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Anna University
Affliated - Anna University
Other Autonomous
- M1 – Matrix & Calculus
- M2 – Numerical Method & Probability
- M3 – Transform Partial Differential & Equation
- Numerical Methods
- Discrete Mathematics
- Transforms & Computational Techniques
- Random Process Variable
- Engineering Mathematics
- Circuit Analysis
- Electro Magnetic Fields
- Control Systems
- Digital Signal Processing
- Statistical Modeling
- Probability & Statistics
- Elements of Operations Research
- Calculus
- Algebra
- Statistics & Probability
- Differential Equation
- Partial Differential Equation
- Probability & Queuing Theory
- Probability & Stochastic Processes
- Probability & Random Variables
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Engineering Core Papers
Circuit Theory
- Concept of Network and Circuit
- Types of elements
- Types of sources
- Source transformation.
- R-L-C Parameters
- Voltage-Current relationship for Passive Elements (for different input signals -Square, Ramp, Saw tooth and Triangular)
- Kirchhoff's Laws
- Network Reduction Techniques-Resistive networks
- Inductive networks and capacitive networks- Series, Parallel, Series-Parallel combinations
- Star-to-Delta and Delta-to-Star Transformation
- Mesh Analysis and Super mesh
- Nodal Analysis and Super node for DC Excitation
- Network topology-Definitions
- Graph
- Tree
- Basic Cut set and Basic Tie set Matrices for Planar Networks
- Average value, R.M.S. value, form factor and peak factor for different periodic wave forms
- J-notation, Complex and Polar forms of representation
- Steady State Analysis of series R-L-C circuits
- Concept of Reactance, Impedance, Susceptance, Admittance, Phase and Phase difference
- Concept of Power Factor, Real, Reactive and Complex power
- Thevenin's
- Norton's
- Maximum Power Transfer
- Superposition
- Reciprocity
- Tellegen's
- Substitution
- Compensation and Milliman's theorems
- Faraday's laws of electromagnetic induction
- concept of self and mutual inductance
- dot convention
- coefficient of coupling
- composite magnetic circuit
- analysis of series and parallel magnetic circuits.
Control system
- Open and closed loop control system Transfer function of a system Need for mathematical modelling
- Representation of mechanical translational and rotational systems using differential equation and determination
of transfer function - Conversions of Mechanical system to Electrical system f-V and f- I electrical analogies
- Block diagram reduction rules and methodology
- Evaluation of transfer function using block diagram
reduction - Signal flowgraphs and evaluation of transfer function
- Block diagram to signal flow conversion
- Standard test signals and their expression
- Type number and order of a system
- Transfer function of First order system for Step, ramp, Impulse and parabolic signal
- General transfer function of second order
system for different damping factor based on step response - Time domain specifications and their significance
- Transient and Steady state error analysis
- Static and dynamic Error coefficients
- Analytical design for PD, PI and PID control systems
- Poles and zeros of a system - Pole zero plot and concept of s plane
- Concept of stability
- Significance of Routh Hurwitz Technique with different cases
- Root locus techniques
- Root locus plot of typical systems
- Design of Compensator using root locus
- Lead Compensation
- Frequency domain specifications
- Bode plot approach and stability analysis
- Rules for sketching Bode plot of typical systems
- Design of Compensator using Bode Plot
- Lag Compensation
- Polar plot and
significance - Sketching the Polar plot of typical systems
- Nyquist stability criterion
- State variable representation-Conversion of state variable models to transfer function
- Conversion of transfer functions to state variable models
- Solution of state equations
- Concepts of Controllability and
Observability - Fuzzy logic
- based control system
- Adaptive Controller
Electro Magnetic Field / Theory
- Introduction to electrostatics
- rectangular co-ordinate
- Cylindrical & Spherical Co-ordinate
- Review of vector calculus
- Coulomb’s Law and field intensity
- Problem based on coulomb’s law
- Electric field due to
continuous charge distribution - Concept
- Derivation of E due Infinite Line charge
- Energy density in electrostatic field- Problem discussion.
- Biot savart law
- Magnetic field intensity due to Infinite line charge
- H- due finite and semi finite line charge
- Ampere’s circuital law and application: Infinite
line current - Infinite Sheet current
- Infinitely long coaxial Transmission line
- Problem based on ACL
- Introduction to EM waves
- Waves in general
- Plane wave in lossless dielectric
- Plane wave in free space
- Plane wave in good conductor
- Problems based on plane waves in lossless, free space and good conductor rectangular
waveguide - rectangular waveguide
- Problems
- Transmission line parameters
- Transmission line equivalent circuit
- Explanation
- Transmission line equation derivation
- Problem discussion.
- Transmission line characteristics: lossless Line
- Distortion lessline
EMI/EMC- Types of EMI/EMC - SE, CE- Susceptibility
- Introduction to impedance matching
- Smith chart Introduction
- Reflection coefficient, Standing wave ratio Input impedance calculation in smith chart
- Practice problems.
- Single stub matching Introduction
- Procedure
for single stub matching - Problems solving in smith chart
Mathematics (M-Series) Papers
ENGINEERING MATHEMATICS – M1
Calculus & Linear Algebra - M1
- Representation of functions
- Limit of a function
- Continuity
- Derivatives
- Differentiation rules
- Maxima and Minima of functions of one variable
- Partial differentiation
- Homogeneous functions and Euler’s theorem
- Total derivative
- Change
of variables - Jacobians
- Partial differentiation of implicit functions
- Taylor’s series for functions
of two variables - Maxima and minima of functions of two variables
- Lagrange’s method of
undetermined multipliers
- Definite and Indefinite integrals
- Substitution rule
- Techniques of Integration
- Integration by
parts - Trigonometric integrals
- Trigonometric substitutions
- Integration of rational functions by
partial fraction - Integration of irrational functions
- Improper integrals
- Double integrals
- Change of order of integration
- Double integrals in polar coordinates
- Area
enclosed by plane curves - Triple integrals
- Volume of solids
- Change of variables in double
and triple integrals
- Higher order linear differential equations with constant coefficients
- Method of variation of
parameters - Homogenous equation of Euler’s and Legendre’s type
- System of simultaneous
linear differential equations with constant coefficients - Method of undetermined coefficients
ENGINEERING MATHEMATICS - M2
Advanced Calculus & Complex Analysis - M2
- Eigenvalues and Eigenvectors of a real matrix
- Characteristic equation
- Properties of Eigenvalues and Eigenvectors
- Cayley-Hamilton theorem
- Diagonalization of matrices
- Reduction of a quadratic form to canonical form by orthogonal transformation
- Nature of quadratic forms
- Gradient and directional derivative
- Divergence and curl
- Vector identities
- Irrotational and Solenoidal vector fields
- Line integral over a plane curve
- Surface integral
- Area of a curved surface
- Volume integral - Green's, Gauss divergence and Stoke's theorems
- Verification and application in evaluating line, surface and volume integrals
- Analytic functions
- Necessary and sufficient conditions for analyticity in Cartesian and polar coordomates
- Properties
- Harmonic Conjugates
- Construction of analytic function
- Conformal
Mapping - Mapping by functions w = Z+C, CZ,-1/Z,Z^2
- Bilinear Transformation
- Line integral
- Cauchy's integral theorem
- Cauchy's integral formula
- Taylor's and Laurent's series
- Singularities
- Residues
- Residue Theorem
- Application of residue theorem for evaluation of real integrals
- Use of circular contour & semicircular contour
- Existence conditions
- Transforms of elementary functions
- Transform of unit step function and unit impulse function
- Basic properties
- Shifting theorems
- Transforms of derivatives and integrals
- Initial and final value theorems
- Inverse transforms
- Convolution theorem
- Transform of periodic functions
- Application to solution of linear second order ordinary differential equations with constant coefficients
ENGINEERING MATHEMATICS - M3
Transforms & Boundary Value Problems - M3
- Formation of partial differential equations by eliminating arbitrary constants & arbitrary functions
- Solutions of standard types of first order partial differential equations
- Lagrange’s linear equation
- Linear partial
differential equations of second and higher order with constant coefficients of homogeneous types
- Dirichlet’s conditions
- General Fourier series
- Odd and even functions
- Half range sine and cosine series
- Parseval’s identity
- Harmonic Analysis
- Classification of second order partial differential equations
- Method of separation of variables
- Solutions of one dimensional wave equation - One dimensional equation of heat conduction (Insulated edges excluded)
- Steady state condition with zero boundary
- Steady state condition with non-zero boundary conditions
- Fourier transform pair
- Properties - Fourier sine and cosine transforms
- Properties– Transforms of simple functions
- Convolution theorem (without proof)
- Parseval’s identity.
- Z - transforms
- Properties of Z transforms
- Inverse Z transforms
- Convolution theorem (without Proof)
- Solution of linear difference equations with constant coefficients using Z-transform
Other Mathematics Papers
Numerical Methods
- Solution of nonlinear equations
- False position method
- Fixed point iteration method
- Newton Raphson method - Solution of linear system of equations: Gaussian elimination method
- Gauss Jacobi method
Gauss Seidel method- Eigenvalues of a matrix by power method
- Curve fitting
- Method of least squares
- Interpolation: Newton’s forward and backward difference
- Divided differences
- Newton’s divided difference
- Lagrange’s interpolation
- Inverse interpolation
- Numerical differentiation by using Newton’s forward, backward and divided differences
- Numerical integration by trapezoidal and Simpson’s 1/3rd and 3/8th rules
- Single step methods: Taylor’s series method, Euler and Improved Euler methods, fourth order Runge
- Kutta method
- Multistep methods: Milne’s predictor
- corrector method
- Finite difference techniques: Solution of two dimensional Laplace’s equations by Liebmann’s iterative process and Poisson’s equations
- Solution of one dimensional heat equation using Bender Schmidt and Crank
Nicholson difference schemes - Solution of one dimensional wave equation by explicit scheme.
Discrete Mathematics
- Sets - Operations on sets
- Laws of set theory
- Partition of a set
- Cartesian product of sets
- Relations
- Properties
- Equivalence relation and partial order relation
- Poset
- Graphs of relations
- Digraphs
- Hasse
diagram - Closures of relations
- Transitive closure and Warshall’s algorithm
- Functions - Types of functions
- Composition of functions
- Properties
- Inverse of functions
- Necessary and sufficient condition for
existence of inverse function - Uniqueness of identity
- Inverse of composition
- Permutation and combination
- Addition and product rules
- Principle of inclusion and exclusion
- Pigeon-hole principle and generalized pigeon-hole principle
- Divisibility and prime numbers
- Fundamental theorem
of arithmetic - Prime factorization
- Division algorithm
- Greatest common divisor - Properties
- Euclid’s algorithm
- Least common multiple
- Propositions and logical operators
- Truth tables
- Converse, inverse and contrapositive
- Tautology and contradiction
- Equivalences
- Implications
- Laws of logic
- Inference theory
- Rules of inference
- Direct
method - CP rule
- Inconsistency
- Indirect method
- Principle of mathematical induction
- Groups
- Permutation group
- Cyclic group
- Properties
- Subgroup
- Group homomorphism
- Properties
- Ring
- Zero divisor
- Integral domain- Field
- Coding theory - Group code
- Hamming codes
- Error correction
using matrices - Error correction - Decoding group codes
- Definitions
- Handshaking theorem
- Some special graphs
- Isomorphism of graphs - Paths, cycles and circuits
- Connectivity in undirected graphs
- Eulerian and Hamiltonian graphs
- Matrix representation of graphs
Isomorphism using adjacency- Digraphs
- Trees
- Properties
- Spanning tree
- Kruskal’s algorithm
- Graph coloring
- Chromatic number
- Four color theorem (statement only)
Transforms and Computational Techniques
- Dirichlet’s conditions
- Fourier Series
- Functions having arbitrary periods
- Odd and even function
- Half range sine and cosine Fourier series
- Parseval’s identity
- Harmonic Analysis
- Fourier transform pair
- Fourier sine and cosine transforms
- Transforms of simple functions
- Convolution theorem (without proof)
- Parseval’s identity
- Z – transforms: Properties of Z transforms
- Inverse Z
transforms - Convolution theorem (without Proof)
- Solution of linear difference equations with constant coefficients using Z-transform
- Classification of second-order partial differential equations
- Linear Partial differential equations of second and higher order with constant coefficients of homogeneous type
- Solutions of one dimensional wave
equation - One dimensional equation of heat conduction
- Steady state conditions with zero boundary
- Solutions of first order simultaneous differential equations by Taylor’s series method
- Euler’s method and its applications
- Runge-Kutta method of fourth order (No proof)
- Trapezoidal rule
- Simpson’s one third
and Simpson’s 3/8th rule
- Classification of Second order PDE
- Solutions of Elliptic Equations
- Solutions of Laplace Equations by Liebmann’s iterative process
- Solutions of Poisson Equations
- Solutions of Parabolic equations by Bender
Schmidt formula- Solutions of Parabolic equations by Crank
- Nicolson formula
- Solutions of Hyperbolic equations by Explicit formula
Probability & Queuing Theory
- Probability concepts
- Discrete and continuous random variables
- Probability distribution function, Cumulative distribution function
- Moments
- Central and raw moments, Expectation and variance
- Moment
generating function (MGF) - Tchebycheff’s inequality
- Function of a random variable
- Discrete distribution
- Binomial distribution, Poisson distribution
- MGF, Mean, Variance, Theoretical frequencies and applications
- Continuous distribution
- Exponential and normal distributions
- MGF, Mean,
Variance and applications
- Joint distributions
- Marginal and conditional distributions
- Covariance
- Correlation and linear regression
- Central limit theorem (for independent and identically distributed random variables)
- Queueing theory
- Characteristics of a queueing Model
- Kendal’s notation
- Poisson queues - (M/M/1): (∞/FIFO) Model
- System characteristics
- Applications (M/M/s):
- (∞/FIFO) Model
- System characteristics
- Applications - (M/M/1): (/FIFO) Model
- System characteristics
- Applications.
- Markov process
- Markov chain
- One step transition probability matrix
- Chapman Kolmogorov theorem
- Limiting probabilities
- Classification of states of a Markov chain
Probability and Stochastic Processes
- One-dimensional random variable: Discrete Case
- Probability function, Cumulative Distribution Function, continuous random variable
- Probability density function, Cumulative distribution function
- properties, Problems
on one - dimensional random variable, Expectation, variance, Moments
- raw and central moments, Binomial distribution
- moments, Binomial distribution-Applications, Poisson distribution
- moments, Poisson
distribution -Applications - Exponential distribution
- moments, Exponential distribution -Applications
- Normal Distribution
- moments, Normal Distribution-Applications
- Uniform Distribution-moments, Uniform
Distribution-Applications - Function of a random variable
- Applications of random variables in engineering
- Two-dimensional random variables
- Discrete cases, Probability function of (X, Y)
- Marginal probability distribution, Conditional probability distribution of (X, Y),
- Problems on discrete random variables, Continuous
random variables - Joint PDF, Marginal Probability distributions, Conditional probability distribution of (X, Y),
- Problems on continuous two-dimensional random variables, Independent random variables, Cumulative
distribution function - properties of F(x, y), Expected values of two-dimensional random variables, Covariance and correlation, Conditional expected values
- Problems on uncorrelated random variables
- Functions of
two-dimensional random variables - Probability density functions of the type Z=XY
- Probability density functions of the type Z=X-Y, Probability density functions of the type Z=X/Y
- Application of two-dimensional
random variables in engineering
- Limit theorems--Markov's inequality, Chebyshev's inequality without proof,
- Chebyshev's inequality - Applications, Chebyshev's inequality
- Applications using Binomial distribution
- Chebyshev's inequality–
Applications using Exponential distribution, - The weak law of large numbers, Central limit theorem without proof,
- Central limit theorem - Applications,
- Central limit theorem- Applications using Poisson random
variables, - Central limit theorem- Applications using Exponential random variables,
- The strong law of large numbers, The strong law of large numbers, One-sided Chebychev's inequality,
- Cauchy Schwartz inequality,
Chernoff bounds, Chernoff bounds for the standard normal variate, Chernoff bounds for the Poisson random variate, Jenson's inequality - Applications of Central Limit Theorem in engineering
- Random Processes
- Introduction, Classification of random processes, Distribution of the process
- Averages of the process, Stationary, SSS, WSS processes
- Problems on stationary and SSS processes, Problem
- Problems on WSS process, Problems on WSS process,
- Autocorrelation function
- properties, Proof of properties, Problems on autocorrelation function,
- Application of autocorrelation function, Cross-correlation properties,
- Proof of properties, Problems on Cross-correlation function, Ergodicity, Mean ergodic process, Mean ergodic theorem,
- Applications of random process in engineering.
- Power spectral density function- properties,
- Proof of properties,
- Problems on power spectral density function,
- Problems on power spectral density function,
- Power density spectrum,
- Problems based on power
density spectrum, - Linear systems with random inputs,
- Representation of system in the form of convolution,
- Unit impulse response of the system, Properties, Applications of unit impulse function,
- Einstein Weiner-
Khinchine Relationship, - Cross-power density spectrum-problems,
- Cross-power density spectrum Cross-power density spectrum,
- Applications of power spectral density functions in engineering, Applications of power
spectral density functions in engineering
Probability & Random Variables
- Probability
- Axioms of probability
- Conditional Probility
- Baye's Theorem
- Discrete and Continuous Random Variables
- Moments - Moment Generating Functions
- Binomial, Poisson, Geometric, Uniform, Exponential & Normal Distributions
- Joint Distributions
- Marginal and Conditional Distributions
- Covariance
- Correlation and Linear Regression
- Transformation of Random Variables
- Central Limit Theorem(for Independent & Identically Distributed Random Variables).
- Classification
- Stationary Process
- Markov Process
- Poisson Process
- Discrete Parameter Markov Chain
- Chapman Kolmogorov Equations
- Limiting Distributions
- Markovian Queues
- Birth & Death Processes
- Single & Multiple Server Queueing Models
- Little's Formula
- Queues with Finite Waiting Rooms
- Queues with Impatient Customers : Balking & Renegining
- Finite Source Models
- M/G/1 Queue
- Pollaczek Khinchin Formula
- M/D/1 as Special Cases
- Series Queues
- Open Jackson Networks
Statistical Modeling
- Random sampling
- Sampling from finite and infinite populations
- Estimates and standard error (sampling with replacement and sampling without replacement)
- Sampling distribution of sample mean
- stratified
random sampling - Systematic sampling and cluster sampling
- Definition of Statistics
- Basic objectives
- Applications in various branches of science with examples
- Collection of Data: Internal and external data, primary and secondary data
- Population and sample
- Representative sample
- Descriptive Statistics: Classification and tabulation of univariate data
- graphical representation
- Frequency curves
- Descriptive measures
- central tendency and dispersion.
- Point estimation
- criteria for good estimates (unbiasedness, consistency)
- Methods of estimation including maximum likelihood estimation
- Sufficient statistic: concept and examples, complete sufficiency and their
application in estimation - Test of hypothesis: concept and formulation
- Type I and Type II errors
- Neyman Pearson lemma
- Comparison with parametric inference
- Use of order statistics
- Sign test
- Wilcoxon signed rank test
- Mann
- Whitney test
- Run test
- Kolmogorov
- Smirnov test
- Spearman’s and Kendall’s test
- Tolerance region
- Basics of Time Series Analysis and Forecasting
- Stationary
- ARIMA Models: Identification
- Estimation and Forecasting
- Applications to industrial problems
Probability and Statistics
- Probability concepts, Types of Events, Axioms and theorems
- Conditional probability, Baye’s theorem (without proof)
- Applications of Baye’s Theorem. Random variables
- Discrete case and continuous case
- Mathematical expectation, Variance
- discrete case and continuous case - Raw Moments
- Central Moments
- Moment generating function (MGF)
- discrete and continuous random variable
- Discrete distributions
- Introduction- Mean and Variance of Binomial Distribution
- Fitting a Binomial distribution
- MGF of Binomial Distribution
- Poisson Distribution
- Mean and Variance of Poisson Distribution
- Fitting
a Poisson distribution - MGF of Poisson distribution
- Geometric distribution mean and variance, Memoryless property
- Continuous distributions
- Introduction- Uniform distribution
- MGF, Mean and Variance-
Exponential distribution - MGF, Mean and Variance, Memoryless property
- Normal distribution
- Sampling Distributions
- Type I and Type II errors
- large sample test
- Test of significance for single proportion
- Test of significance for difference of proportions
- Test of significance for single mean
- Test of significance
for difference of means - Small sample tests
- Student’s t- test for single mean- t- test for the difference of means- Fisher’s F-test
- Test of significance for two sample variances- Chi -square tes for the goodness of
fit- Chi-square test for the independence of attributes.
- Correlation and its Properties
- Karl Pearson’s coefficient of correlation
- Spearman’s rank correlation coefficient for repeated and non-repeated ranks
- Linear Regression lines and Properties
- Relation between
correlation and regression coefficient - Introduction to Analysis of Variance (ANOVA)
- One-way Classification
- two-way classification
- Introduction – Process control
- control charts for variables - X & R, X
and S charts - control charts for attributes: p-chart, np-chart, c- chart and their applications in process control
ELEMENTS OF OPERATIONS RESEARCH
Mathematics Paper (MBA)
- Operations Research
- Meaning
- Definition
- Origin and History
- Characteristic Features
- Need
- Scope
- Steps
- Techniques
- Application
- Limitations
- Meaning
- Requirements
- Assumptions
- Applications
- Formulating Lpp –Advantages
- Limitations Formulating LP Model (Simple Problems Only)
- Obtaining Optimal Solution for Linear Programming Problem (LPP)
- Graphical Method
- Problems
- Simplex Method for Type of LPP and for Slack Variable Case
- Maximization Function
- Minimization Function (Simple Problem Only)
- Meaning
- (Initial Basic Feasible Solution )Assumptions
- Degenerate Solution
- North -West Corner Method- Least Cost Method -Vogels Approximation Method
- Assignment Problems
- Features
- Transportation Problem Vs Assignment Problem
- Hungarian Method (Simple Problems Only)
- Meaning
- Types of Games
- Basic Assumptions
- Finding Value of Game for Pure Strategy
- Mixed Strategy
- Indeterminate Matrix and Average Method
- Graphical Method
- Pure Strategy
- Saddle Point Payoff Matrix Value of Game (Simple Problems Only)
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