Improved Bonferroni Inequalities via Abstract Tubes: Inequalities and Identities of Inclusion-Exclusion Type (Lecture Notes in Mathematics, 1826)

This Introduction To The Recent Theory Of Abstract Tubes Describes The Framework For Establishing Improved Inclusion-exclusion Identities And Bonferroni Inequalities, Which Are Provably At Least As Sharp As Their Classical Counterparts While Involving Fewer Terms. All Necessary Definitions From Graph Theory, Lattice Theory And Topology Are Provided. The Role Of Closure And Kernel Operators Is Emphasized, And Examples Are Provided Throughout To Demonstrate The Applicability Of This New Theory. Applications Are Given To System And Network Reliability, Reliability Covering Problems And Chromatic Graph Theory. Topics Also Covered Include Zeilberger's Abstract Lace Expansion, Matroid Polynomials And Möbius Functions. 1. Introduction And Overview -- 2. Preliminaries -- 3. Bonferroni Inequalities Via Abstract Tubes -- 4. Abstract Tubes Via Closure And Kernel Operators -- 5. Recursive Schemes -- 6. Reliability Applications -- 7. Combinatorial Applications And Related Topics. Klaus Dohmen. Based On Author's Habilitation Thesis--humboldt-university. Includes Bibliographical References (p. [100]-109) And Indexes. Also Available In An Electronic Version. 1 Introduction and Overview....Pages 1-4 2 Preliminaries....Pages 5-8 3 Bonferroni Inequalities via Abstract Tubes....Pages 9-18 4 Abstract Tubes via Closure and Kernel Operators....Pages 19-43 5 Recursive Schemes....Pages 44-46 6 Reliability Applications....Pages 47-81 7 Combinatorial Applications and Related Topics....Pages 82-99 Bibliography....Pages 100-109