Defects of Properties in Mathematics

by Ban, Adrian I.; Gal, Sorin G.

• ISBN13: 9789810249243
• ISBN10: 9810249241
• Format: Hardcover
• Copyright: 2002-07-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Introduces a method of research which can be used in various fields of mathematics.

Table of Contents

Preface	vii
Introduction	1	(56)
General Description of the Topic	1	(2)
On Chapter 2: Defect of Property in Set Theory	3	(5)
On Chapter 3: Defect of Property in Topology	8	(10)
On Chapter 4: Defect of Property in Measure Theory	18	(6)
On Chapter 5: Defect of Property in Real Function Theory	24	(7)
On Chapter 6: Defect of Property in Functional Analysis	31	(14)
On Chapter 7: Defect of Property in Algebra	45	(5)
On Chapter 8: Miscellaneous	50	(7)
Defect of Property in Set Theory	57	(30)
Measures of Fuzziness	57	(10)
Intuitionistic Entropies	67	(9)
Applications	76	(10)
Application to determination of degree of interference	76	(5)
Application to description of the performance of systems	81	(3)
Application to digital image processing	84	(2)
Bibliographical Remarks	86	(1)
Defect of Property in Topology	87	(44)
Measures of Noncompactness for Classical Sets	87	(8)
Random Measures of Noncompactness	95	(3)
Measures of Noncompactness for Fuzzy Subsets in Metric Space	98	(13)
Measures of Noncompactness for Fuzzy Subsets in Topological Space	111	(14)
Defects of Opening and of Closure for Subsets in Metric Space	125	(2)
Bibliographical Remarks and Open Problems	127	(4)
Defect of Property in Measure Theory	131	(54)
Defect of Additivity: Basic Definitions and Properties	131	(17)
Application to calculation of fuzzy integral	142	(3)
Application to best approximation of a fuzzy measure	145	(1)
A metric on the family of fuzzy measures	146	(2)
Defect of Complementarity	148	(16)
Defect of Monotonicity	164	(9)
Defect of Subadditivity and of Superadditivity	173	(8)
Defect of Measurability	181	(3)
Bibliographical Remarks	184	(1)
Defect of Property in Real Function Theory	185	(26)
Defect of Continuity, of Differentiability and of Integrability	185	(8)
Defect of Monotonicity, of Convexity and of Linearity	193	(7)
Defect of Equality for Inequalities	200	(8)
Bibliographical Remarks and Open Problems	208	(3)
Defect of Property in Functional Analysis	211	(66)
Defect of Orthogonality in Real Normed Spaces	211	(23)
Defect of Property for Sets in Normed Spaces	234	(12)
Defect of Property for Functionals	246	(6)
Defect of Property for Linear Operators on Normed Spaces	252	(8)
Defect of Fixed Point	260	(13)
Bibliographical Remarks and Open Problems	273	(4)
Defect of Property in Algebra	277	(28)
Defects of Property for Binary Operations	277	(12)
Calculations of the Defect of Property	289	(4)
Defect of Idempotency and Distributivity of Triangular Norms	293	(7)
Applications	300	(4)
Bibliographical Remarks	304	(1)
Miscellaneous	305	(32)
Defect of Property in Complex Analysis	305	(2)
Defect of Property in Geometry	307	(13)
Defect of Property in Number Theory	320	(7)
Defect of Property in Fuzzy Logic	327	(7)
Bibliographical Remarks and Open Problems	334	(3)
Bibliography	337	(12)
Index	349