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Principles of Mathematics in operations research

Author: Kandiller, Levent Series: International series in operations research and management science Publisher: Springer, 2007Language: EnglishDescription: 297 p. : 24 cm. Ill. ;ISBN: 0387377344 ; 9781441942500Type of document: BookBibliography/Index: Includes bibliographical references and index
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Item type Current location Collection Call number Status Date due Barcode Item holds
Book Asia Campus
Textbook Collection (PhD)
Print T57.6 .K36 2007
(Browse shelf)
900240796
Available 900240796
Book Europe Campus
Main Collection
Print T57.6 .K36 2007
(Browse shelf)
001226046
Available 001226046
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Includes bibliographical references and index

Digitized

Principles of Mathematics in Operations Research Contents 1 Introduction........................................................................................................... 1 1.1 Mathematics and OR ................................................................................. 1 1.2 Mathematics as a language ........................................................................ 2 1.3 The art of making proofs ........................................................................... 5 1.3.1 Forward­Backward method............................................................. 5 1.3.2 Induction Method ............................................................................ 7 1.3.3 Contradiction Method ...................................................................... 8 1.3.4 Theorem of alternatives ................................................................... 9 Problems ............................................................................................................ 9 Web material..................................................................................................... 10 2 Preliminary Linear Algebra.............................................................................. 13 2.1 Vector Spaces............................................................................................. 13 2.1.1 Fields and linear spaces ................................................................. 13 2.1.2 Subspaces ....................................................................................... 14 2.1.3 Bases .............................................................................................. 16 2.2 Linear transformations, matrices and change of basis............................. 17 2.2.1 Matrix multiplication ..................................................................... 17 2.2.2 Linear transformation..................................................................... 18 2.3 Systems of Linear Equations .................................................................... 20 2.3.1 Gaussian elimination...................................................................... 20 2.3.2 Gauss-Jordan method for inverses ................................................ 23 2.3.3 The most general case ................................................................... 24 2.4 The four fundamental subspaces............................................................... 25 2.4.1 The row space of A......................................................................... 25 2.4.2 The column space of A................................................................... 26 2.4.3 The null space (kernel) of A ......................................................... 26 2.4.4 The left null space of A ................................................................. 27 2.4.5 The Fundamental Theorem of Linear Algebra.............................. 27 Problems .......................................................................................................... 28 Web material..................................................................................................... 29 X Contents 3 Orthogonality ...................................................................................................... 33 3.1 Inner Products............................................................................................. 33 3.1.1 Norms ............................................................................................ 33 3.1.2 Orthogonal Spaces......................................................................... 35 3.1.3 Angle between two vectors............................................................ 36 3.1.4 Projection....................................................................................... 37 3.1.5 Symmetric Matrices ...................................................................... 37 3.2 Projections and Least Squares Approximations.......................................... 38 3.2.1 Orthogonal bases............................................................................ 39 3.2.2 Gram-Schmidt Orthogonalization.................................................. 40 3.2.3 Pseudo (Moore-Penrose) Inverse................................................... 42 3.2.4 Singular Value Decomposition...................................................... 43 3.3 Summary for Ax = b.................................................................................... 44 Problems ........................................................................................................... 47 Web material..................................................................................................... 47 4 Eigen Values and Vectors................................................................................... 51 4.1 Determinants .............................................................................................. 51 4.1.1 Preliminaries ................................................................................. 51 4.1.2 Properties ....................................................................................... 52 4.2 Eigen Values and Eigen Vectors ................................................................ 54 4.3 Diagonal Form of a Matrix ......................................................................... 55 4.3.1 All Distinct Eigen Values............................................................... 55 4.3.2 Repeated Eigen Values with Full Kernels .................................... 57 4.3.3 Block Diagonal Form .................................................................... 58 4.4 Powers of A ................................................................................................ 60 4.4.1 Difference equations ..................................................................... 61 4.4.2 Differential Equations.................................................................... 62 4.5 The Complex case....................................................................................... 63 Problems ........................................................................................................... 65 Web material..................................................................................................... 66 5 Positive Definiteness............................................................................................ 71 5.1 Minima, Maxima, Saddle points ................................................................ 71 5.1.1 Scalar Functions ............................................................................ 71 5.1.2 Quadratic forms.............................................................................. 73 5.2 Detecting Positive-Definiteness ................................................................. 74 5.3 Semidefinite Matrices ................................................................................ 75 5.4 Positive Definite Quadratic Forms ............................................................. 76 Problems ........................................................................................................... 77 Web material..................................................................................................... 77 Contents XI 6 Computational Aspects....................................................................................... 81 6.1 Solution of Ax = b...................................................................................... 81 6.1.1 Symmetric and positive definite .................................................... 81 6.1.2 Symmetric and not positive definite............................................... 83 6.1.3 Asymmetric .................................................................................... 83 6.2 Computation of eigen values ..................................................................... 86 Problems .......................................................................................................... 89 Web material..................................................................................................... 90 7 Convex Sets ......................................................................................................... 93 7.1 Preliminaries............................................................................................... 93 7.2 Hyperplanes and Polytopes ....................................................................... 95 7.3 Separating and Supporting Hyperplanes.................................................... 97 7.4 Extreme Points ........................................................................................... 98 Problems .......................................................................................................... 99 Web material..................................................................................................... 100 8 Linear Programming.......................................................................................... 103 8.1 The Simplex Method ................................................................................. 103 8.2 Simplex Tableau ........................................................................................ 107 8.3 Revised Simplex Method ...........................................................................110 8.4 Duality Theory............................................................................................111 8.5 Farkas' Lemma............................................................................................113 Problems .......................................................................................................... 115 Web material..................................................................................................... 117 9 Number Systems .................................................................................................121 9.1 Ordered Sets............................................................................................... 121 9.2 Fields...........................................................................................................123 9.3 The Real Field ........................................................................................... 125 9.4 The Complex Field .................................................................................... 127 9.5 Euclidean Space..........................................................................................128 9.6 Countable and Uncountable Sets ...............................................................129 Problems .......................................................................................................... 133 Web material..................................................................................................... 134 10 Basic Topology ................................................................................................. 137 10.1 Metric Spaces........................................................................................... 137 10.2 Compact Sets ...........................................................................................146 10.3 The Cantor Set ......................................................................................... 150 10.4 Connected Sets......................................................................................... 151 Problems .......................................................................................................... 152 Web material.....................................................................................................154 XII Contents 11 Continuity.......................................................................................................... 157 11.1 Introduction ............................................................................................ 157 11.2 Continuity and Compactness................................................................... 159 11.3 Uniform Continuity................................................................................ 160 11.4 Continuity and Connectedness ............................................................... 161 11.5 Monotonic Functions............................................................................... 164 Problems ......................................................................................................... 166 Web material................................................................................................... 166 12 Differentiation.................................................................................................... 169 12.1 Derivatives .............................................................................................. 169 12.2 Mean Value Theorems ............................................................................ 170 12.3 Higher Order Derivatives ...................................................................... 172 Problems ......................................................................................................... 173 Web material................................................................................................... 173 13 Power Series and Special Functions ............................................................... 175 13.1 Series .......................................................................................................175 13.1.1 Notion of Series .......................................................................... 175 13.1.2 Operations on Series.................................................................... 177 13.1.3 Tests for positive series................................................................ 177 13.2 Sequence of Functions............................................................................. 178 13.3 Power Series ........................................................................................... 179 13.4 Exponential and Logarithmic Functions .................................................180 13.5 Trigonometric Functions......................................................................... 182 13.6 Fourier Series .......................................................................................... 184 13.7 Gamma Function..................................................................................... 185 Problems ........................................................................................................... 186 Web material.................................................................................................... 188 14 Special Transformations .................................................................................. 191 14.1 Differential Equations............................................................................... 191 14.2 Laplace Transforms...................................................................................192 14.3 Difference Equations................................................................................ 197 14.4 Z Transforms ............................................................................................ 199 Problems ........................................................................................................... 201 Web material..................................................................................................... 202 Solutions ................................................................................................................. 205 Index........................................................................................................................ 293

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