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MA 101

MATHEMATICS - I

I year B.Tech. I Semester

(common for all branches)

(to be effective for the batches admitted from 2006 onwards)

 

 

L/T : 4 Hours                                                                                                         Credits : 4

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Matrix Theory:  Elementary row and column operations on a matrix, Rank of matrix – Normal form – Inverse of a matrix using elementary operations –Consistency  and solutions of  systems of linear equations using elementary operations, Gauss Seidal iteration method - linear  dependence and independence of vectors  -  Characteristic roots  and  vectors  of  a  matrix  -  Caley-Hamillton  theorem and its applications, Calculation of dominant  eigen value by iteration - Reduction to diagonal form - Reduction of a quadratic form to canonical form – orthogonal transformation and congruent transformation.                                          (16)

 

Differential  Calculus:   Rolle’s  theorem;  Mean  value  theorem; Taylor’s and Maclaurin’s theorems with remainders, Expansions;  Indeterminate forms; Asymptotes and curvature; Curve tracing; Functions of several variables, Partial Differentiation,   Total Differentiation, Euler’s theorem and generalization,    maxima  and  minima  of functions of several variables (two and three variables) – Lagrange’s method of Multipliers; Change of variables – Jacobians.                                                                                                                              (20)

 

Ordinary  differential  equations of first order:   Formation  of differential equations; Separable equations; equations  reducible to  separable form; exact equations; integrating factors;  linear first   order   equations;   Bernoulli’s   equation;    Orthogonal trajectories.

(8)

 

Ordinary linear differential equations of higher order : Homogeneous linear  equations of arbitrary order with  constant  coefficients - Non-homogeneous  linear  equations  with  constant  coefficients; Euler and Cauchy’s equations; Method of variation of parameters; System of linear differential equations.                                                                            (12)

 

 

Recommended Text Book:

 

R.K.Jain and S.R.K.Iyengar :  Advanced Engineering Mathematics,                                                                                  Narosa Publishing House, 2002.

(Chapters 1,2,3,4,5)

Reference Books:

  1. Erwyn Kreyszig : Advanced Engineering Mathematics,

John Wiley and Sons, 8th Edition.

2.      B.S.Grewal  : Higher Engineering Mathematics, Khanna Publications, 2002.


MA 151

MATHEMATICS - II

I year B.Tech. II Semester

(common for all branches)

(to be effective for the batches admitted from 2006 onwards)

 

 

L/T : 4 Hours                                                                                                         Credits : 4

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Laplace  Transformation:    Laplace transform -  Inverse  Laplace transform - properties of Laplace transforms - Laplace transforms of  unit step function, impulse function and periodic function  - convolution theorem - Solution of ordinary differential equations with constant  coefficients  and  system  of  linear  differential equations with constant coefficients using Laplace transform.                                                                                        (16)

 

Integral Calculus:  Fundamental theorem of integral calculus and mean value theorems; Evaluation of plane areas, volume and surface area of a solid of revolution and lengths. Convergence of Improper integrals – Beta and Gamma integrals – Elementary properties – Differentiation  under integral sign. Double and triple integrals – computation of surface areas and volumes – change of variables in double and triple integrals.                      (20)

 

Vector    Calculus :     Scalar   and   Vector    fields;    Vector Differentiation;  Level surfaces - directional  derivative - Gradient  of   scalar field;  Divergence  and Curl  of a vector field - Laplacian -  Line  and  surface integrals;  Green’s theorem in plane; Gauss  Divergence  theorem; Stokes’ theorem.                                                                                                       (20)

 

 

 

Recommended Text Book:

 

R.K.Jain and S.R.K.Iyengar :  Advanced Engineering Mathematics,                                                                                  Narosa Publishing House, 2002

(Chapters 1,2,8,15)

 

Reference Books:

 

  1. Erwyn Kreyszig : Advanced Engineering Mathematics,

John Wiley and Sons, 8th Edition.

 

2.   B.S.Grewal  : Higher Engineering Mathematics, Khanna Publications, 2002.


MA 201

MATHEMATICS - III

II year B.Tech. I Semester

(common for all branches except ECE)

(to be effective for the batches admitted from 2006 onwards)

 

 

L/T : 4 Hours                                                                                                         Credits : 4

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Fourier  Series:   Expansion of a  function in Fourier series for a given range - Half range sine and  cosine expansions                                                                                                   (10)

 

Fourier Transforms : Complex  form  of  Fourier  series - Fourier transformation  -  sine  and  cosine  transformations  - simple illustrations.                                                                 (6)

 

Z-transforms :  Inverse Z-transfroms – Properties – Initial and final value theorems – convolution theorem -  Difference equations – solution of difference equations using z-transforms                                                                                                                            (10)

 

Partial  Differential  Equations:   Solutions  of  Wave  equation, Heat  equation  and Laplace’s   equation   by  the   method   of separation  of variables and their use in problems  of  vibrating string,  one dimensional unsteady heat flow and two  dimensional steady state heat flow  including polar form.                                                                                   (10)

 

Complex  Variables:  Analytic function - Cauchy  Riemann  equations  - Harmonic functions -   Conjugate functions - complex  integration  -  line integrals in complex plane - Cauchy’s theorem (simple proof only), Cauchy’s  integral  formula - Taylor’s  and  Laurent’s  series expansions - zeros and singularities - Residues - residue theorem, evaluation of real integrals using residue theorem, Bilinear transformations, conformal mapping.              (20)

 

Recommended Text Books:

 

1.  R.K.Jain and S.R.K.Iyengar :  Advanced Engineering Mathematics,                                                                             Narosa Publishing House, 2002

2.  Erwyn Kreyszig : Advanced Engineering Mathematics,

John Wiley and Sons, 8th Edition.

Reference Book:

 

B.S.Grewal  : Higher Engineering Mathematics, Khanna Publications, 2003.


MA 202

SIGNAL TRANSFORMATION TECHNIQUES

II year B.Tech. I Semester

(for ECE)

(to be effective for the batches admitted from 2006 onwards)

 

 

L/T : 4 Hours                                                                                                         Credits : 4

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Complex  Variables:  Analytic function - Cauchy  Riemann  equations  - Harmonic functions -   Conjugate functions - complex  integration  -  line integrals in complex plane - Cauchy’s theorem (simple proof only), Cauchy’s  integral  formula - Taylor’s  and  Laurent’s  series expansions - zeros and singularities - Residues - residue theorem, evaluation of real integrals using residue theorem, Bilinear transformations, conformal mapping.              (16)

 

Fourier  Series:   Expansion of a  function in Fourier series for a given range - Half range sine and  cosine expansions                                                                                                     (8)

 

Fourier Transforms : Complex  form  of  Fourier  series - Fourier transformation  -  sine  and  cosine  transformations  - simple illustrations.                                                                 (8)

 

Z-transforms :  Inverse Z-transfroms – Properties – Initial and final value theorems – convolution theorem -  Difference equations – solution of difference equations using z-transforms                                                                                                                              (8)

 

Linear Transformations:

Linear tranformation, range, domain of a linear transformation, rank and nullity, rank nullity theorem, Silvestor’s theorem, Anhilating space, matrix form of a linear transformation, inverse of a transformation.                                                                          (8)

 

Vector Spaces

Vector space, sub-spaces, properties, span of a set, linear dependence and independence of vectors, basis, changing a vector in a basis, orthogonal bases, Schmidt’s orthogonalisation process, Direct sum of vector spaces.                                                                                                     (8)

 

 

Recommended Text Books:

1.  R.K.Jain and S.R.K.Iyengar :  Advanced Engineering Mathematics,                                                                             Narosa Publishing House, 2002

2. V.Krishnamoorthy et.al. : An introduction to linear algebra, Affiliate East West Press.

 

Reference Book:

1.        Erwyn Kreyszig : Advanced Engineering Mathematics,

John Wiley and Sons, 8th Edition.


CS 201

DISCRETE MATHEMATICS

II year B.Tech. - I semester (Computer Science)

 

L/T : 4 Hours                                                                                                         Credits : 4

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Fundamentals

Sets, relations and functions, fundamentals of logic, logical inferences, first order logic, quantified propositions, mathematical induction.

 

Elementary Combinatorics

Combinations and permutations, Enumeration - with repetitions, with constrained repetitions.

 

Recurrence Relations

Generating functions, coefficients of generating functions, recurrence relations, inhomogeneous recurrence relations.

 

Relations and digraphs

Relations and digraphs, binary relations, equivalence relations, ordering relations, lattices, paths and closures, directed graphs, adjacency matrices.

 

Graphs

Graphs, Isomorphism, trees, spanning trees, binary trees, planar graphs, Euler circuits, Hamiltonian graphs, Chromatic numbers, Four-colour problem.

 

 

Text Book:

J.L.Mott, A.Kandel and T.P.Baker : Discrete Mathematics for Computer Scientists,

Second Edition, Reston, 1986.


 

MA 251

MATHEMATICS - IV

II year B.Tech. II Semester

(common for all branches except ECE)

(to be effective for the batches admitted from 2006 onwards)

L/T : 4 Hours                                                                                                        Credits : 4

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Statistics and Probability:  Probability laws – Addition and Multiplication theorems on probability - Baye’s theorem –Expectation, Moments and Moment generating function of Discrete and continuous distributions, Binomial, Poisson and Normal distributions, fitting these distributions to the given data, Testing of Hypothesis -  Z-test for single mean and difference of means, single proportion and difference of proportions - t-test for single mean and difference of means, F-test for comparison of variances,. Chi-square test for goodness of fit. – Correlation, regression.                                                                                                          (23)

 

Numerical Analysis:

Curve fitting by the method of least  squares.  Fitting of (i) Straight line (ii) Second  degree parabola (iii) Exponential curves.

Lagrange interpolation, Forward, backward and   central   differences, Newton’s forward  and backward  interpolation  formulae,  Gauss’s forward and backward interpolation formulae, Numerical  differentiation  at  the tabulated points with forward backward and central differences. Numerical Integration with Trapezoidal rule, Simpson’s 1/3  rule, Simpson’s 3/8 rule and Romberg integration. Taylor series method, Euler’s  method, modified  Euler’s  method, Runge-Kutta method of 2nd & 4th orders for  solving first  order   ordinary   differential equations, Numerical solution of algebraic and  transcendental  equations by  Regula-Falsi  method  Newton-Raphson’s method.                                        (23)

 

 

Series Solution : Classification of singularities of an ordinary  differential equation -  Series  solution-  Method  of   Frobenius  - indicial equation  - examples

Bessel  and Legendre functions:   Bessel function of  first  kind Recurrence formulae Generating function Orthogonality of Bessel functions Legendre polynomial rodrigue’s formula  Generating function   Recurrence   formula   Orthogonality   of   Legendre

polynomials.                                                                                                                     (10)

 

Recommended Text Books:

1.    S.C.Gupta and V.K.Kapoor : Fundamentals of Mathematical Statistics.

  1. M.K. Jain S.R.K. Iyengar and R.K.Jain:  Numerical methods for                                                                                      Scientific and Engineering Computation
  2. Erwyn Kreyszig : Advanced Engineering Mathematics,

 

Reference Book:

B.S.Grewal  : Higher Engineering Mathematics, Khanna Publications

 

 

 

 

 

MA 252

NUMERICAL TECHNIQUES AND GRAPH THEORY

II year B.Tech. II Semester

(for ECE)

(to be effective for the batches admitted from 2006 onwards)

L/T : 4 Hours                                                                                                         Credits : 4

--------------------------------------------------------------------------------------------------------------------

 

Numerical Analysis:

Curve fitting by the method of least  squares.  Fitting of (i) Straight line (ii) Second  degree parabola (iii) Exponential curves.

Lagrange interpolation, Forward, backward and   central   differences, Newton’s forward  and backward  interpolation  formulae,  Gauss’s forward and backward interpolation formulae, Numerical  differentiation  at  the tabulated points with forward backward and central differences. Numerical Integration with Trapezoidal rule, Simpson’s 1/3  rule, Simpson’s 3/8 rule and Romberg integration. Taylor series method, Euler’s  method, modified  Euler’s  method, Runge-Kutta method of 2nd & 4th orders for  solving first  order   ordinary   differential equations, Numerical solution of algebraic and  transcendental  equations by  Regula-Falsi  method  Newton-Raphson’s method.                                        (23)

 

Graph Theory: Graphs and planar graphs : Basic terminology, multigraphs and weighted graphs, paths and circuits, shortest paths in weighted graphs, Eulerian paths and circuits, Hamiltonian paths and circuits. Colourable graphs, Chromatic numbers, Five colour theorem and Four colour problem. Trees and cut-sets : trees, rooted trees, path lengths in rooted trees, spanning trees and BFS & DFS algorithms, minimum spanning trees and Prims & Kruskal’s algorithms.                                                                                                   (23)

 

Introduction to Queuing Theory:

Poisson process and exponential distribution. Poisson queues - Model (M/M/1):(¥/FIFO) and its characteristics.                                                                                                                  (10)

 

 

Recommended Text Books:

 

1        M.K. Jain S.R.K. Iyengar and R.K.Jain:  Numerical methods for Scientific and

Engineering Computation,Wiley Eastern

  1. Mott, Kandel and Baker : Discrete Mathematics for Computer Scientists.

McGrawHill

3.   Kanti Swarup, Man Mohan & P.K.Gupta: Introduction to Operations Research

S.Chand and & Sons

 

 

 

 

 

 

 

 

 

MA 451/461

OPERATIONS RESEARCH

(Global elective)

(to be effective for the batches admitted from 2006 onwards)

 

L/T : 3 Hours                                                                                                         Credits : 3

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LINEAR PROGRAMMING :

Formulation and graphical solution of LPP’s. The general LPP, slack, surplus and artificial variables.  Reduction of a LPP to the standard form. Simplex computational procedure, Big-M method, Two-phase method. Solution in case of unrestricted variables. Dual linear programming problem. Solution of the primal problem from the solution of the dual problems.

 

TRANSPORTATION PROBLEMS :

Balanced and unbalanced Transportation problems. Initial basic feasible solution using N-W corner rule, row minimum method, column minimum, least cost entry method and Vogel’s approximation method. Optimal solutions. Degenracy in Transportation problems.

 

QUEUEING THEORY :

Poisson process and exponential distribution. Poisson queues - Model (M/M/1):(¥/FIFO) and its characteristics.

 

ELEMENTS OF INVENTORY CONTROL :

Economic lot size problems - Fundamental problems of EOQ. The problem of EOQ with finite rate of replenishment. Problems of EOQ with shortages - production instantaneous, replenishment of the inventory with finite rate. Stochastic problems with uniform demand (discrete case only).

 

Recommended Text Book :

Introduction to Operations Research by Kanti Swarup, Man Mohan & P.K.Gupta

(Pub:  Sultan Chand & Sons)

 

Reference Books :

1.  J.C.Pant  :  Introduction to Operatins Research,   (Jain Brothers, New Delhi)

2.  N.S.Kambo :  Mathematical Programming Techniques (EWP)

(Affiliated East West Press Pvt. Ltd., New Delhi)

 


MA 462

NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS

(Global Elective)

(to be effective for the batches admitted from 2006 onwards)

 

L/T : 3 Hours                                                                                                         Credits : 3

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Ordinary Differential Equations : Multistep (explicit and implicit) methods for initial value problems, Stability and Convergence analysis,  Linear  nonlinear boundary value problems,  Quazilinearization, Shooting methods

 

Finite difference methods :   Finite difference approximations for derivatives, boundary value problems with explicit boundary conditions, implicit boundary conditions, error analysis, stability analysis, convergence analysis.

 

Partial Differential Equations:

Classification of partial differential equations, finite difference approximations for partial derivatives and finite difference schemes for :

 

Parabolic equations :  Schmidt’s two level, multilevel explicit methods, Crank-Nicolson’s two level, multilevel implicit methods, Dirichlet’s  problem, Neumann problem, mixed boundary value problem.

 

Hyperbolic Equations :  Explicit methods, implicit methods, one space dimension, two space dimensions, ADI methods.

 

Elliptic equations :  Laplace equation, Poisson equation, iterative schemes, Dirichlet’s problem, Neumann problem, mixed boundary value problem, ADI methods.

 

 

Recommended Text Books :

1. M.K.Jain  :   Numerical Solution of Differential Equations, Wiley Eastern, New Delhi

2. G.D.Smith :   Numerical Solution of Partial Differential Equations, Oxford Uni. Press

 

 


 

OPTIMIZATION TECHNIQUES

(Global Elective offered for III B.Tech.)

(MA 452)

 

L-T-P : 3-0-0

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Linear Programming :

Introduction and formulation of models, Convexity, simplex method, Big-M method, two-phase method, degeneracy, non-existent and unbounded solutions, duality in LPP, dual simplex method, sensitivity analysis, revised simplex method, transportation and assignment problems, traveling salesman problem.

 

Nonlinear Programming :

Classical optimization methods, equality and inequality constraints, Lagrange multipliers and Kuhn-Tucker conditions, quadratic forms, quadratic programming problem and Beale’s method.

 

Dynamic Programming :

Principle of optimality, recursive relations, solution of LPP

 

Integer Linear Programming :

Gomory’s cutting plane method, Branch and bound algorithm, Knapsack problem, linear 0-1 problem.

 

 

Reference Books :

 

1.      J.C. Pant : Introduction to Optimization,  Jain Brothers, 2006

2.      S.S. Rao :Optimization Theory and applications,  Wiley Eastern Ltd., 2004

3.      K.V.Mittal : Optimization Methods,  Wiley Eastern Ltd., 2003

4.      H.A.Taha : Operations Research, PHI, 2006

 

 

 


FUZZY LOGIC AND FUZZY SYSTEMS

(Global Elective offered for III B.Tech.)

(MA 453)

 

L-T-P : 3-0-0

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Crisp set theory (CST):

Introduction, Relations between sets, Operations on sets, Characteristic functions, Cartesian products of crisp sets, crisp relations on sets.

 

Fuzzy set theory (FST) :

Introduction, concept of fuzzy set (FS), Relation between FS, operations on FS, properties of standard operations, certain numbers associated with a FS, certain crisp sets associated with FS, Certain FS associated with given FS, Extension principle.

 

Propositional Logic (PL1) :

Introduction, Syntax of PL1, Semantics of PL1, certain properties satisfied by connectives, inference rules, Derivation, Resolution.

 

Predicate Logic (PL2) :

Introduction, Syntax of PL2, Semantics of PL2, certain properties satisfied by connectives and quantifiers, inference rules, Derivation, Resolution

 

Fuzzy Relations (FR):

Introduction, Operations on FR, a-cuts of FR, Composition of FR, Projections of FR, Cylindric extensions, Cylindric closure, FR on a domain.

 

Fuzzy Logic (FL):

Introduction, Three-valued logics, N-valued logics and infinite valued logics, Fuzzy logics, Fuzzy propositions and their interpretations in terms of fuzzy sets, Fuzzy rules and their interpretations in terms of FR, fuzzy inference, More on fuzzy inference, Generalizations of FL

 

Switching functions (SF) and Switching circuits (SC):

Introduction, SF, Disjunctive normal form, SC, Relation between SF and SC, Equivalence and simplification of circuits, Introduction of Boolean Algebra BA, Identification, Complete Disjunctive normal form.

 

APPLICATIONS :

Introduction to fuzzy logic controller (FLC), Fuzzy expert systems, classical control theory versus fuzzy control, examples, working of FLC through examples, Details of FLC, Mathematical formulation of FLC, Introduction of fuzzy methods in decision making.

 

Recommended Text Book :

M. Ganesh : Introduction to Fuzzy Sets and Fuzzy Logic, PHI, 2001.

Reference Books:

1. G.J. Klir and B.Yuan: Fuzzy sets and Fuzzy Logic–Theory and Applications, PHI, ‘97.

2. T.J.Ross : Fuzzy Logic with Engineering Applications, McGraw-Hill, 1995.