روش‌های ریاضی در اقتصاد نظری: بنیادهای توپولوژیکی و فضای برداری تحلیل تعادل

Instead of the cover......Page 1 Series......Page 2 Title page......Page 3 Copyright page......Page 4 Dedication......Page 5 CONTENTS......Page 7 Preface......Page 15 Acknowledgments......Page 17 Notation......Page 19 Part I POINT SET TOPOLOGY......Page 21 1.1 Introduction......Page 23 1.2 Preliminary Notions......Page 24 1.3 Functions......Page 26 1.4 Operations with Sets......Page 28 1.5 Algebra of Sets......Page 31 1.6 Duality Principle......Page 32 Problems......Page 34 Notes......Page 36 2.1 Products of Sets......Page 37 2.2 Sets of Points......Page 39 2.3 Metric Spaces......Page 44 2.4 Binary Relations......Page 45 2.5 Classes of Relations......Page 47 Exercises......Page 51 Problems......Page 53 Notes......Page 54 3.1 Preorderings and Orderings......Page 55 3.2 Preordered and Ordered Subsets......Page 58 3.3 Elements of Preordered and Ordered Sets......Page 59 3.4 Preorderings on Product Sets......Page 61 3.5 Lower and Upper Bounds......Page 62 Exercises......Page 64 Problems......Page 65 Notes......Page 66 4.1 Denumerable and Countable Sets......Page 67 4.2 Closedness......Page 70 4.3 Boundedness......Page 80 4.4 Compactness......Page 83 4.5 Connectedness......Page 90 4.6 Convexity......Page 92 4.7 Sum of Sets in $E^n$......Page 94 Exercises......Page 96 Problems......Page 98 Notes......Page 99 5 Point-to-Point Mappings......Page 100 5.1 Single-Valued Functions......Page 101 5.2 Classes of Mappings......Page 104 5.3 Continuous Mappings......Page 109 5.4 Mappings and Set Properties......Page 112 5.5 Convex and Concave Functions......Page 115 5.6 Linear Single-Valued Function......Page 120 Exercises......Page 121 Problems......Page 122 Notes......Page 123 6 Point-to-Set Mappings......Page 124 6.1 Multivalued Function......Page 125 6.2 Continuity......Page 129 6.3 Some Theorems on Maxima......Page 134 Exercises......Page 138 Problems......Page 139 Notes......Page 140 7.1 Introduction......Page 141 7.2 Simplicial Topology: Elementary Concepts......Page 144 7.3 Point-to-Point Mappings......Page 155 7.4 Point-to-Set Mappings......Page 157 Exercises......Page 160 Notes......Page 162 8 Algebraic Structures......Page 163 8.1 Abstract Systems and Homomorphisms......Page 164 8.2 Groups......Page 165 8.3 Rings, Integral Domains, and Fields......Page 167 Exercises......Page 169 Problems......Page 170 Notes......Page 171 9 General Equilibrium For Economics with a Finite Number of Agents and Commodities......Page 172 9.1 Frame of the Problem and Basic Assumption......Page 173 9.2 Producers and Supply......Page 175 9.3 Consumers and Demand......Page 176 9.4 General Equilibrium......Page 182 Exercises......Page 185 Problems......Page 187 Notes......Page 188 Part II VECTOR SPACES AND VECTOR-SPACE HOMOMORPHISMS......Page 189 10 Vector Spaces and Subspaces......Page 191 10.1 Vector Spaces over a Field......Page 192 10.2 Vectors and Operations with Vectors......Page 193 10.3 Metrics......Page 196 10.4 Subspaces......Page 199 10.5 Linearly Independent Set of Vectors......Page 203 10.6 Basis and Dimension......Page 206 10.7 Vector-Space Homomorphisms......Page 209 Exercises......Page 211 Notes......Page 212 11.1 Linear Transformations......Page 213 11.2 Analysis of Linear Transformations......Page 216 11.3 Nonsingular and Inverse Transformations......Page 218 11.4 Linear Functional and Dual Spaces......Page 220 11.5 Transpose of a Linear Transformation......Page 223 11.6 Linear Algebras......Page 225 Exercises......Page 227 Notes......Page 228 12.1 Operational Description of Linear Mappings......Page 229 12.2 Matrix over a Field......Page 231 12.3 Basic Operations with Matrices......Page 232 12.4 The Transpose of a Matrix......Page 237 12.5 Square Matrices......Page 239 12.6 Miscellaneous Square Matrices......Page 242 Exercises......Page 246 Problems......Page 247 Notes......Page 248 13 Rank, Equivalence, and Similarity......Page 249 13.1 The Rank of a Matrix......Page 250 13.2 Matrix Equivalence......Page 252 13.3 Matrix Similarity......Page 259 Exercises......Page 260 Notes......Page 261 14.1 The Determinant Function......Page 262 14.2 Evaluation of Determinants......Page 271 14.3 The Adjoint of a Square Matrix......Page 274 14.4 Methods of Matrix Inversion......Page 275 Exercises......Page 279 Problems......Page 280 Notes......Page 282 15.1 Introduction......Page 283 15.2 General Case: The Nonhomogeneous System......Page 286 15.3 Special Case: The Homogeneous System......Page 291 15.4 A Fundamental Property of Linear Systems of Equations......Page 293 Exercises......Page 294 Problems......Page 296 Notes......Page 297 16 Characteristic Vectors, Diagonalization, and Triangularization......Page 298 16.1 Characteristic Polynomials, Roots, and Vectors......Page 299 16.2 The Cayley-Hamilton Theorem......Page 305 16.3 Diagonalization and Triangularization......Page 307 16.4 Matrices of Functions and Series of Matrices......Page 313 Exercises......Page 317 Problems......Page 318 Notes......Page 319 17 Real Quadratic Forms......Page 320 17.1 Unconstrained Quadratic Forms......Page 321 17.2 Constrained Quadratic Forms......Page 329 Exercises......Page 335 Problems......Page 336 Notes......Page 337 18 Linear Inequalities, Hyperplanes, and Convex Cones......Page 338 18.1 Linear Inequality Systems......Page 339 18.2 Bounding Hyperplanes......Page 343 18.3 Convex Cones......Page 353 Exercises......Page 361 Problems......Page 362 Notes......Page 364 19 Nowtegative Square Matrices......Page 365 19.1 Indecomposable and Decomposable Matrices......Page 366 19.2 Nonnegative Indecomposable Matrices......Page 370 19.3 Nonnegative Decomposable Matrices......Page 371 19.4 The Matrices $\mu I - A$ and $(\mu I - A)^{-1}$......Page 372 Exercises......Page 375 Problems......Page 376 Notes......Page 377 20 Multisectoral Balanced Growth......Page 378 20.1 Framework and Assumptions......Page 379 20.2 Technological Expansion......Page 382 20.4 Von Neumann's Theorem......Page 383 20.5 Indecomposability and Duality......Page 385 Exercises......Page 387 Problems......Page 388 Notes......Page 389 References......Page 390 Index......Page 397