Defects of Properties in Mathematics: Quantitative Characterizations

Adrian I. Ban, Sorin G. Gal9789810249243, 981-02-4924-1

This book introduces a method of reasearch which can be used in various fields of mathematics. It examines, in a systematic way, the quantitative characterizations of the “deviation from a (given) property”, called the “defect of a property”, in: set theory; topology; measure theory; real, complex and functional analysis; algebra; geometry; number theory; fuzzy mathematics”

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Table of contents :
Contents……Page 10
Preface……Page 8
1.1 General Description of the Topic……Page 13
1.2 On Chapter 2: Defect of Property in Set Theory……Page 15
1.3 On Chapter 3: Defect of Property in Topology……Page 20
1.4 On Chapter 4: Defect of Property in Measure Theory……Page 30
1.5 On Chapter 5: Defect of Property in Real Function Theory……Page 36
1.6 On Chapter 6: Defect of Property in Functional Analysis……Page 43
1.7 On Chapter 7: Defect of Property in Algebra……Page 57
1.8 On Chapter 8: Miscellaneous……Page 62
2.1 Measures of Fuzziness……Page 69
2.2 Intuitionistic Entropies……Page 79
2.3.1 Application to determination of degree of interference……Page 88
2.3.2 Application to description of the performance of systems……Page 93
2.3.3 Application to digital image processing……Page 96
2.4 Bibliographical Remarks……Page 98
3.1 Measures of Noncompactness for Classical Sets……Page 99
3.2 Random Measures of Noncompactness……Page 107
3.3 Measures of Noncompactness for Fuzzy Subsets in Metric Space……Page 110
3.4 Measures of Noncompactness for Fuzzy Subsets in Topological Space……Page 123
3.5 Defects of Opening and of Closure for Subsets in Metric Space……Page 137
3.6 Bibliographical Remarks and Open Problems……Page 139
4.1 Defect of Additivity: Basic Definitions and Properties……Page 143
4.1.1 Application to calculation of fuzzy integral……Page 154
4.1.2 Application to best approximation of a fuzzy measure……Page 157
4.1.3 A metric on the family of fuzzy measures……Page 158
4.2 Defect of Complementarity……Page 160
4.3 Defect of Monotonicity……Page 176
4.4 Defect of Subadditivity and of Superadditivity……Page 185
4.5 Defect of Measurability……Page 193
4.6 Bibliographical Remarks……Page 196
5.1 Defect of Continuity of Differentiability and of Integrability……Page 197
5.2 Defect of Monotonicity of Convexity and of Linearity……Page 205
5.3 Defect of Equality for Inequalities……Page 212
5.4 Bibliographical Remarks and Open Problems……Page 220
6.1 Defect of Orthogonality in Real Normed Spaces……Page 223
6.2 Defect of Property for Sets in Normed Spaces……Page 246
6.3 Defect of Property for Functionals……Page 258
6.4 Defect of Property for Linear Operators on Normed Spaces……Page 264
6.5 Defect of Fixed Point……Page 272
6.6 Bibliographical Remarks and Open Problems……Page 285
7.1 Defects of Property for Binary Operations……Page 289
7.2 Calculations of the Defect of Property……Page 301
7.3 Defect of Idempotency and Distributivity of Triangular Norms……Page 305
7.4 Applications……Page 312
7.5 Bibliographical Remarks……Page 316
8.1 Defect of Property in Complex Analysis……Page 317
8.2 Defect of Property in Geometry……Page 319
8.3 Defect of Property in Number Theory……Page 332
8.4 Defect of Property in Fuzzy Logic……Page 339
8.5 Bibliographical Remarks and Open Problems……Page 346
Bibliography……Page 349
Index……Page 361