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

✍ Scribed by Levent Kandiller

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 302
• Series: International Series in Operations Research & Management Science
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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