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Improved Bonferroni inequalities via abstract tubes : inequalities and identities of inclusion-exclusion type / Klaus Dohmen.

By: Dohmen, Klaus, 1968-Material type: TextTextSeries: Lecture notes in mathematics (Springer-Verlag) ; 1826.Publication details: Berlin ; New York : Springer-Verlag, ©2003. Description: 1 online resource (viii, 109 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9783540393993; 3540393994Subject(s): Inequalities (Mathematics) | Distribution (Probability theory) | Combinatorial analysis | Combinatorial probabilities | Combinatorial analysis | Combinatorial probabilities | Distribution (Probability theory) | Inequalities (Mathematics)Genre/Form: Electronic books. | Electronic books. Additional physical formats: Print version:Dohmen, Klaus, 1968-: Improved Bonferroni inequalities via abstract tubes : inequalities and identities of inclusion-exclusion typeDDC classification: 510 s | 512.9/7 LOC classification: QA3 | .L28 no. 1826 | QA295Online resources: Click here to access online
Contents:
Intro; Title; Preface; Contents; 2.1 Graphs and posets; 2.2 Topology; 3.1 An outline of abstract tube theory; 3.1.1 The notion of an abstract tube; 3.1.2 Two fundamental theorems on abstract tubes; 3.1.3 An importance sampling scheme; 3.2 Examples of abstract tubes; 3.2.1 Abstract tubes for convex polyhedra; 3.2.2 Abstract tubes for unions of balls and spherical caps; 4.1 Abstract tubes via closure operators; 4.1.1 Closure operators; 4.1.2 How closure operators lead to abstract tubes; 4.2 Abstract tubes via kernel operators; 4.2.1 Kernel operators
4.2.2 How kernel operators lead to abstract tubes4.3 Alternative proofs; 4.3.1 Improved identities via closure operators; 4.3.2 Improved inequalities via kernel operators; 4.3.3 Additional proofs of identities; 4.4 The chordal graph sieve; 5.1 Recursive schemes for semilattices; 5.2 Dynamic programming approach; 6.1 System reliability; 6.1.1 Terminology; 6.1.2 Abstract tubes for system reliability; 6.1.3 Shier's pseudopolynomial algorithm; 6.1.4 Inclusion-exclusion and domination theory; 6.2 Special reliability systems; 6.2.1 Communications networks; 6.2.2 k-out-of-n systems
6.2.3 Consecutive k-out-of-n systems6.3 Reliability covering problems; 6.3.1 The hypergraph model; 6.3.2 Abstract tubes and polynomial algorithms; 6.4 Chapter notes; 7.1 Inclusion-exclusion on partition lattices; 7.2 Chromatic polynomials and broken circuits; 7.2.1 The usual chromatic polynomial; 7.2.2 The generalized chromatic polynomial; 7.3 Sums over partially ordered sets; 7.3.1 A general theorem on sums; 7.3.2 Application to inclusion-exclusion; 7.4 Matroid polynomials and the B invariant; 7.4.1 The Tutte polynomial; 7.4.2 The characteristic polynomial; 7.4.3 The B invariant
7.5 Euler characteristics and Möbius functions7.5.1 Euler characteristics; 7.5.2 Möbius functions; Bibliography; Author Index; Subject Index
Summary: 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.
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Based on author's habilitation thesis--Humboldt-University.

Includes bibliographical references and indexes.

Intro; Title; Preface; Contents; 2.1 Graphs and posets; 2.2 Topology; 3.1 An outline of abstract tube theory; 3.1.1 The notion of an abstract tube; 3.1.2 Two fundamental theorems on abstract tubes; 3.1.3 An importance sampling scheme; 3.2 Examples of abstract tubes; 3.2.1 Abstract tubes for convex polyhedra; 3.2.2 Abstract tubes for unions of balls and spherical caps; 4.1 Abstract tubes via closure operators; 4.1.1 Closure operators; 4.1.2 How closure operators lead to abstract tubes; 4.2 Abstract tubes via kernel operators; 4.2.1 Kernel operators

4.2.2 How kernel operators lead to abstract tubes4.3 Alternative proofs; 4.3.1 Improved identities via closure operators; 4.3.2 Improved inequalities via kernel operators; 4.3.3 Additional proofs of identities; 4.4 The chordal graph sieve; 5.1 Recursive schemes for semilattices; 5.2 Dynamic programming approach; 6.1 System reliability; 6.1.1 Terminology; 6.1.2 Abstract tubes for system reliability; 6.1.3 Shier's pseudopolynomial algorithm; 6.1.4 Inclusion-exclusion and domination theory; 6.2 Special reliability systems; 6.2.1 Communications networks; 6.2.2 k-out-of-n systems

6.2.3 Consecutive k-out-of-n systems6.3 Reliability covering problems; 6.3.1 The hypergraph model; 6.3.2 Abstract tubes and polynomial algorithms; 6.4 Chapter notes; 7.1 Inclusion-exclusion on partition lattices; 7.2 Chromatic polynomials and broken circuits; 7.2.1 The usual chromatic polynomial; 7.2.2 The generalized chromatic polynomial; 7.3 Sums over partially ordered sets; 7.3.1 A general theorem on sums; 7.3.2 Application to inclusion-exclusion; 7.4 Matroid polynomials and the B invariant; 7.4.1 The Tutte polynomial; 7.4.2 The characteristic polynomial; 7.4.3 The B invariant

7.5 Euler characteristics and Möbius functions7.5.1 Euler characteristics; 7.5.2 Möbius functions; Bibliography; Author Index; Subject Index

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.

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