MA 202 Probability Theory & Stochastic Processes L T P...

MA 101 Mathematics-I L T P C

First Semester ( All Branch ) 3 1 0 8

Differential Calculus :

Successive differentiation, Leibnitz’s theorem & its application. Indeterminate forms,

L. Hospital’s Rules. Rolle’s Theorem, Lagrange’s Mean value theorem, Taylor’s &

Maclaurin’s theorems with Lagrange’s form of remainder for a function of one variable.

Curvature, radius & centre of curvature for Cartesian and polar curves.. Partial

differentiation, change of variables, Euler’s theorem & Jacobian.

Integral Calculus :

Reduction Formulae. Asymptotes for Cartesian and polar curves. Curve tracing. Area & length of

plane curves. Volume & surface area of solids of revolution (for Cartesian and polar curves).

Differential Equation :

Solution of ordinary differential equations of first order & of first degree: Homogeneous

equation, Exact differential equation, Integrating factors, Leibnitz’s linear equation,

Bernoulli’s equation.

Differential equation of first order but of higher degree, Clairaut’s equation. Differential

equations of second & higher order with constant coefficients. Homogeneous Linear

equations.

Reference Books:

1. Differential Calculus Das & Mukherjee U.N. Dhur & Sons Pvt. Ltd.

2. Integral Calculus Das & Mukherjee U.N. Dhur & Sons Pvt. Ltd.

3. Elementary Engineering Mathematics B.S. Grewal Khanna Publisher

4. Engineering Mathematics-II Santi Narayan S. Chand & Co.

MA 102 Mathematics-II L T P C

Second Semester ( All Branch ) 3 1 0 8

Matrices :

Rank of a matrix , Elementary transformations, Consistency and solutions of a system of

linear equations by matrix methods .Eigen values & eigen vectors. Caley-Hamilton’s

theorem & its applications.

Solid geometry :

Straight lines. Shortest distance between skew lines . Sphere , cone, cylinder and conicoid .

Infinite & Fourier Series:

Convergence of infinite series & simple tests of convergence . Fourier series in any interval .Half

range sine & cosine series .

Complex Analysis :

Function of a complex variable, Analytic functions , Cauchy-Reimann equations, Complex line

integral , Cauchy’s theorem , Cauchy’s Integral formula. Singularities and residues, Cauchy’s

Residue theorem and its application to evaluate real integrals.

Differential calculus :

Taylor’s & Maclaurin’s theorems with Lagrange’s form of remainder for a function of two

variables, Expansions of functions of two variables . Errors & approximations. Extreme values of

functions of two & more variables

.

.

Reference Books:

1. Matrices Frank Ayres Mc Graw Hills

2. Solid Geometry Santi Narayan S. Chand & Co.

3. Laplace Transforms M.R.Spiegel Mc Graw Hills

4. Higher Engineering Mathematics B.S. Grewal Khanna Publisher

5. Engineering Mathematics Bali & Iyengar Laxmi Publications Ltd.

6..Advanced Mathematical Analysis Malik & Arrora S. Chand & Co.

MA 201 Mathematics-III L T P C

Third Semester ( All Branch ) 3 1 0 8

Integral & Vector Calculus :Double & triple integrals, Beta & Gamma functions . Differentiation of vector functions of

scalar variables. Gradient of a scalar field , Divergence & Curl of a vector field and their

properties, directional derivatives. Line & surface integrals. Green’s theorem , Stokes’

theorem & Gauss’ theorem both in vector & Cartesian forms ( statement only) with

simple applications.

Integral transforms :

Laplace transform : Transform of elementary functions , Inverse Laplace transforms. Solution of

ordinary differential equation using Laplace transform.

Fourier transforms: Definition, Fourier sine and cosine transforms, properties, relation between

Fourier and Laplace transforms.

Z-transform: Definition, standard z-transforms, properties, initial and final value theorems,

convolution theorem. Inverse z-transform, application to difference equation.

Partial Differential Equation:

Formation of partial differential equations (PDE), Solution of PDE by direct integration.

Lagrange’s linear equation . Non-linear PDE of first order. Method of separation of

variables. Heat, Wave & Laplace’s equations (Two dimensional Polar & Cartesian Co-

ordinates).

Reference Books:

1.Vector Analysis Frank Ayres Mc Graw Hills

2.Advanced Mathematical Analysis Malik & Arrora S. Chand & Co.

3. Advanced Differential Equations M.D.Rai Singhaniya S. Chand & Co.

4. Complex Analysis. M.R.Spiegel. Schuam’s out line Series



MA 202 Probability Theory & Stochastic Processes L T P C Third Semester ( ET & CS Branch ) 3 1 0 8

Probability :

Introduction, joint probability, conditional probability , total probability, Baye’s theorem , multiple events, independent events.

Random Variable:

Introduction, discrete and continuous random variables, distribution function , mass / density function , Binomial, Poisson , Uniform, Exponential, Gaussian and Gamma random variables, conditional distribution and density function, function of a random variable .

Bivariate distributions, joint distribution and density, marginal distribution and density functions, conditional distribution and density, statistical independence, distribution and density of a sum of random variables.

Operation on one Random variable:

Expected value of a random variable , conditional expected value , moments about the origin , central moments , moment generating function, variance , skewness and Kurtosis,covariance, correlation and regression, monotonic and non-monotonic transformation of a random variable (both discrete and continuous).

Stochastic Processes:

Definition of a stochastic process , classification of states, Random walk, Markov chains, poisson process, Wiener process ,stationary and independence, distribution and density functions, statistical independence, Kolmogorov equations, first order stationary processes, second order and wide sense stationary, time averages and ergodicity , correlation functions, covariance function.

Spectral characteristics of random processes:

Power density spectrum and its properties, bandwidth of the power density spectrum, relationship between power spectrum and autocorrelation function, cross power spectral density and its properties.

Noise:

White Noise , shot noise, thermal noise, noise equivalent bandwidth.

Reference Books:

1. An Introduction to Probability Theory and its Applications (Vol. 1 & II)-W Feller (John Wiley & Sons).

2. Probability , Random

Variables & Stochastic processes Papoulis McGraw Hill

3. Probability & Stochastic processes C.W.Helstrom McMillan, New York

for engineers

4. Probability & Random processes A.Leon-Garcia Addison Wisley

for electrical engineers

5. The Theory of Stochastic Processes – D R Cox and H D Miller ( Chapman & Hall Ltd.)


MA 203 Discrete Mathematics L T P C

Fourth Semester ( CS Branch ) 3 1 0 8

Boolean algebra:- Binary relation , equivalence relation , Partial order relations, PO-set, Totally

ordered set, Maximal and Minimal elements. Well ordered set. Lattice, bounded lattices,

sublattice, distributive lattice, modular lattice, irreducible elements, complemented lattice.

Boolean Algebra, Boolean functions & expression , minimization of Boolean functions &

expressions.(Algebraic method and Karnaugh map method)

Logic gates :- Introduction, Design of digital circuits and application of Boolean algebra in

switching circuits.

Graph theory:-. Introduction, Basic definition, incidence and degree, adjacency, paths and cycles, matrix representation of graphs( directed and non-directed). Digraphs. Trees. Mathematical Logic:

Statement Calculus- sentential connectives, Truth tables, Logical equivalence, Deduction theorem.

Predicate Calculus- Symbolizing everyday language., validity and consequence.

Modern Algebra:

Algebraic structures, Semi group, Monoid, Group, Cyclic group, Subgroup, Normal subgroup,

Quotient group, Homomorphism of groups.

Ring, Integral domain, Field. Vector space , Linear dependence & independence . Basis &

Dimension.

Recurrence relations & Generating functions.

Reference Books:

1. Set Theory and Logic R.R Stoll. S. Chand.& Co.

2. Discrete Mathematical Structures G. S. Rao New age International

3.Discrete Mathematics and Structures S. Balgupta Laxmi Publications

4. Modern Algebra Herstein New age International

5. Graph theory Harary Narosa Publishing House

MA 204 Mathematics-IV L T P C Fourth Semester ( CE & ME Branch ) 2 1 0 6

Statistics :

Measures of central tendency, dispersion, moments, skewness & kurtosis.

Probability density function, distribution function, Binomial, Poisson & Normal distributions.

Curve fitting- Method of Least squares, fitting of straight line & parabola.

Correlation & Regression- determination of correlation & regression coefficients &

determination of lines of regression.

Numerical Analysis:

Finite differences, Interpolation & extrapolation. Newton’s forward & backward formulae,

Lagrange’s formula & Newton’s divided difference formula for unequal intervals.(statements &

applications of the formulae only)

Numerical differentiation & integration, Trapezoidal rule, Simpson’s 1/3rd & 3/8th rules.

Numerical solution of transcendental & algebraic equations- Method of Iteration &

Newton-Raphson method.

Solution of system of linear equations :

Gaussian elimination method, Gauss Seidal method, LU decomposition & Cholesky

decomposition.& their application in solving system of linear equations, matrix inversion by

Gauss-Jordan method .

Reference Books : 1. Numerical Mathematical Analysis James B Scarborough Oxford & IBH Publishing

2. Numerical Analysis B.S. Grewal Khanna Publishers

3. Finite Differences H.C. Sexena S. Chand & Co.

& Numerical Analysis

4. Probability & Statistics M.R. Spiegel Mc Graw Hill


MA 101 Mathematics-I L T P C

First Semester ( All Branch ) 3 1 0 8

Differential Calculus :

Successive differentiation, Leibnitz’s theorem & its application. Indeterminate forms,

L. Hospital’s Rules. Rolle’s Theorem, Lagrange’s Mean value theorem, Taylor’s &

Maclaurin’s theorems with Lagrange’s form of remainder for a function of one variable.

Curvature, radius & centre of curvature for Cartesian and polar curves.. Partial

differentiation, change of variables, Euler’s theorem & Jacobian.

Integral Calculus :

Reduction Formulae. Asymptotes for Cartesian and polar curves. Curve tracing. Area & length of

plane curves. Volume & surface area of solids of revolution (for Cartesian and polar curves).

Differential Equation :

Solution of ordinary differential equations of first order & of first degree: Homogeneous

equation, Exact differential equation, Integrating factors, Leibnitz’s linear equation,

Bernoulli’s equation.

Differential equation of first order but of higher degree, Clairaut’s equation. Differential

equations of second & higher order with constant coefficients. Homogeneous Linear

equations.

Reference Books:

1. Differential Calculus Das & Mukherjee U.N. Dhur & Sons Pvt. Ltd.

2. Integral Calculus Das & Mukherjee U.N. Dhur & Sons Pvt. Ltd.

3. Elementary Engineering Mathematics B.S. Grewal Khanna Publisher

4. Engineering Mathematics-II Santi Narayan S. Chand & Co.

MA 102 Mathematics-II L T P C

Second Semester ( All Branch ) 3 1 0 8

Matrices :

Rank of a matrix , Elementary transformations, Consistency and solutions of a system of

linear equations by matrix methods .Eigen values & eigen vectors. Caley-Hamilton’s

theorem & its applications.

Solid geometry :

Straight lines. Shortest distance between skew lines . Sphere , cone, cylinder and conicoid .

Infinite & Fourier Series:

Convergence of infinite series & simple tests of convergence . Fourier series in any interval .Half

range sine & cosine series .

Complex Analysis :

Function of a complex variable, Analytic functions , Cauchy-Reimann equations, Complex line

integral , Cauchy’s theorem , Cauchy’s Integral formula. Singularities and residues, Cauchy’s

Residue theorem and its application to evaluate real integrals.

Differential calculus :

Taylor’s & Maclaurin’s theorems with Lagrange’s form of remainder for a function of two

variables, Expansions of functions of two variables . Errors & approximations. Extreme values of

functions of two & more variables

.

.

Reference Books:

1. Matrices Frank Ayres Mc Graw Hills

2. Solid Geometry Santi Narayan S. Chand & Co.

3. Laplace Transforms M.R.Spiegel Mc Graw Hills



6..Advanced Mathematical Analysis Malik & Arrora S. Chand & Co.

MA 201 Mathematics-III L T P C

Third Semester ( All Branch ) 3 1 0 8

Integral & Vector Calculus :Double & triple integrals, Beta & Gamma functions . Differentiation of vector functions of

scalar variables. Gradient of a scalar field , Divergence & Curl of a vector field and their

properties, directional derivatives. Line & surface integrals. Green’s theorem , Stokes’

theorem & Gauss’ theorem both in vector & Cartesian forms ( statement only) with

simple applications.

Integral transforms :

Laplace transform : Transform of elementary functions , Inverse Laplace transforms. Solution of

ordinary differential equation using Laplace transform.

Fourier transforms: Definition, Fourier sine and cosine transforms, properties, relation between

Fourier and Laplace transforms.

Z-transform: Definition, standard z-transforms, properties, initial and final value theorems,

convolution theorem. Inverse z-transform, application to difference equation.

Partial Differential Equation:

Formation of partial differential equations (PDE), Solution of PDE by direct integration.

Lagrange’s linear equation . Non-linear PDE of first order. Method of separation of

variables. Heat, Wave & Laplace’s equations (Two dimensional Polar & Cartesian Co-

ordinates).

Reference Books:

1.Vector Analysis Frank Ayres Mc Graw Hills

2.Advanced Mathematical Analysis Malik & Arrora S. Chand & Co.

3. Advanced Differential Equations M.D.Rai Singhaniya S. Chand & Co.

4. Complex Analysis. M.R.Spiegel. Schuam’s out line Series



MA 202 Probability Theory & Stochastic Processes L T P C Third Semester ( ET & CS Branch ) 3 1 0 8

Probability :

Introduction, joint probability, conditional probability , total probability, Baye’s theorem , multiple events, independent events.

Random Variable:

Introduction, discrete and continuous random variables, distribution function , mass / density function , Binomial, Poisson , Uniform, Exponential, Gaussian and Gamma random variables, conditional distribution and density function, function of a random variable .

Bivariate distributions, joint distribution and density, marginal distribution and density functions, conditional distribution and density, statistical independence, distribution and density of a sum of random variables.

Operation on one Random variable:

Expected value of a random variable , conditional expected value , moments about the origin , central moments , moment generating function, variance , skewness and Kurtosis,covariance, correlation and regression, monotonic and non-monotonic transformation of a random variable (both discrete and continuous).

Stochastic Processes:

Definition of a stochastic process , classification of states, Random walk, Markov chains, poisson process, Wiener process ,stationary and independence, distribution and density functions, statistical independence, Kolmogorov equations, first order stationary processes, second order and wide sense stationary, time averages and ergodicity , correlation functions, covariance function.

Spectral characteristics of random processes:

Power density spectrum and its properties, bandwidth of the power density spectrum, relationship between power spectrum and autocorrelation function, cross power spectral density and its properties.

Noise:

White Noise , shot noise, thermal noise, noise equivalent bandwidth.

Reference Books:


2. Probability , Random

Variables & Stochastic processes Papoulis McGraw Hill

3. Probability & Stochastic processes C.W.Helstrom McMillan, New York

for engineers

4. Probability & Random processes A.Leon-Garcia Addison Wisley

for electrical engineers

5. The Theory of Stochastic Processes – D R Cox and H D Miller ( Chapman & Hall Ltd.)


MA 203 Discrete Mathematics L T P C

Fourth Semester ( CS Branch ) 3 1 0 8

Boolean algebra:- Binary relation , equivalence relation , Partial order relations, PO-set, Totally

ordered set, Maximal and Minimal elements. Well ordered set. Lattice, bounded lattices,

sublattice, distributive lattice, modular lattice, irreducible elements, complemented lattice.

Boolean Algebra, Boolean functions & expression , minimization of Boolean functions &

expressions.(Algebraic method and Karnaugh map method)

Logic gates :- Introduction, Design of digital circuits and application of Boolean algebra in

switching circuits.

Graph theory:-. Introduction, Basic definition, incidence and degree, adjacency, paths and cycles, matrix representation of graphs( directed and non-directed). Digraphs. Trees. Mathematical Logic:

Statement Calculus- sentential connectives, Truth tables, Logical equivalence, Deduction theorem.

Predicate Calculus- Symbolizing everyday language., validity and consequence.

Modern Algebra:

Algebraic structures, Semi group, Monoid, Group, Cyclic group, Subgroup, Normal subgroup,

Quotient group, Homomorphism of groups.

Ring, Integral domain, Field. Vector space , Linear dependence & independence . Basis &

Dimension.

Recurrence relations & Generating functions.

Reference Books:

1. Set Theory and Logic R.R Stoll. S. Chand.& Co.

2. Discrete Mathematical Structures G. S. Rao New age International

3.Discrete Mathematics and Structures S. Balgupta Laxmi Publications

4. Modern Algebra Herstein New age International

5. Graph theory Harary Narosa Publishing House

MA 204 Mathematics-IV L T P C Fourth Semester ( CE & ME Branch ) 2 1 0 6

Statistics :

Measures of central tendency, dispersion, moments, skewness & kurtosis.

Probability density function, distribution function, Binomial, Poisson & Normal distributions.

Curve fitting- Method of Least squares, fitting of straight line & parabola.

Correlation & Regression- determination of correlation & regression coefficients &

determination of lines of regression.

Numerical Analysis:

Finite differences, Interpolation & extrapolation. Newton’s forward & backward formulae,

Lagrange’s formula & Newton’s divided difference formula for unequal intervals.(statements &

applications of the formulae only)

Numerical differentiation & integration, Trapezoidal rule, Simpson’s 1/3rd & 3/8th rules.

Numerical solution of transcendental & algebraic equations- Method of Iteration &

Newton-Raphson method.

Solution of system of linear equations :

Gaussian elimination method, Gauss Seidal method, LU decomposition & Cholesky

decomposition.& their application in solving system of linear equations, matrix inversion by

Gauss-Jordan method .

Reference Books : 1. Numerical Mathematical Analysis James B Scarborough Oxford & IBH Publishing



& Numerical Analysis

4. Probability & Statistics M.R. Spiegel Mc Graw Hill


MA 301 Numerical Methods & L T P C Computations

Fifth Semester ( ET & CS Branch ) 2 1 0 6

Numerical Analysis:

Finite differences, Interpolation & extrapolation. Newton’s forward & backward formulae, Lagrange’s formula & Newton’s divided difference formula for unequal intervals.(statements & applications of the formulae only), evaluation of functions , minimization & maximization of functions .

Numerical differentiation & integration, Newton’s general quadrature formula, Trapezoidal rule, Simpson’s 1/3rd & 3/8th rules.

Numerical solution of transcendental & algebraic equations:- Method of Iteration & Newton-Raphson method.

Numerical Solution of a system of linear equations :

Gaussian elimination method with pivoting strategies , Gauss-Jordan method & Gauss-Seidel

method. LU decomposition & Cholesky decomposition.& their application in solving system of

linear equations. Matrix inversion by Gauss-Jordan method .

Numerical solution of ordinary differential equations with initial value:

Taylor’s series method , Eulers & modified Eulers method , Runge-Kutta method of 4th order.

Reference Books :

1. Numerical Mathematical Analysis James B Scarborough Oxford & IBH Publishing



& Numerical Analysis.


MA 441 Modern Algebra L T P C

Eighth Semester (Elective –III, Open ) 3 0 0 6

Posets & Lattices :

Partial order relations, Po-set, Lattices & Boolean algebra.

Groups :

Groups , Subgroups , Normal subgroups , Permutation group . Lagrange’s Theorem . Cyclic groups Quotient group , Homomorphism of groups , First three isomorphism theorems . Inner Automorphism . Normalizer / Centralizer of an element , Centre of a group . Conjugacy relation , Class equation , Sylow’s Theorems. Subnormal & Normal series , Solvable group , Commutators . Nilpotent groups .Free groups.

Rings :

Ring , Integral domain , Field . Ideals & Quotient rings , Homomorphism of Rings , Maximal Ideal , Minimal Ideal , Prime Ideal , Principal Ideal , Principal Ideal Ring / Domain (PIR / PID) , Euclidean Domain , Polynomial Rings. Field of quotient of an integral domain. Field extensions.

1. Modern Algebra Surjit singh & Zameeruddin Vikas Publishing House

2. Modern Algebra I.N. Herstein New age International

3. Modern Algebra Khanna & Bhamri Vikas Publishing House

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MA 442 Functional Analysis L T P C Eighth Semester (Elective –III, Open) 3 0 0 6

Matric Space :

Definition and Examples of metric space . Open Sphere, Open Set & Closed Set. Convergence of sequences, Cauchy sequence, Complete Metric Spaces, Sequentially Compact Metric Space, Continuous mappings.

Topological Space :

Definition and examples, Trivial and non-trivial topology, Cofinite topology, Usual Topology with special reference to R. Continuity and homeomorphism.

Functional Analysis :

Linear space, subspace, basis, dimension, normed linear space, Banach space, continuous linear transformation, Conjugate space, Inner product spaces, Hillbest space, Orthogenality, orthonormal sets, Cauchy’s Schwartz’s inequality, Bessel’s in equality.

Linear operators, Self adjoint operator, normal and unitary operators, Projections, Spectrum of an operator. The spectral theorem.

******************************

Reference Books:

1. Introduction to Topology and

Modern Analysis

Simmon G.F. Tata McGraw Hill

2. Functional Analysis B.K. Lahiri World Press Pvt. Ltd.

3. General Topology Lipschutz Schaum Outline Series, McGraw

Hill Book Company.

MA 443 Mathematical Modeling L T P C Eighth Semester (Elective –III, Open ) 3 0 0 6

Mathematical modelling techniques, classification with simple illustration.

Mathematical modelling through ordinary differential equations.

Modelling through difference equations.

Modelling through partial differential equations.

Modelling through integral and differential - difference equations.

Modelling through calculus of variations and dynamic programming.

Modelling through mathematical programming, maximum principle and maximum entropy

principle.

***********************************************

Reference Books:

1. Mathematical Modelling J.N. Kapur New age International

2. Advanced Engineering Mathematics

E. Kreyszig New age International

3. Higher Engineering Mathematics B.S. Grewal Khanna Publishers

4. Operations Research, Methods

and Practice

5. Numerical Methods for Engineering Problems

C.K. Mustafi

N.K. Raju & K.U. Muthu

Wiley Eastern

Macmillan India Limited

MA 444 Operation Research L T P C

Eighth Semester (Elective –III, Open ) 3 0 0 6

Introduction to Operation Research (O.R):

Meaning of O.R. Principles of Modelling. Features and Phases of O.R.

Linear Programming:

Introduction, Formulation of Linear Programming Problems (L.P.P), Graphical solution procedure. Idea of Convex set & convex combination of two points, Fundamental Theorem of L.P.P. (proof not required ). Solutions of L.P.P. Simplex Method . Big-M methods.

Transportation Problems(T.P):

Introduction. Mathematical formulation. Definitions of Balanced, Unbalanced T. P. Rules to find initial Basic feasible Solution (B.F.S) of a T.P.- North West Corner Rule, Vogel’s approximation Method. Solution algorithm of T.P. Solution technique for unbalanced

T. P. Resolution of degeneracy. Examples.

Assignment Problems(A.P):

Introduction , Mathematical Formulation. Reduction theorem ( proof not required). Definitions of Balanced and Unbalanced A.P. Hungarian Algorithm for solving A.P. Solution technique for unbalanced A.P. Examples.

Sequencing Problems:

Introduction. Definition. Solution of Sequencing problems. Processing n jobs through 2 machines, 2 jobs through m machines ( Graphical method), Processing n jobs through m machines.

Integer Programming Problems(I.P.P):

Introduction. Pure and mixed integer programming problems. Gomory’s Cutting Plane technique for solving I.P.P. Examples.

Reference Book:

1.Operations Research Kanti Swarup Sultan Chand & Sons

2.Operations Research S.D. Sharma Khanna Publishers

3.Operations Research J.K. Sharma MacMillan India Ltd

4.Operations Research Hira and Gupta Sultan Chand & Sons

5.Operations Research Hamady A Taha Prentice Hall of India

6. Linear Programming P.M.Karak New Central Book Agency

MA 202 Probability Theory & Stochastic Processes L T P...

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