Principles of Mathematics in Operations Research (International Series in Operations Research & Management Science (97))

This book is a comprehensive survey of the mathematical concepts and principles of industrial mathematics. Its purpose is to provide students and professionals with an understanding of the fundamental mathematical principles used in Industrial Mathematics/OR in modeling problems and application solutions. All the concepts presented in each chapter have undergone the learning scrutiny of the author and his students. The illustrative material throughout the book was refined for student comprehension as the manuscript developed through its iterations, and the chapter exercises are refined from the previous year's exercises. Principles of Mathematics in Operations Research - Levent Kandiller (0387377344)......Page 1 Preface......Page 6 Contents......Page 8 1.1 Mathematics and OR......Page 12 1.2 Mathematics as a language......Page 13 1.3.1 Forward-Backward method......Page 16 1.3.2 Induction Method......Page 18 1.3.3 Contradiction Method......Page 19 Problems......Page 20 Web material......Page 21 2.1.1 Fields and linear spaces......Page 24 2.1.2 Subspaces......Page 25 2.1.3 Bases......Page 27 2.2.1 Matrix multiplication......Page 28 2.2.2 Linear transformation......Page 29 2.3.1 Gaussian elimination......Page 31 2.3.2 Gauss-Jordan method for inverses......Page 34 2.3.3 The most general case......Page 35 2.4.1 The row space of A......Page 36 2.4.3 The null space (kernel) of A......Page 37 2.4.5 The Fundamental Theorem of Linear Algebra......Page 38 Problems......Page 39 Web material......Page 40 3.1.1 Norms......Page 44 3.1.2 Orthogonal Spaces......Page 46 3.1.3 Angle between two vectors......Page 47 3.1.5 Symmetric Matrices......Page 48 3.2 Projections and Least Squares Approximations......Page 49 3.2.1 Orthogonal bases......Page 50 3.2.2 Gram-Schmidt Orthogonalization......Page 51 3.2.3 Pseudo (Moore-Penrose) Inverse......Page 53 3.2.4 Singular Value Decomposition......Page 54 3.3 Summary for Ax = b......Page 55 Web material......Page 58 4.1.1 Preliminaries......Page 62 4.1.2 Properties......Page 63 4.2 Eigen Values and Eigen Vectors......Page 65 4.3.1 All Distinct Eigen Values......Page 66 4.3.2 Repeated Eigen Values with Full Kernels......Page 68 4.3.3 Block Diagonal Form......Page 69 4.4 Powers of A......Page 71 4.4.1 Difference equations......Page 72 4.4.2 Differential Equations......Page 73 4.5 The Complex case......Page 74 Problems......Page 76 Web material......Page 77 5.1.1 Scalar Functions......Page 81 5.1.2 Quadratic forms......Page 83 5.2 Detecting Positive-Definiteness......Page 84 5.3 Semidefinite Matrices......Page 85 5.4 Positive Definite Quadratic Forms......Page 86 Web material......Page 87 6.1.1 Symmetric and positive definite......Page 90 6.1.3 Asymmetric......Page 92 6.2 Computation of eigen values......Page 95 Problems......Page 98 Web material......Page 99 7.1 Preliminaries......Page 101 7.2 Hyperplanes and Polytopes......Page 103 7.3 Separating and Supporting Hyperplanes......Page 105 7.4 Extreme Points......Page 106 Problems......Page 107 Web material......Page 108 8.1 The Simplex Method......Page 111 8.2 Simplex Tableau......Page 115 8.3 Revised Simplex Method......Page 118 8.4 Duality Theory......Page 119 8.5 Farkas' Lemma......Page 121 Problems......Page 123 Web material......Page 125 9.1 Ordered Sets......Page 128 9.2 Fields......Page 130 9.3 The Real Field......Page 132 9.4 The Complex Field......Page 134 9.5 Euclidean Space......Page 135 9.6 Countable and Uncountable Sets......Page 136 Problems......Page 140 Web material......Page 141 10.1 Metric Spaces......Page 143 10.2 Compact Sets......Page 152 10.3 The Cantor Set......Page 156 10.4 Connected Sets......Page 157 Problems......Page 158 Web material......Page 160 11.1 Introduction......Page 162 11.2 Continuity and Compactness......Page 164 11.3 Uniform Continuity......Page 165 11.4 Continuity and Connectedness......Page 166 11.5 Monotonic Functions......Page 169 Web material......Page 171 12.1 Derivatives......Page 174 12.2 Mean Value Theorems......Page 175 12.3 Higher Order Derivatives......Page 177 Web material......Page 178 13.1.1 Notion of Series......Page 180 13.1.3 Tests for positive series......Page 182 13.2 Sequence of Functions......Page 183 13.3 Power Series......Page 184 13.4 Exponential and Logarithmic Functions......Page 185 13.5 Trigonometric Functions......Page 187 13.6 Fourier Series......Page 189 13.7 Gamma Function......Page 190 Problems......Page 191 Web material......Page 193 14.1 Differential Equations......Page 195 14.2 Laplace Transforms......Page 196 14.3 Difference Equations......Page 201 14.4 Z Transforms......Page 203 Problems......Page 205 Web material......Page 206 Solutions......Page 208 Problems of Chapter 1......Page 209 Problems of Chapter 2......Page 213 Problems of Chapter 3......Page 220 Problems of Chapter 4......Page 227 Problems of Chapter 5......Page 233 Problems of Chapter 6......Page 237 Problems of Chapter 7......Page 244 Problems of Chapter 8......Page 249 Problems of Chapter 9......Page 269 Problems of Chapter 10......Page 274 Problems of Chapter 11......Page 282 Problems of Chapter 12......Page 283 Problems of Chapter 13......Page 287 Problems of Chapter 14......Page 293 Index......Page 296 Principles of Mathematics in Operations Research is a comprehensive survey of the mathematical concepts and principles of industrial mathematics. Its purpose is to provide students and professionals with an understanding of the fundamental mathematical principles used in Industrial Mathematics/OR in modeling problems and application solutions. All the concepts presented in each chapter have undergone the learning scrutiny of the author and his students. The conceptual relationships within the chapter material have been developed in the classroom experience working with the students'level of understanding. The illustrative material throughout the book (i.e., worked-out problems and examples of the mathematical principles) was refined for student comprehension as the manuscript developed through its iterations, and the chapter exercises are refined from the previous year's exercises. In sum, the author has carefully developed a pedagogically strong survey textbook of OR and Industrial Mathematics. Operations Research Is The Application Of Scientific Models, Mathematical And Statistical Ones, To Decision Making Problems, And Principles Of Mathematics In Operations Research Is A Comprehensive Survey Of The Mathematical Concepts And Principles Of Industrial Mathematics. Its Purpose Is To Provide Students And Professionals With An Understanding Of The Fundamental Mathematical Principles Used In Industrial Mathematics/or In Modeling Problems And Application Solutions.--jacket. Introduction -- Preliminary Linear Algebra -- Orthogonality -- Eigen Values And Vectors -- Positive Definiteness -- Computational Aspects -- Convex Sets -- Linear Programming -- Number Systems -- Basic Topology -- Continuity -- Differentiation -- Power Series And Special Functions -- Special Transformations. Levent Kandiller. Series No. From P. [4] Of Cover.