Contents. Chapter 1 Vector Spaces. Foreword... (vii) Message...(ix) Preface...(xi)
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1 (xiii) Contents Foreword... (vii) Message...(ix) Preface...(xi) Chapter 1 Vector Spaces Vector space... 1 General Properties of vector spaces... 5 Vector Subspaces... 7 Algebra of subspaces Linear combination Linear span Linear dependence and linear independence Basis of vector space Linear transformation Range and null space of a Linear Transformation Rank and nullity of a Linear transformation Representation of transformation by matrices... 22
2 (xiv) Chapter 2 Hermite Polynomials Recurrence Relations The Rodrigues formula Hn (x) Brafman generation function The Hermite polynomial as 2 Fo Intrgral Representation Curzon s Integral for Pn (x) Orthagonal Property Expansion of polynomials More Generating function Bilinear Generating function Chapter 3 Functions Hash Function Applications of Hash Function Properties of Hash Function Perfect Hashing Minimal Perfect hashing Origin of from Hash Heaviside step Functions Error Function Inverse error Function Modular Mathematics Simultaneous equations Chapter 4 Solution of Partial Differential Equations Method of Separation of variable One dimensional heat how equation... 62
3 (xv) Solution of one dimensional wave equation (by separation of variable) De-Almbert s solution Solution of one dimensional heat flow equation Two dimensional wave equation (Vibrating membrane) Solution of two dimensional wave equation (Rectangular membrane) Solution of two dimensional wave equation (Circular membrane) Laplace equation in three dimensions Chapter 5 Numerical Solution of Partial Differential Equations Classification of second order partial differential equations Finite differential approximations to partial derivatives Elliptic equations Solution of Laplace equation Solution of Poisson s equation Solution of elliptic by relaxation method Parabolic Equation (Heat equation) Schmidt method Crank Nicolson method Iterative method Du-ford and frankel method Solution of two dimensional heat equation Hyperbolic equations Chapter 6 Fourier Transform Fourier Integral formula Fourier sine and cosine integrals Fourier Transforms Properties of Fourier Transform Inversion theorem for complex Fourier Transform Multiple Fourier Transforms
4 (xvi) Convolution Convolution theorem for Fourier Transform (Felting theorem) Perseval s Identity for Fourier Transforms Relationship between Fourier and Laplace Transforms Fourier Transforms of the derivatives of a Function Finite Fourier sine Transforms Finite Fourier cosine Transforms Application of Fourier Transforms to boundary value problems Chapter 7 The Discrete Fourier Transform The Discrete Fourier Transform The DFT as a Linear Transformation Properties of the DFT Modulo N-Operation Chapter 8 Wavelet and Haar Transform Introduction Merlot s wavelet Mother wavelet The continuance wave transform (CWT) Application of wavelet transform Haar Transform Chapter 9 Theory of Probability Introduction Random Experiment Sample Point Sample space
5 (xvii) Discrete sample space Continuous sample space Event Type of events Addition theorem of probabilities Conditional Probability Multiplication theorem (Theorem of compound probability) Probability compound Event theorem Bay s Theorem Discrete Random variable Continuous Random variable Probability functions of a Discrete Random variable Mathematical expectation Chapter 10 Theoretical Distribution Theoretical Distribution Binomial Distributions Constants of Binomial Distribution Recurrence Formula Moment Generating Function Cumulate Generation Function Poisson Distribution Constants of Poisson Distribution Recurrence formula for Poisson distribution Normal Distribution Properties of the normal distribution Constant of normal distribution Moment Generating function Fitting of Normal Distribution
6 (xviii) Chapter 11 Sampling Distribution (Large Samples) Introduction Type of sampling Sample of Attributes Simple sampling Mean and Standard deviation in simple sampling of Attributes Test of significance for large samples Standard error Probable error Comparison of two large samples Sampling Distribution Standard error of sampling distribution of means Distribution of the difference between two sample means Test of significance for means Test of significance of the means of two large samples Fiducial or confidence limits Some standard error of other parameters Chapter 12 Theory of Estimation Introduction Point Estimation Interval estimation Properties of best estimator Unbiased estimator Consistent estimator Efficient estimator Sufficient estimator Maximum likelihood parameter
7 (xix) Properties of maximum likelihood estimators Method of moments Properties of the moment method Chapter 13 Theory of Testing a Hypothesis A Statistical Hypothesis Null Hypothesis Composite Hypothesis Critical Region and Acceptance Region Type of errors Level of significance Power Function of a test Best critical Region Procedure for testing a hypothesis Recurred relation Chapter 14 Markov Analysis Introduction Stochastic process Markov Process Classification of Markov processes Transition probability Transition probability matrix n-step Transition probabilities Diagrammatic Representation of Transition probabilities Multi-period Transition probabilities First order and Higher order Markov Process Markov chain
8 (xx) Steady state (Equilibrium) condition Method for determining steady state condition Characteristics of a Markov chain Chapter 15 Queuing Theory Introduction Important definition in Queuing Theory Queuing system Transient and steady states Traffic Intensity (Utilization factor) Probability distributions in Queuing systems Distribution of Arrival (Pure Birth Process) Distribution of Inter-arrival times (Exponential Process) Distribution of Departures (Pure Death Process) Distribution of service Times Concepts of Queuing Models Solution of Queuing Models Model II (M/M/I): (N/ /FCFS) Measures of Model II Model III (M/M/I): ( / /FCFS) Measures for Model III Chapter 16 Fuzzy Sets Introduction Fuzzy versus crisp Fuzzy Sets Power of a Fuzzy Set Product of a Fuzzy with a crisp number
9 (xxi) Properties of Fuzzy sets Fuzzy Relations Fuzzy Cartesian product Composition of relations Binary relations Operations of Fuzzy relations Fuzzy logic Fuzzy proposition Types of Fuzzy propositions Fuzzy connectives Fuzzy Quantifiers Fuzzy Inference Fuzzy Relation Equations Defuzzification Chapter 17 Decision Theory Introduction Basic concept of Decision Theory Type of Decision making Environment Decision making under uncertainty Decision making under Risk Chapter 18 Calculus of Variation Introduction Functionals Euler s Equation Functional dependent on more than one independent variable (Euler Ostrogradsky equation) Variation Problems in Parametric from (Euler Langrange equation) Rayleigh Ritz method
10 (xxii) Galerkin s method Discretization Finite element method Variational formulation Chapter 19 Theory of Reliability and Fault Tolerance Introduction Definition of Reliability Failure rate (Hazard rate) Reliability Functions Properties of reliability Mean time to Failure (MTTF) Mean time between Failure (MTBF) Relation between Reliability and mean Time between Failures Maintainability Availability System Reliability The Importance of fault Tolerance Chapter 20 Goal Programming Introduction Goal Programming model formulation Single goal models Multiple goal models The general goal programming model Graphical solution of GP problems Simplex method of GP
11 (xxiii) Chapter 21 MATLAB Introduction Starting MATLAB The MATLAB environment Useful functions and operation in MATLAB Obtaining help on MATLAB Commands Chapter 22 Hankel and Mellin Transforms Hankel Transforms Inversion Hankel Transform Linear Property Some useful results Hankel Transform of the Derivatives The Finite Hankel Transforms d f 1 df n Hankel Transform of + f dx x dx x Applications of Hankel Transform Mellin Transforms Inversion Mellin Transform Linear Property Some Elementary Properties Mellin Transform of Derivatives Mellin Transform of Integrals Convolution (or Falting) Theorem Practice Problems
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