. Fuzzy inference engine. Fuzzification.
Defuzzification and the various defuzzification methods (the center of area, the center of maxima, and the mean of maxima methods).
Decision Making in Fuzzy Environment – Individual decision making. Multiperson decision making. Multicriteria decision making. Multistage decision making. Fuzzy ranking methods. Fuzzy linear programming.
References
1. H.J. Zimmermann, Fuzzy set theory and its Applications, Allied Publishers Ltd., New Delhi, 1991.
2. G.J. Klir and B. Yuan – Fuzzy sets and fuzzy logic, Prentice-Hall of India, New Delhi. 1995. Top