Defects of Properties in Mathematics : Quantitative Characterizations.
Title:
Defects of Properties in Mathematics : Quantitative Characterizations.
Author:
Ban, Adrian I.
ISBN:
9789812777645
Personal Author:
Physical Description:
1 online resource (365 pages)
Series:
Series on Concrete and Applicable Mathematics ; v.5
Series on Concrete and Applicable Mathematics
Contents:
Contents -- Preface -- Chapter 1 Introduction -- 1.1 General Description of the Topic -- 1.2 On Chapter 2: Defect of Property in Set Theory -- 1.3 On Chapter 3: Defect of Property in Topology -- 1.4 On Chapter 4: Defect of Property in Measure Theory -- 1.5 On Chapter 5: Defect of Property in Real Function Theory -- 1.6 On Chapter 6: Defect of Property in Functional Analysis -- 1.7 On Chapter 7: Defect of Property in Algebra -- 1.8 On Chapter 8: Miscellaneous -- Chapter 2 Defect of Property in Set Theory -- 2.1 Measures of Fuzziness -- 2.2 Intuitionistic Entropies -- 2.3 Applications -- 2.3.1 Application to determination of degree of interference -- 2.3.2 Application to description of the performance of systems -- 2.3.3 Application to digital image processing -- 2.4 Bibliographical Remarks -- Chapter 3 Defect of Property in Topology -- 3.1 Measures of Noncompactness for Classical Sets -- 3.2 Random Measures of Noncompactness -- 3.3 Measures of Noncompactness for Fuzzy Subsets in Metric Space -- 3.4 Measures of Noncompactness for Fuzzy Subsets in Topological Space -- 3.5 Defects of Opening and of Closure for Subsets in Metric Space -- 3.6 Bibliographical Remarks and Open Problems -- Chapter 4 Defect of Property in Measure Theory -- 4.1 Defect of Additivity: Basic Definitions and Properties -- 4.1.1 Application to calculation of fuzzy integral -- 4.1.2 Application to best approximation of a fuzzy measure -- 4.1.3 A metric on the family of fuzzy measures -- 4.2 Defect of Complementarity -- 4.3 Defect of Monotonicity -- 4.4 Defect of Subadditivity and of Superadditivity -- 4.5 Defect of Measurability -- 4.6 Bibliographical Remarks -- Chapter 5 Defect of Property in Real Function Theory -- 5.1 Defect of Continuity of Differentiability and of Integrability.
5.2 Defect of Monotonicity of Convexity and of Linearity -- 5.3 Defect of Equality for Inequalities -- 5.4 Bibliographical Remarks and Open Problems -- Chapter 6 Defect of Property in Functional Analysis -- 6.1 Defect of Orthogonality in Real Normed Spaces -- 6.2 Defect of Property for Sets in Normed Spaces -- 6.3 Defect of Property for Functionals -- 6.4 Defect of Property for Linear Operators on Normed Spaces -- 6.5 Defect of Fixed Point -- 6.6 Bibliographical Remarks and Open Problems -- Chapter 7 Defect of Property in Algebra -- 7.1 Defects of Property for Binary Operations -- 7.2 Calculations of the Defect of Property -- 7.3 Defect of Idempotency and Distributivity of Triangular Norms -- 7.4 Applications -- 7.5 Bibliographical Remarks -- Chapter 8 Miscellaneous -- 8.1 Defect of Property in Complex Analysis -- 8.2 Defect of Property in Geometry -- 8.3 Defect of Property in Number Theory -- 8.4 Defect of Property in Fuzzy Logic -- 8.5 Bibliographical Remarks and Open Problems -- Bibliography -- Index.
Abstract:
This book introduces a method of research 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. Besides well-known "defects", the book introduces and studies new ones, such as: measures of noncompactness for fuzzy sets; fuzzy and intuitionistic entropies; the defect of (sub, super)additivity; complementarity; monotonicity for set functions; the defect of convexity; monotonicity; differentiability for real functions; the defect of equality for inequalities; the defect of orthogonality for sets and defects of properties for linear operators in normed spaces; defects of properties (commutativity, associativity, etc.) for binary operations; defects of orthogonality and parallelness in Euclidean and non-Euclidean geometries; defects of integer, perfect, prime and amicable numbers; the defect of tautology in fuzzy logic. Contents: Defect of Property in Set Theory; Defect of Property in Topology; Defect of Property in Measure Theory; Defect of Property in Real Function Theory; Defect of Property in Functional Analysis; Defect of Property in Algebra; Miscellaneous. Readership: Upper level undergraduates, graduate students and researchers interested in measure theory, real and functional analysis, fuzzy mathematics, topology and algebra.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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